KoreanFoodie's Study
https://koreanfoodie.me/
Introducing Korean Food and Attractions!koFri, 23 Aug 2019 14:28:17 +0900TISTORYhashnutKoreanFoodie's Studyhttps://tistory3.daumcdn.net/tistory/3091872/attach/a5e22cb2fa7d4955b8878834acb5513a
https://koreanfoodie.me
Introducing Korean Food and Attractions![Seoul, Jongno, Gangnam] Tacos! Dos Tacos! (Dos Tacos, 도스타코스)
https://koreanfoodie.me/97
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</script></div><p style="margin-top: 20px; margin-right: 0px; margin-bottom: 30px; line-height: 1.6; color: #333333; font-family: AppleSDGothicNeo-Regular, 'Malgun Gothic', '맑은 고딕', dotum, 돋움, sans-serif; font-size: 16px;"><b><i>I'm introducing fine Korean restaurants/places where actual local Koreans usually go, not just tourist-targeted restaurants.</i></b></p>
<hr style="border: none; font-size: 0px; line-height: 0; height: 18px; margin: 20px auto; background: url('https://t1.daumcdn.net/tistory_admin/static/content/divider-line.svg') 0px -12px / 200px 153px no-repeat; cursor: pointer !important; width: 67px; color: rgba(0, 0, 0, 0.84); font-family: 'Spoqa Han Sans', Dotum, 돋움, Helvetica, 'Apple SD Gothic Neo', sans-serif; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-style: initial; text-decoration-color: initial;" contenteditable="false" data-ke-type="hr" data-ke-style="style2" />
<p><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/baLdt5/btqv8lNT4I1/re87VuicT9JmlhM88sLfUk/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/baLdt5/btqv8lNT4I1/re87VuicT9JmlhM88sLfUk/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FbaLdt5%2Fbtqv8lNT4I1%2Fre87VuicT9JmlhM88sLfUk%2Fimg.jpg' data-filename="2019-06-14-12-15-58.jpg"></span></figure></p>
<hr style="border: none; font-size: 0px; line-height: 0; height: 18px; margin: 20px auto 0px; background: url('../image/divider-line.svg') 0px -12px / 200px 153px no-repeat; cursor: pointer !important; width: 67px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-style: initial; text-decoration-color: initial; z-index: 1; color: #333333; font-family: AppleSDGothicNeo-Regular, 'Malgun Gothic', '맑은 고딕', dotum, 돋움, sans-serif;" contenteditable="false" data-ke-type="hr" data-ke-style="style2" />
<h3 style="margin: 1em 0px 20px; font-size: 1.62em; line-height: 1.6; font-weight: normal; letter-spacing: -1px; font-family: AppleSDGothicNeo-Regular, 'Malgun Gothic', '맑은 고딕', dotum, 돋움, sans-serif;"><b>Brief Review :</b></h3>
<p style="margin: 10px 0px 30px; line-height: 1.6; color: #333333; font-family: AppleSDGothicNeo-Regular, 'Malgun Gothic', '맑은 고딕', dotum, 돋움, sans-serif; font-size: 16px;">- <b>Name</b> : Dos Tacos</p>
<p style="margin: 10px 0px 30px; line-height: 1.6; color: #333333; font-family: AppleSDGothicNeo-Regular, 'Malgun Gothic', '맑은 고딕', dotum, 돋움, sans-serif; font-size: 16px;">- <b>Summary</b> : <b>Fine</b> place to eat nice mexican foods, especially tacos. Well, it's actually street food, but I'm more of a begger than a prince, so street food would suffice to me. Oh, I love chimichanga. *Deadpool: TIME TO MAKE THE CHIMI-****ING-CHANGAS!</p>
<p style="margin: 1px auto 30px; line-height: 1.6; color: #333333; font-family: AppleSDGothicNeo-Regular, 'Malgun Gothic', '맑은 고딕', dotum, 돋움, sans-serif; font-size: 16px;">- <b>Prices</b> : 6 ~ $</p>
<p style="margin: 1px auto 30px; line-height: 1.6; color: #333333; font-family: AppleSDGothicNeo-Regular, 'Malgun Gothic', '맑은 고딕', dotum, 돋움, sans-serif; font-size: 16px;">- <b>Taste</b> : 4 / 5</p>
<p style="margin: 1px auto 30px; line-height: 1.6; color: #333333; font-family: AppleSDGothicNeo-Regular, 'Malgun Gothic', '맑은 고딕', dotum, 돋움, sans-serif; font-size: 16px;">-<b> Recommendation/How to order</b> : Chimichanga(Beef), <span style="color: #333333;">Chimichanga(Pork)</span></p>
<hr style="border: none; font-size: 0px; line-height: 0; height: 18px; margin: 20px auto; background: url('https://t1.daumcdn.net/tistory_admin/static/content/divider-line.svg') 0px -12px / 200px 153px no-repeat; cursor: pointer !important; width: 67px; color: rgba(0, 0, 0, 0.84); font-family: 'Spoqa Han Sans', Dotum, 돋움, Helvetica, 'Apple SD Gothic Neo', sans-serif; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-style: initial; text-decoration-color: initial;" contenteditable="false" data-ke-type="hr" data-ke-style="style2" />
<p style="margin: 20px 0px 30px; line-height: 1.6; color: #333333; font-family: AppleSDGothicNeo-Regular, 'Malgun Gothic', '맑은 고딕', dotum, 돋움, sans-serif; font-size: 16px;">For your convenience, I added three places in Seoul : Jongno, Gangnam, SNU.</p>
<center><iframe src="https://www.google.com/maps/embed?pb=!1m18!1m12!1m3!1d50624.57352595291!2d126.97865393194189!3d37.53065229619925!2m3!1f0!2f0!3f0!3m2!1i1024!2i768!4f13.1!3m3!1m2!1s0x357ca2e6191aa671%3A0x43ae74b7bd4bb205!2sDos+Tacos!5e0!3m2!1sen!2skr!4v1560775764629!5m2!1sen!2skr" width="600" height="450" frameborder="0" allowfullscreen=""></iframe></center><center><iframe src="https://www.google.com/maps/embed?pb=!1m18!1m12!1m3!1d35808.701672258314!2d126.995355987826!3d37.50622061996461!2m3!1f0!2f0!3f0!3m2!1i1024!2i768!4f13.1!3m3!1m2!1s0x0%3A0xf1ed8d01403e5ff!2sDos+Tacos!5e0!3m2!1sen!2skr!4v1560775782496!5m2!1sen!2skr" width="600" height="450" frameborder="0" allowfullscreen=""></iframe></center><center><iframe src="https://www.google.com/maps/embed?pb=!1m18!1m12!1m3!1d3166.2217789528377!2d126.95075484963594!3d37.479092494987825!2m3!1f0!2f0!3f0!3m2!1i1024!2i768!4f13.1!3m3!1m2!1s0x357c9f89ed440001%3A0xea83d2cc9cb4bb6e!2sDos+Tacos!5e0!3m2!1sen!2skr!4v1560775848621!5m2!1sen!2skr" width="600" height="450" frameborder="0" allowfullscreen=""></iframe></center>
<p style="margin: 1px auto 30px; line-height: 1.6; color: #333333; font-family: AppleSDGothicNeo-Regular, 'Malgun Gothic', '맑은 고딕', dotum, 돋움, sans-serif; font-size: 16px;"> </p>
<p style="margin: 1px auto 30px; line-height: 1.6; color: #333333; font-family: AppleSDGothicNeo-Regular, 'Malgun Gothic', '맑은 고딕', dotum, 돋움, sans-serif; font-size: 16px;"> </p>
<p><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/4aP28/btqv8Z4GpTO/0uRKOcUAJ4fdHdQEu2OTc1/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/4aP28/btqv8Z4GpTO/0uRKOcUAJ4fdHdQEu2OTc1/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2F4aP28%2Fbtqv8Z4GpTO%2F0uRKOcUAJ4fdHdQEu2OTc1%2Fimg.jpg' data-filename="2019-06-12-11-31-07.jpg"></span></figure></p>
<p style="margin: 1px auto 30px; line-height: 1.6; color: #333333; font-family: AppleSDGothicNeo-Regular, 'Malgun Gothic', '맑은 고딕', dotum, 돋움, sans-serif; font-size: 16px;"> </p>
<p><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/bdMcI4/btqv95J8WGa/Qc5wv8sRGCmuSQr8nBZqSk/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/bdMcI4/btqv95J8WGa/Qc5wv8sRGCmuSQr8nBZqSk/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FbdMcI4%2Fbtqv95J8WGa%2FQc5wv8sRGCmuSQr8nBZqSk%2Fimg.jpg' data-filename="2019-06-12-11-32-26.jpg"></span></figure></p>
<p>Sick glow, huh?</p>
<p><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/bivx3X/btqwahXQBEx/tLdnvQkzFL3Q2J6UrAZ491/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/bivx3X/btqwahXQBEx/tLdnvQkzFL3Q2J6UrAZ491/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2Fbivx3X%2FbtqwahXQBEx%2FtLdnvQkzFL3Q2J6UrAZ491%2Fimg.jpg' data-filename="2019-06-12-11-31-52.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/JaUGx/btqv7QOkoOe/a22prTJ4ljm5WI4frXZh00/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/JaUGx/btqv7QOkoOe/a22prTJ4ljm5WI4frXZh00/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FJaUGx%2Fbtqv7QOkoOe%2Fa22prTJ4ljm5WI4frXZh00%2Fimg.jpg' data-filename="2019-06-12-11-31-58.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/wpKq7/btqv81amg59/jeLNxFUE6jVxxBYKqkAZGK/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/wpKq7/btqv81amg59/jeLNxFUE6jVxxBYKqkAZGK/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FwpKq7%2Fbtqv81amg59%2FjeLNxFUE6jVxxBYKqkAZGK%2Fimg.jpg' data-filename="2019-06-12-11-32-02.jpg"></span></figure></p>
<p>This is the menu plate.</p>
<p><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/M87ay/btqv96hYe6e/aivWZNzdc7EucMKg3rjob1/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/M87ay/btqv96hYe6e/aivWZNzdc7EucMKg3rjob1/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FM87ay%2Fbtqv96hYe6e%2FaivWZNzdc7EucMKg3rjob1%2Fimg.jpg' data-filename="2019-06-12-11-40-07-01.jpeg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/bGPxOk/btqv8ZKmkD9/EGgf8uUdzIvckMvhjv2w41/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/bGPxOk/btqv8ZKmkD9/EGgf8uUdzIvckMvhjv2w41/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FbGPxOk%2Fbtqv8ZKmkD9%2FEGgf8uUdzIvckMvhjv2w41%2Fimg.jpg' data-filename="2019-06-12-11-40-19-01.jpeg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/bRuowY/btqv8ZRdyFV/vKltkAWG7hzhNwbVUTCcyK/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/bRuowY/btqv8ZRdyFV/vKltkAWG7hzhNwbVUTCcyK/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FbRuowY%2Fbtqv8ZRdyFV%2FvKltkAWG7hzhNwbVUTCcyK%2Fimg.jpg' data-filename="2019-06-12-11-40-27-01.jpeg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/qX2Ts/btqwaHPxLb7/lxdG9EziCXC2X3NwvlKVZk/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/qX2Ts/btqwaHPxLb7/lxdG9EziCXC2X3NwvlKVZk/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FqX2Ts%2FbtqwaHPxLb7%2FlxdG9EziCXC2X3NwvlKVZk%2Fimg.jpg' data-filename="2019-06-12-11-40-39-01.jpeg"></span></figure></p>
<p>Oh my lovely chimichanga..</p>
<p><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/7T9zX/btqv7Yk5hil/iSeGWSoGEtGCNR3Drdx3D1/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/7T9zX/btqv7Yk5hil/iSeGWSoGEtGCNR3Drdx3D1/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2F7T9zX%2Fbtqv7Yk5hil%2FiSeGWSoGEtGCNR3Drdx3D1%2Fimg.jpg' data-filename="2019-06-12-11-40-47-01.jpeg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/pnHpk/btqv9rUbTHa/QAden2A3oWHcLSE2JrKmIk/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/pnHpk/btqv9rUbTHa/QAden2A3oWHcLSE2JrKmIk/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FpnHpk%2Fbtqv9rUbTHa%2FQAden2A3oWHcLSE2JrKmIk%2Fimg.jpg' data-filename="2019-06-12-11-50-24-01.jpeg"></span></figure></p>
<p>I ate the whole plate on my own. Chi---mi--changa!</p>
<p><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/tUjaq/btqv7PWh7oB/nvw4P5sJi0Wdc3TyhAJRE1/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/tUjaq/btqv7PWh7oB/nvw4P5sJi0Wdc3TyhAJRE1/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FtUjaq%2Fbtqv7PWh7oB%2Fnvw4P5sJi0Wdc3TyhAJRE1%2Fimg.jpg' data-filename="2019-06-14-12-37-56-01.jpeg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/lNVPw/btqwaHPxMor/ZE29YTrnhLXwMN3yAC2tt1/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/lNVPw/btqwaHPxMor/ZE29YTrnhLXwMN3yAC2tt1/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FlNVPw%2FbtqwaHPxMor%2FZE29YTrnhLXwMN3yAC2tt1%2Fimg.jpg' data-filename="2019-06-14-12-38-04-01.jpeg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/cXiDJF/btqv8mMPslR/2wiSKPKKM7CwXg6oDAX5GK/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/cXiDJF/btqv8mMPslR/2wiSKPKKM7CwXg6oDAX5GK/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FcXiDJF%2Fbtqv8mMPslR%2F2wiSKPKKM7CwXg6oDAX5GK%2Fimg.jpg' data-filename="2019-06-14-12-38-10-01.jpeg"></span></figure></p>
<p>This is Al Pastol Taco. I hated it ceause of coriander.I really hate coriander. I hope coriander would distict tomorrow so I don't have to smell that shitty scent. Coriander tastes like .. well, not shit, to be fair. It tastes like warm, clean mob which has been never used. Ok, It is basically not better than a shit.</p>
<p>But taco was ok other than that. :)</p>
<p><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/uViow/btqwaHvfwkD/pCtpDAFbSWcmqEFkWtyMu1/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/uViow/btqwaHvfwkD/pCtpDAFbSWcmqEFkWtyMu1/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FuViow%2FbtqwaHvfwkD%2FpCtpDAFbSWcmqEFkWtyMu1%2Fimg.jpg' data-filename="2019-06-12-11-32-38.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/bVPJ3A/btqv7YFq9G0/HASqfkvQ8MKGLIj5PGZfNK/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/bVPJ3A/btqv7YFq9G0/HASqfkvQ8MKGLIj5PGZfNK/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FbVPJ3A%2Fbtqv7YFq9G0%2FHASqfkvQ8MKGLIj5PGZfNK%2Fimg.jpg' data-filename="2019-06-12-11-34-04.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/FA7jN/btqv8nkFS35/9jDqXhlHETROvvBeAlfHZ0/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/FA7jN/btqv8nkFS35/9jDqXhlHETROvvBeAlfHZ0/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FFA7jN%2Fbtqv8nkFS35%2F9jDqXhlHETROvvBeAlfHZ0%2Fimg.jpg' data-filename="2019-06-12-11-34-34.jpg"></span></figure></p>
<p>How about trying a nice taco in Korea? Despacito!</p>
<p> </p>
<hr style="border: none; font-size: 0px; line-height: 0; height: 18px; margin: 20px auto; background: url('https://t1.daumcdn.net/tistory_admin/static/content/divider-line.svg') 0px -12px / 200px 153px no-repeat; cursor: pointer !important; width: 67px; color: rgba(0, 0, 0, 0.84); font-family: 'Spoqa Han Sans', Dotum, 돋움, Helvetica, 'Apple SD Gothic Neo', sans-serif; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-style: initial; text-decoration-color: initial;" contenteditable="false" data-ke-type="hr" data-ke-style="style2" />
<p style="margin: 1px auto 30px; line-height: 1.6; color: #333333; font-family: AppleSDGothicNeo-Regular, 'Malgun Gothic', '맑은 고딕', dotum, 돋움, sans-serif; font-size: 16px;"> </p>
<p style="margin: 20px 0px 30px; line-height: 1.6; color: #333333; font-family: AppleSDGothicNeo-Regular, 'Malgun Gothic', '맑은 고딕', dotum, 돋움, sans-serif; font-size: 16px;"><span style="font-size: 14pt;"><b>Recommendable restaurants : Find more at the bottom section!</b></span></p>
<p style="margin-top: 20px; margin-right: 0px; margin-bottom: 30px; line-height: 1.6;"><span><span style="font-size: 12pt;"><b>It would be very much appreciated if you share my post if it was helpful :)</b></span></span></p>
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<p><a style="text-decoration: none;" href="https://iamfoodie.tistory.com"><span> </span></a> <a style="text-decoration: none;" href="https://iamfoodie.tistory.com/category/Korean%20Cuisine%20and%20Attractions/Korean%20Culture%20Overview"><span> </span></a><a style="text-decoration: none;" href="https://iamfoodie.tistory.com/category/Korean%20Cuisine%20and%20Attractions/GANGNAM"><span> </span></a><a style="text-decoration: none;" href="https://iamfoodie.tistory.com/category/Korean%20Cuisine%20and%20Attractions/HONGDAE%2C%20SINCHON"><span> </span></a><a style="text-decoration: none;" href="https://iamfoodie.tistory.com/category/Korean%20Cuisine%20and%20Attractions/ITAEWON"><span> </span></a><a style="text-decoration: none;" href="https://iamfoodie.tistory.com/category/Korean%20Cuisine%20and%20Attractions"><span> </span></a><a style="text-decoration: none;" href="https://iamfoodie.tistory.com/category/Korean%20Cuisine%20and%20Attractions/MYEONGDONG"><span> </span></a><a style="text-decoration: none;" href="https://iamfoodie.tistory.com/category/Korean%20Cuisine%20and%20Attractions/SNU"><span> </span></a><a style="text-decoration: none;" href="https://iamfoodie.tistory.com/category/Korean%20Cuisine%20and%20Attractions/JAMSIL"><span> </span></a><span style="text-decoration: none;"><a style="text-decoration: none;" href="https://iamfoodie.tistory.com/category/Korean%20Cuisine%20and%20Attractions/BUSAN "> </a></span></p><div style="text-align:center;margin:10px 0 10px 0;clear:both"><div style="display:inline;text-align:center;"><script async src="https://pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script>
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JONGNOGangnamJongnomexicanSNUTacohashnuthttps://koreanfoodie.me/97https://koreanfoodie.me/97#entry97commentMon, 17 Jun 2019 22:02:32 +0900[Seoul, Franchise] Cheapest Pizza Franchise In Korea. Worth the price! (Pizza School, 피자 스쿨)
https://koreanfoodie.me/96
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</script></div><p style="margin-top: 20px; margin-right: 0px; margin-bottom: 30px; line-height: 1.6; color: #333333; font-family: AppleSDGothicNeo-Regular, 'Malgun Gothic', '맑은 고딕', dotum, 돋움, sans-serif; font-size: 16px;"><b><i>I'm introducing fine Korean restaurants/places where actual local Koreans usually go, not just tourist-targeted restaurants.</i></b></p>
<hr style="border: none; font-size: 0px; line-height: 0; height: 18px; margin: 20px auto; background: url('https://t1.daumcdn.net/tistory_admin/static/content/divider-line.svg') 0px -12px / 200px 153px no-repeat; cursor: pointer !important; width: 67px; color: rgba(0, 0, 0, 0.84); font-family: 'Spoqa Han Sans', Dotum, 돋움, Helvetica, 'Apple SD Gothic Neo', sans-serif; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-style: initial; text-decoration-color: initial;" contenteditable="false" data-ke-type="hr" data-ke-style="style2" />
<p><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/bCrqmW/btqv7X6Kpew/1xy4xQmivKNal5E9hDC6L1/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/bCrqmW/btqv7X6Kpew/1xy4xQmivKNal5E9hDC6L1/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FbCrqmW%2Fbtqv7X6Kpew%2F1xy4xQmivKNal5E9hDC6L1%2Fimg.jpg' data-filename="2019-06-09-22-14-29.jpg"></span></figure></p>
<hr style="border: none; font-size: 0px; line-height: 0; height: 18px; margin: 20px auto 0px; background: url('../image/divider-line.svg') 0px -12px / 200px 153px no-repeat; cursor: pointer !important; width: 67px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-style: initial; text-decoration-color: initial; z-index: 1; color: #333333; font-family: AppleSDGothicNeo-Regular, 'Malgun Gothic', '맑은 고딕', dotum, 돋움, sans-serif;" contenteditable="false" data-ke-type="hr" data-ke-style="style2" />
<h3 style="margin: 1em 0px 20px; font-size: 1.62em; line-height: 1.6; font-weight: normal; letter-spacing: -1px; font-family: AppleSDGothicNeo-Regular, 'Malgun Gothic', '맑은 고딕', dotum, 돋움, sans-serif;"><b>Brief Review :</b></h3>
<p style="margin: 10px 0px 30px; line-height: 1.6; color: #333333; font-family: AppleSDGothicNeo-Regular, 'Malgun Gothic', '맑은 고딕', dotum, 돋움, sans-serif; font-size: 16px;">- <b>Name</b> : Pizza School (피자 스쿨)</p>
<p style="margin: 10px 0px 30px; line-height: 1.6; color: #333333; font-family: AppleSDGothicNeo-Regular, 'Malgun Gothic', '맑은 고딕', dotum, 돋움, sans-serif; font-size: 16px;">- <b>Summary</b> : <b>Fine</b> place to have cheap pizza, but It's worth it! It's the cheapest franchise, serving pizza and spaghetti. The cheapest one only costs 6000 Korean Won, which is about 5$. But I personally recommend pepperoni pizza, and it costs 7000 Korean Won(5.7$). They started as a only-takeout restaurant, but as they expand the buisness, they made some room for guests who eat inside the restaurant.</p>
<p style="margin: 1px auto 30px; line-height: 1.6; color: #333333; font-family: AppleSDGothicNeo-Regular, 'Malgun Gothic', '맑은 고딕', dotum, 돋움, sans-serif; font-size: 16px;">- <b>Prices</b> : 5~$</p>
<p style="margin: 1px auto 30px; line-height: 1.6; color: #333333; font-family: AppleSDGothicNeo-Regular, 'Malgun Gothic', '맑은 고딕', dotum, 돋움, sans-serif; font-size: 16px;">- <b>Taste</b> : 3.5/5</p>
<p style="margin: 1px auto 30px; line-height: 1.6; color: #333333; font-family: AppleSDGothicNeo-Regular, 'Malgun Gothic', '맑은 고딕', dotum, 돋움, sans-serif; font-size: 16px;">-<b> Recommendation/How to order</b> : I like Pepperoni pizza. And also pineapple pizza(Deal with it).</p>
<hr style="border: none; font-size: 0px; line-height: 0; height: 18px; margin: 20px auto; background: url('https://t1.daumcdn.net/tistory_admin/static/content/divider-line.svg') 0px -12px / 200px 153px no-repeat; cursor: pointer !important; width: 67px; color: rgba(0, 0, 0, 0.84); font-family: 'Spoqa Han Sans', Dotum, 돋움, Helvetica, 'Apple SD Gothic Neo', sans-serif; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-style: initial; text-decoration-color: initial;" contenteditable="false" data-ke-type="hr" data-ke-style="style2" />
<p style="margin: 20px 0px 30px; line-height: 1.6; color: #333333; font-family: AppleSDGothicNeo-Regular, 'Malgun Gothic', '맑은 고딕', dotum, 돋움, sans-serif; font-size: 16px;">You can find them in any place like CV. For your convenience, I added four places, Hongdae, Gangnam, Jongno, Jamsil.</p>
<center><iframe src="https://www.google.com/maps/embed?pb=!1m18!1m12!1m3!1d12653.010383933626!2d126.9344405782676!3d37.54911217166376!2m3!1f0!2f0!3f0!3m2!1i1024!2i768!4f13.1!3m3!1m2!1s0x0%3A0xa4b5b3fcffe71c5c!2sPizza+School!5e0!3m2!1sen!2skr!4v1560701692841!5m2!1sen!2skr" width="600" height="450" frameborder="0" allowfullscreen=""></iframe></center><center><iframe src="https://www.google.com/maps/embed?pb=!1m18!1m12!1m3!1d7528.997380454142!2d127.03229018948035!3d37.49486275433657!2m3!1f0!2f0!3f0!3m2!1i1024!2i768!4f13.1!3m3!1m2!1s0x0%3A0x95455fd9439fbca3!2sPizza+School!5e0!3m2!1sen!2skr!4v1560701835507!5m2!1sen!2skr" width="600" height="450" frameborder="0" allowfullscreen=""></iframe></center><center><iframe src="https://www.google.com/maps/embed?pb=!1m18!1m12!1m3!1d2235.9157478551133!2d126.97286381637336!3d37.57714888511063!2m3!1f0!2f0!3f0!3m2!1i1024!2i768!4f13.1!3m3!1m2!1s0x0%3A0x6f149a500b98c751!2sPizza+School+Gyeongbokgung+points!5e0!3m2!1sen!2skr!4v1560701883290!5m2!1sen!2skr" width="600" height="450" frameborder="0" allowfullscreen=""></iframe></center><center><iframe src="https://www.google.com/maps/embed?pb=!1m18!1m12!1m3!1d7527.536253093118!2d127.09198093376098!3d37.50935388053643!2m3!1f0!2f0!3f0!3m2!1i1024!2i768!4f13.1!3m3!1m2!1s0x0%3A0x7c8a34a4b22823ef!2z7ZS87J6Q7Iqk7L-o!5e0!3m2!1sen!2skr!4v1560701910349!5m2!1sen!2skr" width="600" height="450" frameborder="0" allowfullscreen=""></iframe></center>
<p style="margin: 20px 0px 30px; line-height: 1.6; color: #333333; font-family: AppleSDGothicNeo-Regular, 'Malgun Gothic', '맑은 고딕', dotum, 돋움, sans-serif; font-size: 16px;"> </p>
<p><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/clNjHQ/btqv7gsGAok/TFZdhorLW3Iv6yiZN1jBIk/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/clNjHQ/btqv7gsGAok/TFZdhorLW3Iv6yiZN1jBIk/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FclNjHQ%2Fbtqv7gsGAok%2FTFZdhorLW3Iv6yiZN1jBIk%2Fimg.jpg' data-filename="2019-06-09-22-15-40.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/b8SPUy/btqv6EAveoK/cB1uJa34RfD2UpzkFZm83K/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/b8SPUy/btqv6EAveoK/cB1uJa34RfD2UpzkFZm83K/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2Fb8SPUy%2Fbtqv6EAveoK%2FcB1uJa34RfD2UpzkFZm83K%2Fimg.jpg' data-filename="2019-06-09-22-16-00.jpg"></span></figure></p>
<p>This is the menu plate.</p>
<p><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/cSZAo1/btqv7X6KvQl/F7zsvDxqOV3Dm98LWzwhD1/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/cSZAo1/btqv7X6KvQl/F7zsvDxqOV3Dm98LWzwhD1/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FcSZAo1%2Fbtqv7X6KvQl%2FF7zsvDxqOV3Dm98LWzwhD1%2Fimg.jpg' data-filename="2019-06-09-22-16-10.jpg"></span></figure></p>
<p>Why they work really hard. Like busy bee... buzz, buzz.</p>
<p><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/ezj3kd/btqv6EUVeI3/RdFAUDB1G7QpqthM05lpck/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/ezj3kd/btqv6EUVeI3/RdFAUDB1G7QpqthM05lpck/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2Fezj3kd%2Fbtqv6EUVeI3%2FRdFAUDB1G7QpqthM05lpck%2Fimg.jpg' data-filename="2019-06-09-22-17-33.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/80ria/btqv7ChJkQo/hTlaidYKZGhaLBreyOlfik/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/80ria/btqv7ChJkQo/hTlaidYKZGhaLBreyOlfik/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2F80ria%2Fbtqv7ChJkQo%2FhTlaidYKZGhaLBreyOlfik%2Fimg.jpg' data-filename="2019-06-09-22-27-44.jpg"></span></figure></p>
<p>Waiting in line is always boring.. without pokemon dolls. Hehe..</p>
<p><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/dHngJa/btqv7O3h9kQ/A3RkvAKEYvdojDwAlrw6rK/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/dHngJa/btqv7O3h9kQ/A3RkvAKEYvdojDwAlrw6rK/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FdHngJa%2Fbtqv7O3h9kQ%2FA3RkvAKEYvdojDwAlrw6rK%2Fimg.jpg' data-filename="2019-06-09-22-29-04.jpg"></span></figure></p>
<p>Thanks for the meal.</p>
<p><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/dL21BD/btqv7pppr3J/PZBe8AF3CLvt7tBtI8eeBK/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/dL21BD/btqv7pppr3J/PZBe8AF3CLvt7tBtI8eeBK/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FdL21BD%2Fbtqv7pppr3J%2FPZBe8AF3CLvt7tBtI8eeBK%2Fimg.jpg' data-filename="2019-06-09-22-34-50-01.jpeg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/cheO3b/btqv8Zv231f/F8EpIcjC5LI2aYlvVxinF1/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/cheO3b/btqv8Zv231f/F8EpIcjC5LI2aYlvVxinF1/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FcheO3b%2Fbtqv8Zv231f%2FF8EpIcjC5LI2aYlvVxinF1%2Fimg.jpg' data-filename="2019-06-09-22-34-55-01.jpeg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/ZnKvK/btqv7hSDnVC/Z9TlKHAV49kDI0Ww9KgNKk/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/ZnKvK/btqv7hSDnVC/Z9TlKHAV49kDI0Ww9KgNKk/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FZnKvK%2Fbtqv7hSDnVC%2FZ9TlKHAV49kDI0Ww9KgNKk%2Fimg.jpg' data-filename="2019-06-09-22-37-12-01.jpeg"></span></figure></p>
<p>Well, It was not pepperoni, as you can see. It's 'combination pizza', they say some vegetables are added to the toppings, however, I'm not gonna fooled by their sugarcoating. I proudly announce I'll order pepperoni pizza only from now on if I visit this place.</p>
<hr style="border: none; font-size: 0px; line-height: 0; height: 18px; margin: 20px auto; background: url('https://t1.daumcdn.net/tistory_admin/static/content/divider-line.svg') 0px -12px / 200px 153px no-repeat; cursor: pointer !important; width: 67px; color: rgba(0, 0, 0, 0.84); font-family: 'Spoqa Han Sans', Dotum, 돋움, Helvetica, 'Apple SD Gothic Neo', sans-serif; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-style: initial; text-decoration-color: initial;" contenteditable="false" data-ke-type="hr" data-ke-style="style2" />
<p style="margin: 1px auto 30px; line-height: 1.6; color: #333333; font-family: AppleSDGothicNeo-Regular, 'Malgun Gothic', '맑은 고딕', dotum, 돋움, sans-serif; font-size: 16px;"> </p>
<p style="margin: 20px 0px 30px; line-height: 1.6; color: #333333; font-family: AppleSDGothicNeo-Regular, 'Malgun Gothic', '맑은 고딕', dotum, 돋움, sans-serif; font-size: 16px;"><span style="font-size: 14pt;"><b>Recommendable restaurants : Find more at the bottom section!</b></span></p>
<p style="margin-top: 20px; margin-right: 0px; margin-bottom: 30px; line-height: 1.6;"><span><span style="font-size: 12pt;"><b>It would be very much appreciated if you share my post if it was helpful :)</b></span></span></p>
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HONGDAE, SINCHONCheapest pizzafranchiseGangnamHONGDAEjamsilJongnoPizzahashnuthttps://koreanfoodie.me/96https://koreanfoodie.me/96#entry96commentMon, 17 Jun 2019 01:25:04 +0900Solutions to Linear Algebra, Stephen H. Friedberg, Fourth Edition (Chapter 2)
https://koreanfoodie.me/95
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</script></div><p><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/beX0d7/btqv8mduQzu/VykgBqjpekL7YFQh0h00Kk/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/beX0d7/btqv8mduQzu/VykgBqjpekL7YFQh0h00Kk/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FbeX0d7%2Fbtqv8mduQzu%2FVykgBqjpekL7YFQh0h00Kk%2Fimg.jpg' data-filename="chapter7_1.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/dAeqWW/btqv8mYSexM/tuTsa3rT53EfmGS5yJgp71/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/dAeqWW/btqv8mYSexM/tuTsa3rT53EfmGS5yJgp71/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FdAeqWW%2Fbtqv8mYSexM%2FtuTsa3rT53EfmGS5yJgp71%2Fimg.jpg' data-filename="chapter7_2.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/bnTujQ/btqv7pbaeyo/ajiXRYlAvogpJWaJFj3I91/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/bnTujQ/btqv7pbaeyo/ajiXRYlAvogpJWaJFj3I91/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FbnTujQ%2Fbtqv7pbaeyo%2FajiXRYlAvogpJWaJFj3I91%2Fimg.jpg' data-filename="chapter7_3.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/triMX/btqv7O2Df6a/2DhVTsjeFyq9RURdXi8RNK/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/triMX/btqv7O2Df6a/2DhVTsjeFyq9RURdXi8RNK/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FtriMX%2Fbtqv7O2Df6a%2F2DhVTsjeFyq9RURdXi8RNK%2Fimg.jpg' data-filename="chapter7_4.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/qOL5B/btqv7Q7dMt3/QkFcjUpQKnnFEa2rDdXND0/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/qOL5B/btqv7Q7dMt3/QkFcjUpQKnnFEa2rDdXND0/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FqOL5B%2Fbtqv7Q7dMt3%2FQkFcjUpQKnnFEa2rDdXND0%2Fimg.jpg' data-filename="chapter7_5.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/zR4Yt/btqv8mLkGLk/NmREmBCGyXSkC7kQROpBm1/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/zR4Yt/btqv8mLkGLk/NmREmBCGyXSkC7kQROpBm1/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FzR4Yt%2Fbtqv8mLkGLk%2FNmREmBCGyXSkC7kQROpBm1%2Fimg.jpg' data-filename="chapter7_6.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/XKLv5/btqv5hd27cg/KMV4DaYPZqNjWyJcInDBkk/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/XKLv5/btqv5hd27cg/KMV4DaYPZqNjWyJcInDBkk/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FXKLv5%2Fbtqv5hd27cg%2FKMV4DaYPZqNjWyJcInDBkk%2Fimg.jpg' data-filename="chapter7_7.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/RtEfp/btqv7O2DgcZ/PS0rKegObpb5wM2p62KSc0/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/RtEfp/btqv7O2DgcZ/PS0rKegObpb5wM2p62KSc0/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FRtEfp%2Fbtqv7O2DgcZ%2FPS0rKegObpb5wM2p62KSc0%2Fimg.jpg' data-filename="chapter7_8.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/k6Ez6/btqv6D8Pj5h/bSBOAvcAoPeZEcOKKOzjt0/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/k6Ez6/btqv6D8Pj5h/bSBOAvcAoPeZEcOKKOzjt0/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2Fk6Ez6%2Fbtqv6D8Pj5h%2FbSBOAvcAoPeZEcOKKOzjt0%2Fimg.jpg' data-filename="chapter7_9.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/bQDHaq/btqv8mLkGRY/GTxj9wLMKZb4um0gqKckTk/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/bQDHaq/btqv8mLkGRY/GTxj9wLMKZb4um0gqKckTk/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FbQDHaq%2Fbtqv8mLkGRY%2FGTxj9wLMKZb4um0gqKckTk%2Fimg.jpg' data-filename="chapter7_10.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/bNzC3u/btqv6C9SjeQ/3E5NWULGfOxVIrgRXeHbO0/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/bNzC3u/btqv6C9SjeQ/3E5NWULGfOxVIrgRXeHbO0/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FbNzC3u%2Fbtqv6C9SjeQ%2F3E5NWULGfOxVIrgRXeHbO0%2Fimg.jpg' data-filename="chapter7_11.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/dMJv9y/btqv7hLcKHE/KcoOdVo8EMlRaXNakRwQU0/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/dMJv9y/btqv7hLcKHE/KcoOdVo8EMlRaXNakRwQU0/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FdMJv9y%2Fbtqv7hLcKHE%2FKcoOdVo8EMlRaXNakRwQU0%2Fimg.jpg' data-filename="chapter7_12.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/btdLoW/btqv7YDVtYq/ziyRs3i8dpTNIhzMQpVQkK/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/btdLoW/btqv7YDVtYq/ziyRs3i8dpTNIhzMQpVQkK/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FbtdLoW%2Fbtqv7YDVtYq%2FziyRs3i8dpTNIhzMQpVQkK%2Fimg.jpg' data-filename="chapter7_13.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/cAhT5L/btqv6EzPIyL/jwly6XbIrihRuNER1Ag7E0/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/cAhT5L/btqv6EzPIyL/jwly6XbIrihRuNER1Ag7E0/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FcAhT5L%2Fbtqv6EzPIyL%2Fjwly6XbIrihRuNER1Ag7E0%2Fimg.jpg' data-filename="chapter7_14.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/dW0Zyk/btqv6DOu6FG/9WfwA25IOZ6EjiS9nyEEZ1/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/dW0Zyk/btqv6DOu6FG/9WfwA25IOZ6EjiS9nyEEZ1/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FdW0Zyk%2Fbtqv6DOu6FG%2F9WfwA25IOZ6EjiS9nyEEZ1%2Fimg.jpg' data-filename="chapter7_15.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/tUkNu/btqv5gsK0Jz/k8LLkpeOwOo5aImqF4qRZ1/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/tUkNu/btqv5gsK0Jz/k8LLkpeOwOo5aImqF4qRZ1/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FtUkNu%2Fbtqv5gsK0Jz%2Fk8LLkpeOwOo5aImqF4qRZ1%2Fimg.jpg' data-filename="chapter7_16.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/IhQjv/btqv7X547fe/02omo8dnzzDmZU204kVlK0/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/IhQjv/btqv7X547fe/02omo8dnzzDmZU204kVlK0/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FIhQjv%2Fbtqv7X547fe%2F02omo8dnzzDmZU204kVlK0%2Fimg.jpg' data-filename="chapter7_17.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/ule5K/btqv7B3vccW/K5wkjlF3id8EK0A4S7bOZ1/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/ule5K/btqv7B3vccW/K5wkjlF3id8EK0A4S7bOZ1/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2Fule5K%2Fbtqv7B3vccW%2FK5wkjlF3id8EK0A4S7bOZ1%2Fimg.jpg' data-filename="chapter7_18.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/wGBvo/btqv5V9LpDO/5itLHVl7itc5LDDr9cKfo1/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/wGBvo/btqv5V9LpDO/5itLHVl7itc5LDDr9cKfo1/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FwGBvo%2Fbtqv5V9LpDO%2F5itLHVl7itc5LDDr9cKfo1%2Fimg.jpg' data-filename="chapter7_19.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/b6S6Jn/btqv7hj8g15/hClbAzEjIJHUepSUuhGdp1/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/b6S6Jn/btqv7hj8g15/hClbAzEjIJHUepSUuhGdp1/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2Fb6S6Jn%2Fbtqv7hj8g15%2FhClbAzEjIJHUepSUuhGdp1%2Fimg.jpg' data-filename="chapter7_20.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/SA3Bj/btqv6DufFSB/UW60TfZlPMCdL6H3LvjUVk/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/SA3Bj/btqv6DufFSB/UW60TfZlPMCdL6H3LvjUVk/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FSA3Bj%2Fbtqv6DufFSB%2FUW60TfZlPMCdL6H3LvjUVk%2Fimg.jpg' data-filename="chapter7_21.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/bTEDfT/btqv5Wt08CC/NuIIi49fvKHxu0QvwDdOE1/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/bTEDfT/btqv5Wt08CC/NuIIi49fvKHxu0QvwDdOE1/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FbTEDfT%2Fbtqv5Wt08CC%2FNuIIi49fvKHxu0QvwDdOE1%2Fimg.jpg' data-filename="chapter7_22.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/cL7Sy1/btqv7pWwQCS/D3HKiCqbnk2RLCI9v5Aqz0/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/cL7Sy1/btqv7pWwQCS/D3HKiCqbnk2RLCI9v5Aqz0/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FcL7Sy1%2Fbtqv7pWwQCS%2FD3HKiCqbnk2RLCI9v5Aqz0%2Fimg.jpg' data-filename="chapter7_23.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/bwmWA6/btqv7Y40x5U/Xk6GkW47DBST3kuXxzr10k/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/bwmWA6/btqv7Y40x5U/Xk6GkW47DBST3kuXxzr10k/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FbwmWA6%2Fbtqv7Y40x5U%2FXk6GkW47DBST3kuXxzr10k%2Fimg.jpg' data-filename="chapter7_24.jpg"></span></figure><figure class='imageblock alignCenter'><span 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<p><br />Solution maual to Linear Algebra, Fourth Edition, Stephen H. Friedberg. (Chapter 7)<br /><br />Solutions to Linear Algebra, Fourth Edition, Stephen H. Friedberg. (Chapter 7)<br /><br />Linear Algebra solution manual, Fourth Edition, Stephen H. Friedberg. (Chapter 7)<br /><br />Linear Algebra solutions Friedberg. (Chapter 7)</p>
<div style="width:100%; height:1px; overflow:auto"><font size="1px" color=#FFFFFF>1.Label(a)(b)(c)(d)(e)(f )(g)(h)the following statements as true or false.
Eigenvectors of a linear operator T are also generalized eigenvec-
tors of T.
It is possible for a generalized eigenvector of a linear operator T
to correspond to a scalar that is not an eigenvalue of T.
Any linear operator on a finite-dimensional vector space has a Jor-
dan canonical form.
A cycle of generalized eigenvectors is linearly independent.
There is exactly one cycle of generalized eigenvectors correspond-
ing to each eigenvalue of a linear operator on a finite-dimensional
vector space.
Let T be a linear operator on a finite-dimensional vector space
whose characteristic polynomial splits, and let λ1 , λ2 , . . . , λk be
the distinct eigenvalues of T. If, for each i, βi is a basis for Kλi ,
then β1 ∪ β2 ∪ · · · ∪ βk is a Jordan canonical basis for T.
For any Jordan block J, the operator LJ has Jordan canonical
form J.
Let T be a linear operator on an n-dimensional vector space whose
characteristic polynomial splits. Then, for any eigenvalue λ of T,
Kλ = N((T − λI)n ).
Sec. 7.1
The Jordan Canonical Form I
495
2. For each matrix A, find a basis for each generalized eigenspace of LA
consisting of a union of disjoint cycles of generalized eigenvectors. Then
find a Jordan canonical form J of A.
1 1
1 2
(a) A =
(b) A =
−1 3
3 2
⎛
⎞
⎛
⎞
2 1
0 0
11 −4
−5
⎜0 2
1 0⎟
(c) A = ⎝21 −8 −11⎠
(d) A = ⎜
⎟
⎝0 0
3 0⎠
3 −1
0
0 1 −1 3
3. For each linear operator T, find a basis for each generalized eigenspace
of T consisting of a union of disjoint cycles of generalized eigenvectors.
Then find a Jordan canonical form J of T.
(a) T is the linear operator on P2 (R) defined by T(f (x)) = 2f (x) −
f (x)
(b) V is the real vector space of functions spanned by the set of real
valued functions {1, t, t2 , et , tet }, and T is the linear operator on V
defined by T(f ) = f .
1 1
(c) T is the linear operator on M2×2 (R) defined by T(A) =
· A
0 1
for all A ∈ M2×2 (R).
(d) T(A) = 2A + At for all A ∈ M2×2 (R).
4. † Let T be a linear operator on a vector space V, and let γ be a cycle
of generalized eigenvectors that corresponds to the eigenvalue λ. Prove
that span(γ) is a T-invariant subspace of V.
5.Let γ1 , γ2 , . . . , γp be cycles of generalized eigenvectors of a linear op-
erator T corresponding to an eigenvalue λ. Prove that if the initial
eigenvectors are distinct, then the cycles are disjoint.
6.Let T : V → W be a linear transformation. Prove the following results.
(a) N(T) = N(−T).
(b) N(Tk ) = N((−T)k ).
(c) If V = W (so that T is a linear operator on V) and λ is an eigen-
value of T, then for any positive integer k
7.N((T − λIV )k ) = N((λIV − T)k ).
Let U be a linear operator on a finite-dimensional vector space V. Prove
the following results.
(a) N(U) ⊆ N(U2 ) ⊆ · · · ⊆ N(U k ) ⊆ N(Uk+1 ) ⊆ · · · .
496
Chap. 7
Canonical Forms
(b)(c)(d)(e)(f )If rank(Um ) = rank(Um+1 ) for some positive integer m, then
rank(Um ) = rank(Uk ) for any positive integer k ≥ m.
If rank(Um ) = rank(Um+1 ) for some positive integer m, then
N(Um ) = N(Uk ) for any positive integer k ≥ m.
Let T be a linear operator on V, and let λ be an eigenvalue of T.
Prove that if rank((T−λI)m ) = rank((T−λI)m+1 ) for some integer
m, then Kλ = N((T − λI)m ).
Second Test for Diagonalizability. Let T be a linear operator on
V whose characteristic polynomial splits, and let λ1 , λ 2 , . . . , λk be
the distinct eigenvalues of T. Then T is diagonalizable if and only
if rank(T − λI) = rank((T − λI)2 ) for 1 ≤ i ≤ k.
Use (e) to obtain a simpler proof of Exercise 24 of Section 5.4: If
T is a diagonalizable linear operator on a finite-dimensional vec-
tor space V and W is a T-invariant subspace of V, then TW is
diagonalizable.
8.Use Theorem 7.4 to prove that the vectors v1 , v2 , . . . , vk in the statement
of Theorem 7.3 are unique.
9.Let T be a linear operator on a finite-dimensional vector space V whose
characteristic polynomial splits.
(a) Prove Theorem 7.5(b).
(b) Suppose that β is a Jordan canonical basis for T, and let λ be an
eigenvalue of T. Let β = β ∩ Kλ . Prove that β is a basis for Kλ .
10.11.Let T be a linear operator on a finite-dimensional vector space whose
characteristic polynomial splits, and let λ be an eigenvalue of T.
(a)(b)Suppose that γ is a basis for Kλ consisting of the union of q disjoint
cycles of generalized eigenvectors. Prove that q ≤ dim(Eλ ).
Let β be a Jordan canonical basis for T, and suppose that J = [T]β
has q Jordan blocks with λ in the diagonal positions. Prove that
q ≤ dim(Eλ ).
Prove Corollary 2 to Theorem 7.7.
Exercises 12 and 13 are concerned with direct sums of matrices, defined in
Section 5.4 on page 320.
12. Prove Theorem 7.8.
13.Let T be a linear operator on a finite-dimensional vector space V such
that the characteristic polynomial of T splits, and let λ1 , λ2 , . . . , λk be
the distinct eigenvalues of T. For each i, let Ji be the Jordan canonical
form of the restriction of T to Kλi . Prove that
J = J1 ⊕ J2 ⊕ · · · ⊕ Jk
is the Jordan canonical form of J.
1.Label the following statements as true or false. Assume that the char-
acteristic polynomial of the matrix or linear operator splits.
(a) The Jordan canonical form of a diagonal matrix is the matrix itself.
(b) Let T be a linear operator on a finite-dimensional vector space V
that has a Jordan canonical form J. If β is any basis for V, then
the Jordan canonical form of [T]β is J.
(c) Linear operators having the same characteristic polynomial are
similar.
(d) Matrices having the same Jordan canonical form are similar.
(e) Every matrix is similar to its Jordan canonical form.
(f ) Every linear operator with the characteristic polynomial
(−1)n (t − λ)n has the same Jordan canonical form.
(g) Every linear operator on a finite-dimensional vector space has a
unique Jordan canonical basis.
(h) The dot diagrams of a linear operator on a finite-dimensional vec-
tor space are unique.
510
Chap. 7
Canonical Forms
2. Let T be a linear operator on a finite-dimensional vector space V such
that the characteristic polynomial of T splits. Suppose that λ1 = 2,
λ2 = 4, and λ3 = −3 are the distinct eigenvalues of T and that the dot
diagrams for the restriction of T to Kλi (i = 1, 2, 3) are as follows:
λ1 = 2
λ2 = 4
λ3 = −3
•
•
•
•
•
•
•
•
•
•
•
•
Find the Jordan canonical form J of T.
3. Let T be a linear operator
on a finite-dimensional vector space V with
Jordan canonical form
⎛
⎞
2
1
0
0
0
0
0
⎜ 0
2
1
0
0
0
0
⎟
⎜
⎟
⎜ 0
0
2
0
0
0
0
⎟
⎜
⎟
⎜ 0
0
0
2
1
0
0
⎟ .
⎜
⎟
⎜ 0
0
0
0
2
0
0
⎟
⎜
⎟
⎝ 0
0
0
0
0
3
0
⎠
0
0
0
0
0
0
3
(a)
(b)
(c)
(d)
(e)Find the characteristic polynomial of T.
Find the dot diagram corresponding to each eigenvalue of T.
For which eigenvalues λi , if any, does Eλi = Kλi ?
For each eigenvalue λi , find the smallest positive integer pi for
which K λi = N((T − λi I)pi ).
Compute the following numbers for each i, where Ui denotes the
restriction of T − λi I to Kλi .
(i) rank(Ui )
(ii) rank(U2 )
i(iii) nullity(Ui )
(iv) nullity(U2 )
i4. For each of the matrices A that follow, find a Jordan canonical form
J and an invertible matrix Q such that J = Q−1 AQ. Notice that the
matrices in (a), (b), and (c) are those used in Example 5.
⎛
⎞
⎛
⎞
−3
3 −2
0 1 −1
(a) A = ⎝−7
6 −3⎠
(b) A = ⎝−4 4 −2⎠
1 −1
2
−2 1
1
⎛
⎞
⎛
⎞
0 −3
1 2
(c) A = ⎝−3 0 −1 −1 −2⎠
−1
(d) A = ⎜
⎜−2
1 −1 2⎟
⎟
⎝−2
1 −1 2⎠
7
5
6
−2 −3
1 4
Sec. 7.2
The Jordan Canonical Form II
511
5.For each linear operator T, find a Jordan canonical form J of T and a
Jordan canonical basis β for T.
(a) V is the real vector space of functions spanned by the set of real-
valued functions {et , tet , t2 et , e2t }, and T is the linear operator on
V defined by T(f ) = f .
(b) T is the linear operator on P3 (R) defined by T(f (x)) = xf (x).
(c) T is the linear operator on P3 (R) defined by
T(f (x)) = f (x) + 2f (x).
(d) T is the linear operator on M2×2 (R) defined by
3 1
T(A) =
· A − At .
0 3
(e)T is the linear operator on M2×2 (R) defined by
3 1
T(A) =
· (A − At ).
0 3
(f )V is the vector space of polynomial functions in two real variables
x and y of degree at most 2, as defined in Example 4, and T is the
linear operator on V defined by
∂
∂
T(f (x, y)) =
f (x, y) +
f (x, y).
∂x
∂y
6.Let A be an n × n matrix whose characteristic polynomial splits. Prove
that A and At have the same Jordan canonical form, and conclude that
A and At are similar. Hint: For any eigenvalue λ of A and At and any
positive integer r, show that rank((A − λI)r ) = rank((At − λI)r ).
7.Let A be an n × n matrix whose characteristic polynomial splits, γ be
a cycle of generalized eigenvectors corresponding to an eigenvalue λ,
and W be the subspace spanned by γ. Define γ to be the ordered set
obtained from γ by reversing the order of the vectors in γ.
t
(a) Prove that [TW ]γ = ([TW ]γ ) .
(b) Let J be the Jordan canonical form of A. Use (a) to prove that J
and J t are similar.
(c) Use (b) to prove that A and At are similar.
8.Let T be a linear operator on a finite-dimensional vector space, and
suppose that the characteristic polynomial of T splits. Let β be a Jordan
canonical basis for T.
(a) Prove that for any nonzero scalar c, {cx : x ∈ β} is a Jordan canon-
ical basis for T.
512
Chap. 7
Canonical Forms
(b)(c)Suppose that γ is one of the cycles of generalized eigenvectors that
forms β, and suppose that γ corresponds to the eigenvalue λ and
has length greater than 1. Let x be the end vector of γ, and let y
be a nonzero vector in Eλ . Let γ be the ordered set obtained from
γ by replacing x by x + y. Prove that γ is a cycle of generalized
eigenvectors corresponding to λ, and that if γ replaces γ in the
union that defines β, then the new union is also a Jordan canonical
basis for T.
Apply (b) to obtain a Jordan canonical basis for LA , where A is the
matrix given in Example 2, that is different from the basis given
in the example.
9.Suppose that a dot diagram has k columns and m rows with pj dots in
column j and ri dots in row i. Prove the following results.
(a)(b)(c)(d)m = p1 and k = r1 .
pj = max {i : ri ≥ j} for 1 ≤ j ≤ k and ri = max {j : pj ≥ i} for
1 ≤ i ≤ m. Hint: Use mathematical induction on m.
r1 ≥ r2 ≥ · · · ≥ rm .
Deduce that the number of dots in each column of a dot diagram
is completely determined by the number of dots in the rows.
10.Let T be a linear operator whose characteristic polynomial splits, and
let λ be an eigenvalue of T.
(a) Prove that dim(Kλ ) is the sum of the lengths of all the blocks
corresponding to λ in the Jordan canonical form of T.
(b) Deduce that Eλ = Kλ if and only if all the Jordan blocks corre-
sponding to λ are 1 × 1 matrices.
The following definitions are used in Exercises 11–19.
Definitions. A linear operator T on a vector space V is called nilpotent
if Tp = T0 for some positive integer p. An n × n matrix A is called nilpotent
if Ap = O for some positive integer p.
11.Let T be a linear operator on a finite-dimensional vector space V, and
let β be an ordered basis for V. Prove that T is nilpotent if and only if
[T]β is nilpotent.
12.Prove that any square upper triangular matrix with each diagonal entry
equal to zero is nilpotent.
13.Let T be a nilpotent operator on an n-dimensional vector space V, and
suppose that p is the smallest positive integer for which Tp = T0 . Prove
the following results.
(a) N(Ti ) ⊆ N(Ti+1 ) for every positive integer i.
Sec. 7.2
The Jordan Canonical Form II
513
(b)(c)(d)There is a sequence of ordered bases β1 , β2 , . . . , βp such that βi is
a basis for N(Ti ) and βi+1 contains βi for 1 ≤ i ≤ p − 1.
Let β = βp be the ordered basis for N(Tp ) = V in (b). Then [T]β
is an upper triangular matrix with each diagonal entry equal to
zero.
The characteristic polynomial of T is (−1)n tn . Hence the charac-
teristic polynomial of T splits, and 0 is the only eigenvalue of T.
14.Prove the converse of Exercise 13(d): If T is a linear operator on an n-
dimensional vector space V and (−1)n tn is the characteristic polynomial
of T, then T is nilpotent.
15.Give an example of a linear operator T on a finite-dimensional vector
space such that T is not nilpotent, but zero is the only eigenvalue of T.
Characterize all such operators.
16.Let T be a nilpotent linear operator on a finite-dimensional vector space
V. Recall from Exercise 13 that λ = 0 is the only eigenvalue of T, and
hence V = Kλ . Let β be a Jordan canonical basis for T. Prove that for
any positive integer i, if we delete from β the vectors corresponding to
the last i dots in each column of a dot diagram of β, the resulting set is
a basis for R(Ti ). (If a column of the dot diagram contains fewer than i
dots, all the vectors associated with that column are removed from β.)
17.Let T be a linear operator on a finite-dimensional vector space V such
that the characteristic polynomial of T splits, and let λ1 , λ2 , . . . , λk be
the distinct eigenvalues of T. Let S : V → V be the mapping defined by
S(x) = λ1 v1 + λ2 v2 + · · · + λk vk ,
where, for each i, vi is the unique vector in Kλi such that x = v1 +
v2 + · · · + vk . (This unique representation is guaranteed by Theorem 7.3
(p. 486) and Exercise 8 of Section 7.1.)
(a) Prove that S is a diagonalizable linear operator on V.
(b) Let U = T − S. Prove that U is nilpotent and commutes with S,
that is, SU = US.
18.Let T be a linear operator on a finite-dimensional vector space V, and
let J be the Jordan canonical form of T. Let D be the diagonal matrix
whose diagonal entries are the diagonal entries of J, and let M = J −D.
Prove the following results.
(a)(b)M is nilpotent.
M D = DM .
514
Chap. 7
Canonical Forms
(c) If p is the smallest positive integer for which M p = O, then, for
any positive integer r < p,
r r−1 r(r − 1) r−2 2
r−1 rJ = Dr + rD M +
D M + · · · + rDM + M ,
2!
and, for any positive integer r ≥ p,
r r−1 r(r − 1) r−2 2
J = Dr + rD M +
D M + ···
2!
r!
r−p+1 p−1+
D M .
(r − p + 1)!(p − 1)!
19. Let
⎛
⎞
λ
1
0
···
0
⎜ ⎜
0
λ
1
···
0 ⎟
⎟
⎜ ⎜
0
0
λ
···
0 ⎟
⎟
J = ⎜ ⎜ .
..
..
.
..
.
.. . ⎟
⎟
⎜
⎟
⎝ 0
0
0
· · · 1 ⎠
0
0
0
··· λ
be the m × m Jordan block corresponding to λ, and let N = J − λIm .
Prove the following results:
(a) N m = O, and for 1 ≤ r < m,
1 if j = i + r
r
Nij =
0 otherwise.
(b) For any integer r ≥ m,
⎛
⎞
r(r − 1) r−2
r(r − 1) · · · (r − m + 2) r−m+1
⎜ λr rλr−1
λ
···
λ
⎟
⎜
2!
(m − 1)!
⎟
⎜
⎟
⎜
⎟
⎜
r(r
−
1)
·
·
·
(r
−
m
+
3)
⎟
r
⎜ 0 λr
rλr−1 · · ·
λr−m+2 ⎟
J = ⎜
(m − 2)!
⎟.
⎜
⎟
⎜ . .
⎟
⎜ ⎜
.. ..
..
.
..
.
⎟
⎟
⎝
⎠
0 0
0
···
λr
(c)
lim J r exists if and only if one of the following holds:
r→∞
(i) |λ| < 1.
(ii) λ = 1 and m = 1.
Sec. 7.2
The Jordan Canonical Form II
515
(Note that lim λr exists under these conditions. See the discus-
r→∞
r
sion preceding Theorem 5.13 on page 285.) Furthermore, lim Jr→∞
is the zero matrix if condition (i) holds and is the 1 × 1 matrix (1)
if condition (ii) holds.
(d) Prove Theorem 5.13 on page 285.
The following definition is used in Exercises 20 and 21.
Definition.For any A ∈ Mn×n(C), define the norm of A by
A = max {|Aij | : 1 ≤ i, j ≤ n}.
20.Let A, B ∈ Mn×n (C). Prove the following results.
(a) A ≥ 0 and A = 0 if and only if A = O.
(b) cA = |c|· A for any scalar c.
(c) A + B ≤ A + B.
(d) AB ≤ nAB.
21.Let A ∈ Mn×n (C) be a transition matrix. (See Section 5.3.) Since C is
an algebraically closed field, A has a Jordan canonical form J to which
A is similar. Let P be an invertible matrix such that P −1 AP = J.
Prove the following results.
(a) Am ≤ 1 for every positive integer m.
(b) There exists a positive number c such that J m ≤ c for every
positive integer m.
(c) Each Jordan block of J corresponding to the eigenvalue λ = 1 is a
1 × 1 matrix.
(d) lim Am exists if and only if 1 is the only eigenvalue of A with
m→∞
absolute value 1.
(e) Theorem 5.20(a) using (c) and Theorem 5.19.
The next exercise requires knowledge of absolutely convergent series as well
as the definition of eA for a matrix A. (See page 312.)
22.23.Use Exercise 20(d) to prove that eA exists for every A ∈ Mn×n (C).
Let x = Ax be a system of n linear differential equations, where x is
an n-tuple of differentiable functions x1 (t), x2 (t), . . . , x n (t) of the real
variable t, and A is an n × n coefficient matrix as in Exercise 15 of
Section 5.2. In contrast to that exercise, however, do not assume that
A is diagonalizable, but assume that the characteristic polynomial of A
splits. Let λ1 , λ2 , . . . , λk be the distinct eigenvalues of A.
516
Chap. 7
Canonical Forms
(a)(b)Prove that if u is the end vector of a cycle of generalized eigenvec-
tors of LA of length p and u corresponds to the eigenvalue λi , then
for any polynomial f (t) of degree less than p, the function
eλi t [f (t)(A − λiI)p−1 + f (t)(A − λi I)p−2 + · · · + f (p−1) (t)]u
is a solution to the system x = Ax.
Prove that the general solution to x = Ax is a sum of the functions
of the form given in (a), where the vectors u are the end vectors of
the distinct cycles that constitute a fixed Jordan canonical basis
for LA .
24. Use Exercise 23 to find the general solution to each of the following sys-
tems of linear equations, where x, y, and z are real-valued differentiable
functions of the real variable t.
x = 2x + y
x = 2x + y
(a) y
=
2y − z
(b) y =
2y + z
z =
3z
z =
2z
1.Label the following statements as true or false. Assume that all vector
spaces are finite-dimensional.
(a) Every linear operator T has a polynomial p(t) of largest degree for
which p(T) = T0 .
(b) Every linear operator has a unique minimal polynomial.
(c) The characteristic polynomial of a linear operator divides the min-
imal polynomial of that operator.
(d) The minimal and the characteristic polynomials of any diagonal-
izable operator are equal.
(e) Let T be a linear operator on an n-dimensional vector space V, p(t)
be the minimal polynomial of T, and f (t) be the characteristic
polynomial of T. Suppose that f (t) splits. Then f (t) divides
[p(t)]n .
(f ) The minimal polynomial of a linear operator always has the same
degree as the characteristic polynomial of the operator.
(g) A linear operator is diagonalizable if its minimal polynomial splits.
(h) Let T be a linear operator on a vector space V such that V is a
T-cyclic subspace of itself. Then the degree of the minimal poly-
nomial of T equals dim(V).
(i) Let T be a linear operator on a vector space V such that T has n
distinct eigenvalues, where n = dim(V). Then the degree of the
minimal polynomial of T equals n.
2. Find the minimal polynomial of each
of the following matrices.
2 1
1 1
(a)
(b)
1 2
0 1
⎛
⎞
⎛
⎞
4 −14 5
3 0
1
(c) ⎝1
−4 2⎠
(d) ⎝ 2 2
2⎠
1
−6 4
−1 0
1
3.For each linear operator T on V, find the minimal polynomial of T.
(a)
V = R2 and T(a, b) = (a + b, a − b)
(b)
V = P2 (R) and T(g(x)) = g (x) + 2g(x)
(c)
V = P2 (R) and T(f (x)) = −xf (x) + f (x) + 2f (x)
(d)
V = Mn×n (R) and T(A) = At . Hint: Note that T2 = I.
4.Determine which of the matrices and operators in Exercises 2 and 3 are
diagonalizable.
5.Describe all linear operators T on R2 such that T is diagonalizable and
T3 − 2T2 + T = T0 .
Sec. 7.3
The Minimal Polynomial
523
6. Prove Theorem 7.13 and its corollary.
7. Prove the corollary to Theorem 7.14.
8.Let T be a linear operator on a finite-dimensional vector space, and let
p(t) be the minimal polynomial of T. Prove the following results.
(a) T is invertible if and only if p(0) = 0.
(b) If T is invertible and p(t) = tn + an−1 tn−1 + · · · + a1 t + a0 , then
1 n−1
T−1 = −
T
+ an−1 Tn−2 + · · · + a2 T + a1 I .
a0
9.Let T be a diagonalizable linear operator on a finite-dimensional vector
space V. Prove that V is a T-cyclic subspace if and only if each of the
eigenspaces of T is one-dimensional.
10.Let T be a linear operator on a finite-dimensional vector space V, and
suppose that W is a T-invariant subspace of V. Prove that the minimal
polynomial of TW divides the minimal polynomial of T.
11.Let g(t) be the auxiliary polynomial associated with a homogeneous lin-
ear differential equation with constant coefficients (as defined in Section
2.7), and let V denote the solution space of this differential equation.
Prove the following results.
(a)(b)(c)V is a D-invariant subspace, where D is the differentiation operator
on C∞ .
The minimal polynomial of DV (the restriction of D to V) is g(t).
If the degree of g(t) is n, then the characteristic polynomial of DV
is (−1)n g(t).
Hint: Use Theorem 2.32 (p. 135) for (b) and (c).
12.Let D be the differentiation operator on P(R), the space of polynomials
over R. Prove that there exists no polynomial g(t) for which g(D) = T0 .
Hence D has no minimal polynomial.
13.Let T be a linear operator on a finite-dimensional vector space, and
suppose that the characteristic polynomial of T splits. Let λ1 , λ2 , . . . , λk
be the distinct eigenvalues of T, and for each i let pi be the order of the
largest Jordan block corresponding to λi in a Jordan canonical form of
T. Prove that the minimal polynomial of T is
(t − λ1 )p1 (t − λ2 )p2 · · · (t − λk )pk .
The following exercise requires knowledge of direct sums (see Section 5.2).
524
Chap. 7
Canonical Forms
14.Let T be linear operator on a finite-dimensional vector space V, and
let W1 and W2 be T-invariant subspaces of V such that V = W1 ⊕ W2 .
Suppose that p1 (t) and p2 (t) are the minimal polynomials of TW1 and
TW2 , respectively. Prove or disprove that p1 (t)p2 (t) is the minimal
polynomial of T.
Exercise 15 uses the following definition.
Definition. Let T be a linear operator on a finite-dimensional vector
space V, and let x be a nonzero vector in V. The polynomial p(t) is called
a T-annihilator of x if p(t) is a monic polynomial of least degree for which
p(T)(x) = 0 .
15. † Let T be a linear operator on a finite-dimensional vector space V, and
let x be a nonzero vector in V. Prove the following results.
(a) The vector x has a unique T-annihilator.
(b) The T-annihilator of x divides any polynomial g(t) for which
g(T) = T0 .
(c) If p(t) is the T-annihilator of x and W is the T-cyclic subspace
generated by x, then p(t) is the minimal polynomial of TW , and
dim(W) equals the degree of p(t).
(d) The degree of the T-annihilator of x is 1 if and only if x is an
eigenvector of T.
16.T be a linear operator on a finite-dimensional vector space V, and let
W1 be a T-invariant subspace of V. Let x ∈ V such that x ∈
/ W1 . Prove
the following results.
(a)(b)(c)(d)There exists a unique monic polynomial g1 (t) of least positive de-
gree such that g1 (T)(x) ∈ W1 .
If h(t) is a polynomial for which h(T)(x) ∈ W1 , then g1 (t) divides
h(t).
g1 (t) divides the minimal and the characteristic polynomials of T.
Let W2 be a T-invariant subspace of V such that W2 ⊆ W1 , and
let g2 (t) be the unique monic polynomial of least degree such that
g2 (T)(x) ∈ W2 . Then g1 (t) divides g2 (t).
1.EXERCISES
Label the following statements as true or false.
(a) Every rational canonical basis for a linear operator T is the union
of T-cyclic bases.
546
Chap. 7
Canonical Forms
(b) If a basis is the union of T-cyclic bases for a linear operator T,
then it is a rational canonical basis for T.
(c) There exist square matrices having no rational canonical form.
(d) A square matrix is similar to its rational canonical form.
(e) For any linear operator T on a finite-dimensional vector space, any
irreducible factor of the characteristic polynomial of T divides the
minimal polynomial of T.
(f ) Let φ(t) be an irreducible monic divisor of the characteristic poly-
nomial of a linear operator T. The dots in the diagram used to
compute the rational canonical form of the restriction of T to Kφ
are in one-to-one correspondence with the vectors in a basis for
Kφ .
(g) If a matrix has a Jordan canonical form, then its Jordan canonical
form and rational canonical form are similar.
2. For each of the following matrices A ∈ Mn×n (F ), find the rational
canonical form C of A and a matrix Q ∈ Mn×n (F ) such that Q−1 AQ =
C.
⎛
⎞
3 1 0
0 −1
(a) A = ⎝
0 3 1
⎠
F = R
(b) A =
F = R
1 −1
0 0 3
0 −1
(c) A =
F = C
1 −1
⎛
⎞
0 −7 14 −6
(d) A = ⎜1 ⎜
−4
6 −3⎟
⎟ F = R
⎝0 −4
9 −4⎠
0 −4 11 −5
⎛
⎞
0 −4 12 −7
(e) A = ⎜1 ⎜
−1
3 −3⎟
⎟ F = R
⎝0 −1
6 −4⎠
0 −1
8 −5
3. For each of the following linear operators T, find the elementary divisors,
the rational canonical form C, and a rational canonical basis β.
(a) T is the linear operator on P3 (R) defined by
T(f (x)) = f (0)x − f (1).
(b) Let S = {sin x, cos x, x sin x, x cos x}, a subset of F(R, R), and let
V = span(S). Define T to be the linear operator on V such that
T(f ) = f .
(c) T is the linear operator on M2×2 (R) defined by
Sec. 7.4
The Rational Canonical Form
547
0
1
T(A) =
· A.
−1
1
(d) Let S = {sin x sin y, sin x cos y, cos x sin y, cos x cos y}, a subset of
F(R × R, R), and let V = span(S). Define T to be the linear
operator on V such that
∂f (x, y) ∂f (x, y)
T(f )(x, y) =
+
.
∂x
∂y
4.Let T be a linear operator on a finite-dimensional vector space V with
minimal polynomial (φ(t))m for some positive integer m.
(a) Prove that R(φ(T)) ⊆ N((φ(T))m−1 ).
(b) Give an example to show that the subspaces in (a) need not be
equal.
(c) Prove that the minimal polynomial of the restriction of T to
R(φ(T)) equals (φ(t))m−1 .
5.Let T be a linear operator on a finite-dimensional vector space. Prove
that the rational canonical form of T is a diagonal matrix if and only if
T is diagonalizable.
6.Let T be a linear operator on a finite-dimensional vector space V with
characteristic polynomial f (t) = (−1)n φ1 (t)φ2 (t), where φ1 (t) and φ2 (t)
are distinct irreducible monic polynomials and n = dim(V).
(a) Prove that there exist v1 , v2 ∈ V such that v1 has T-annihilator
φ1 (t), v2 has T-annihilator φ2 (t), and βv1 ∪ βv2 is a basis for V.
(b) Prove that there is a vector v3 ∈ V with T-annihilator φ1 (t)φ2 (t)
such that βv3 is a basis for V.
(c) Describe the difference between the matrix representation of T
with respect to βv1 ∪ βv2 and the matrix representation of T with
respect to βv3 .
Thus, to assure the uniqueness of the rational canonical form, we re-
quire that the generators of the T-cyclic bases that constitute a rational
canonical basis have T-annihilators equal to powers of irreducible monic
factors of the characteristic polynomial of T.
7.Let T be a linear operator on a finite-dimensional vector space with
minimal polynomial
f (t) = (φ1 (t))m1 (φ2 (t))m2 · · · (φk (t))mk ,
where the φi (t)’s are distinct irreducible monic factors of f (t). Prove
that for each i, mi is the number of entries in the first column of the
dot diagram for φi (t).
548
Chap. 7
Canonical Forms
8.Let T be a linear operator on a finite-dimensional vector space V. Prove
that for any irreducible polynomial φ(t), if φ(T) is not one-to-one, then
φ(t) divides the characteristic polynomial of T. Hint: Apply Exercise 15
of Section 7.3.
9.Let V be a vector space and β1 , β2 , . . . , βk be disjoint subsets of V whose
union is a basis for V. Now suppose that γ1 , γ2 , . . . , γk are linearly
independent subsets of V such that span(γi ) = span(βi ) for all i. Prove
that γ1 ∪ γ2 ∪ · · · ∪ γk is also a basis for V.
10.Let T be a linear operator on a finite-dimensional vector space, and
suppose that φ(t) is an irreducible monic factor of the characteristic
polynomial of T. Prove that if φ(t) is the T-annihilator of vectors x and
y, then x ∈ Cy if and only if Cx = Cy .
Exercises 11 and 12 are concerned with direct sums.
11. Prove Theorem 7.25.
12. Prove Theorem 7.26.
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Linear Algebrahashnuthttps://koreanfoodie.me/95https://koreanfoodie.me/95#entry95commentSat, 15 Jun 2019 13:24:44 +0900Solutions to Linear Algebra, Stephen H. Friedberg, Fourth Edition (Chapter 6)
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data-url='https://k.kakaocdn.net/dn/bHt3Ym/btqv6EUeFe2/bCLcxs1LNQcKf54Kvas6c1/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/bHt3Ym/btqv6EUeFe2/bCLcxs1LNQcKf54Kvas6c1/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FbHt3Ym%2Fbtqv6EUeFe2%2FbCLcxs1LNQcKf54Kvas6c1%2Fimg.jpg' data-filename="chapter6_11.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/JSkwa/btqv7QsAZjI/xPKijVg8bUGsZ6oCeYaZz0/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/JSkwa/btqv7QsAZjI/xPKijVg8bUGsZ6oCeYaZz0/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FJSkwa%2Fbtqv7QsAZjI%2FxPKijVg8bUGsZ6oCeYaZz0%2Fimg.jpg' data-filename="chapter6_12.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/bvkQIN/btqv5V2WnCF/t3oUFlwNo3FaMC58E7qH60/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/bvkQIN/btqv5V2WnCF/t3oUFlwNo3FaMC58E7qH60/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FbvkQIN%2Fbtqv5V2WnCF%2Ft3oUFlwNo3FaMC58E7qH60%2Fimg.jpg' data-filename="chapter6_13.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/BMSfl/btqv5W8CjQY/5QKjlbLbNJjujdzVubvOZ1/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/BMSfl/btqv5W8CjQY/5QKjlbLbNJjujdzVubvOZ1/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FBMSfl%2Fbtqv5W8CjQY%2F5QKjlbLbNJjujdzVubvOZ1%2Fimg.jpg' data-filename="chapter6_14.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/bShmtl/btqv4QAZNqX/DmsTJqmCWpD47DqQhE3KP1/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/bShmtl/btqv4QAZNqX/DmsTJqmCWpD47DqQhE3KP1/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FbShmtl%2Fbtqv4QAZNqX%2FDmsTJqmCWpD47DqQhE3KP1%2Fimg.jpg' data-filename="chapter6_15.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/bNQ3o5/btqv7YxbhIc/5kx2AZtuHd0k23EuVKY5IK/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/bNQ3o5/btqv7YxbhIc/5kx2AZtuHd0k23EuVKY5IK/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FbNQ3o5%2Fbtqv7YxbhIc%2F5kx2AZtuHd0k23EuVKY5IK%2Fimg.jpg' data-filename="chapter6_16.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/b05AHt/btqv5gM5vRH/uK4o6mQFkqP8HCkaPKpm6k/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/b05AHt/btqv5gM5vRH/uK4o6mQFkqP8HCkaPKpm6k/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2Fb05AHt%2Fbtqv5gM5vRH%2FuK4o6mQFkqP8HCkaPKpm6k%2Fimg.jpg' data-filename="chapter6_17.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/dFYed2/btqv7ZiwpRk/wPUxPSfcDx1BxRLhKjnHf1/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/dFYed2/btqv7ZiwpRk/wPUxPSfcDx1BxRLhKjnHf1/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FdFYed2%2Fbtqv7ZiwpRk%2FwPUxPSfcDx1BxRLhKjnHf1%2Fimg.jpg' data-filename="chapter6_18.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/ApEl0/btqv7O9obYg/iqoeQNsBpdug3jLpYjfTxk/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/ApEl0/btqv7O9obYg/iqoeQNsBpdug3jLpYjfTxk/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FApEl0%2Fbtqv7O9obYg%2FiqoeQNsBpdug3jLpYjfTxk%2Fimg.jpg' data-filename="chapter6_19.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/bghJVM/btqv7XZkrua/8McxGkiUGqSamDjNAojs8K/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/bghJVM/btqv7XZkrua/8McxGkiUGqSamDjNAojs8K/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FbghJVM%2Fbtqv7XZkrua%2F8McxGkiUGqSamDjNAojs8K%2Fimg.jpg' data-filename="chapter6_20.jpg"></span></figure><figure class='imageblock alignCenter'><span 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data-url='https://k.kakaocdn.net/dn/bE8ItC/btqv7pa98dy/HJE6LM5sL23SKno2Tz9ye1/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/bE8ItC/btqv7pa98dy/HJE6LM5sL23SKno2Tz9ye1/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FbE8ItC%2Fbtqv7pa98dy%2FHJE6LM5sL23SKno2Tz9ye1%2Fimg.jpg' data-filename="chapter6_23.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/bXDxdM/btqv7p93yJW/lEzyXX6nDt90TCWMX5VfSk/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/bXDxdM/btqv7p93yJW/lEzyXX6nDt90TCWMX5VfSk/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FbXDxdM%2Fbtqv7p93yJW%2FlEzyXX6nDt90TCWMX5VfSk%2Fimg.jpg' data-filename="chapter6_24.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/ASDA0/btqv4PCavNU/BopWpdrkMREcKCIQ4Ov8Vk/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/ASDA0/btqv4PCavNU/BopWpdrkMREcKCIQ4Ov8Vk/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FASDA0%2Fbtqv4PCavNU%2FBopWpdrkMREcKCIQ4Ov8Vk%2Fimg.jpg' data-filename="chapter6_25.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/lPhDL/btqv4QODynb/fcMJScV6KHkjhLnXwNtKS0/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/lPhDL/btqv4QODynb/fcMJScV6KHkjhLnXwNtKS0/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FlPhDL%2Fbtqv4QODynb%2FfcMJScV6KHkjhLnXwNtKS0%2Fimg.jpg' data-filename="chapter6_26.jpg"></span></figure><figure class='imageblock alignCenter'><span 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data-url='https://k.kakaocdn.net/dn/cSjPX5/btqv7DUy4XD/H0P45TBvgM96PzuOgKDGI0/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/cSjPX5/btqv7DUy4XD/H0P45TBvgM96PzuOgKDGI0/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FcSjPX5%2Fbtqv7DUy4XD%2FH0P45TBvgM96PzuOgKDGI0%2Fimg.jpg' data-filename="chapter6_43.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/cFQTVD/btqv7YKGyyt/IKzaXOJN5KAyli4VMpQJM1/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/cFQTVD/btqv7YKGyyt/IKzaXOJN5KAyli4VMpQJM1/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FcFQTVD%2Fbtqv7YKGyyt%2FIKzaXOJN5KAyli4VMpQJM1%2Fimg.jpg' data-filename="chapter6_44.jpg"></span></figure><figure class='imageblock alignCenter'><span 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data-url='https://k.kakaocdn.net/dn/c7qBKp/btqv4PPCCZk/RWixFraf1SGzC9sK64tfak/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/c7qBKp/btqv4PPCCZk/RWixFraf1SGzC9sK64tfak/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2Fc7qBKp%2Fbtqv4PPCCZk%2FRWixFraf1SGzC9sK64tfak%2Fimg.jpg' data-filename="chapter6_67.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/cQvknP/btqv7YKGyJT/o2xWwkhSF3B351nNhKkRN0/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/cQvknP/btqv7YKGyJT/o2xWwkhSF3B351nNhKkRN0/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FcQvknP%2Fbtqv7YKGyJT%2Fo2xWwkhSF3B351nNhKkRN0%2Fimg.jpg' data-filename="chapter6_68.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/nLWJD/btqv4QgKiAW/wEKIzivsNSIdn3XPVq7DAK/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/nLWJD/btqv4QgKiAW/wEKIzivsNSIdn3XPVq7DAK/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FnLWJD%2Fbtqv4QgKiAW%2FwEKIzivsNSIdn3XPVq7DAK%2Fimg.jpg' data-filename="chapter6_69.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/8zq3D/btqv5fHou9m/wnef9TwKqKOvYj3UUKJI0k/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/8zq3D/btqv5fHou9m/wnef9TwKqKOvYj3UUKJI0k/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2F8zq3D%2Fbtqv5fHou9m%2Fwnef9TwKqKOvYj3UUKJI0k%2Fimg.jpg' data-filename="chapter6_70.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/bX22eU/btqv7Y40rFR/rdKirKUlKCOiJt8BPSfat0/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/bX22eU/btqv7Y40rFR/rdKirKUlKCOiJt8BPSfat0/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FbX22eU%2Fbtqv7Y40rFR%2FrdKirKUlKCOiJt8BPSfat0%2Fimg.jpg' data-filename="chapter6_71.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/v27Yg/btqv7Pf8Cfm/QWXiqC2irsa3maAjHJigX1/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/v27Yg/btqv7Pf8Cfm/QWXiqC2irsa3maAjHJigX1/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2Fv27Yg%2Fbtqv7Pf8Cfm%2FQWXiqC2irsa3maAjHJigX1%2Fimg.jpg' data-filename="chapter6_72.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/GPxS6/btqv4Q2cWhC/SabVvRmlHs99dBTRTHbKNk/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/GPxS6/btqv4Q2cWhC/SabVvRmlHs99dBTRTHbKNk/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FGPxS6%2Fbtqv4Q2cWhC%2FSabVvRmlHs99dBTRTHbKNk%2Fimg.jpg' data-filename="chapter6_73.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/wh2hb/btqv6DufCmt/iOv9ADnkwUPSyPRU3qFBg1/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/wh2hb/btqv6DufCmt/iOv9ADnkwUPSyPRU3qFBg1/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2Fwh2hb%2Fbtqv6DufCmt%2FiOv9ADnkwUPSyPRU3qFBg1%2Fimg.jpg' data-filename="chapter6_74.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/befBR1/btqv7gMhpSz/BqFzdLKHiZW6POPtq6FAVK/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/befBR1/btqv7gMhpSz/BqFzdLKHiZW6POPtq6FAVK/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FbefBR1%2Fbtqv7gMhpSz%2FBqFzdLKHiZW6POPtq6FAVK%2Fimg.jpg' data-filename="chapter6_75.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/UG3B4/btqv7XLMXsz/SjT9rK3bI42LngdKOoOp0k/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/UG3B4/btqv7XLMXsz/SjT9rK3bI42LngdKOoOp0k/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FUG3B4%2Fbtqv7XLMXsz%2FSjT9rK3bI42LngdKOoOp0k%2Fimg.jpg' data-filename="chapter6_76.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/bC9t2k/btqv5hFde5D/wigGBT4r8Z0KccEXkNjjHk/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/bC9t2k/btqv5hFde5D/wigGBT4r8Z0KccEXkNjjHk/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FbC9t2k%2Fbtqv5hFde5D%2FwigGBT4r8Z0KccEXkNjjHk%2Fimg.jpg' data-filename="chapter6_77.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/t24Pa/btqv5gTKdRK/SDc2qkkLiHapk7akkWm0S1/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/t24Pa/btqv5gTKdRK/SDc2qkkLiHapk7akkWm0S1/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2Ft24Pa%2Fbtqv5gTKdRK%2FSDc2qkkLiHapk7akkWm0S1%2Fimg.jpg' data-filename="chapter6_78.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/boGaQY/btqv5fUSzhe/wc4G6elWCmqgthZBqcuuXK/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/boGaQY/btqv5fUSzhe/wc4G6elWCmqgthZBqcuuXK/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FboGaQY%2Fbtqv5fUSzhe%2Fwc4G6elWCmqgthZBqcuuXK%2Fimg.jpg' data-filename="chapter6_79.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/HgK4W/btqv7C2ogGs/1vqRYK10qBifBH9h4YWq2K/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/HgK4W/btqv7C2ogGs/1vqRYK10qBifBH9h4YWq2K/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FHgK4W%2Fbtqv7C2ogGs%2F1vqRYK10qBifBH9h4YWq2K%2Fimg.jpg' data-filename="chapter6_80.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/dfOZMG/btqv6C9Sev6/VJ4e3cBkPjgbt1b0HbIRv0/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/dfOZMG/btqv6C9Sev6/VJ4e3cBkPjgbt1b0HbIRv0/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FdfOZMG%2Fbtqv6C9Sev6%2FVJ4e3cBkPjgbt1b0HbIRv0%2Fimg.jpg' data-filename="chapter6_81.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/vFOqc/btqv7ZQm7b5/cV7KUF3x07GwURytMGUVI0/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/vFOqc/btqv7ZQm7b5/cV7KUF3x07GwURytMGUVI0/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FvFOqc%2Fbtqv7ZQm7b5%2FcV7KUF3x07GwURytMGUVI0%2Fimg.jpg' data-filename="chapter6_82.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/marAP/btqv8ncoZZP/6TU9O2UMCnKZZN2DSig6l0/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/marAP/btqv8ncoZZP/6TU9O2UMCnKZZN2DSig6l0/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FmarAP%2Fbtqv8ncoZZP%2F6TU9O2UMCnKZZN2DSig6l0%2Fimg.jpg' data-filename="chapter6_83.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/eKLnkG/btqv7p93ySO/AVDZn2BAAfQHqc7IJINmhk/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/eKLnkG/btqv7p93ySO/AVDZn2BAAfQHqc7IJINmhk/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FeKLnkG%2Fbtqv7p93ySO%2FAVDZn2BAAfQHqc7IJINmhk%2Fimg.jpg' data-filename="chapter6_84.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/eAtDKC/btqv7O9ocp8/trHRlutLknk6Idqp0JsSIK/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/eAtDKC/btqv7O9ocp8/trHRlutLknk6Idqp0JsSIK/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FeAtDKC%2Fbtqv7O9ocp8%2FtrHRlutLknk6Idqp0JsSIK%2Fimg.jpg' data-filename="chapter6_85.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/py5Yo/btqv4P3efuO/MCC3kxdsZKFHimKX1dxDx1/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/py5Yo/btqv4P3efuO/MCC3kxdsZKFHimKX1dxDx1/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2Fpy5Yo%2Fbtqv4P3efuO%2FMCC3kxdsZKFHimKX1dxDx1%2Fimg.jpg' data-filename="chapter6_86.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/Qdoqv/btqv7DUy5hp/8mEGKuQKAKI7aYNLVNDAdk/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/Qdoqv/btqv7DUy5hp/8mEGKuQKAKI7aYNLVNDAdk/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FQdoqv%2Fbtqv7DUy5hp%2F8mEGKuQKAKI7aYNLVNDAdk%2Fimg.jpg' data-filename="chapter6_87.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/SPcrM/btqv8mxNyIP/rJ2JOUAr3gNBZ7FUv6ZuCK/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/SPcrM/btqv8mxNyIP/rJ2JOUAr3gNBZ7FUv6ZuCK/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FSPcrM%2Fbtqv8mxNyIP%2FrJ2JOUAr3gNBZ7FUv6ZuCK%2Fimg.jpg' data-filename="chapter6_88.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/bQ3v6i/btqv8ncoZ52/bK0EszesgmSgfwMikwSqEk/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/bQ3v6i/btqv8ncoZ52/bK0EszesgmSgfwMikwSqEk/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FbQ3v6i%2Fbtqv8ncoZ52%2FbK0EszesgmSgfwMikwSqEk%2Fimg.jpg' data-filename="chapter6_89.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/nwkDz/btqv7Yxbh5r/zvLvA8ALPknV8DHv18E9lk/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/nwkDz/btqv7Yxbh5r/zvLvA8ALPknV8DHv18E9lk/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FnwkDz%2Fbtqv7Yxbh5r%2FzvLvA8ALPknV8DHv18E9lk%2Fimg.jpg' data-filename="chapter6_90.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/blJXDw/btqv7Y40rLg/jpxaIKL60ePJ6s0nlf54KK/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/blJXDw/btqv7Y40rLg/jpxaIKL60ePJ6s0nlf54KK/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FblJXDw%2Fbtqv7Y40rLg%2FjpxaIKL60ePJ6s0nlf54KK%2Fimg.jpg' data-filename="chapter6_91.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/btOFFb/btqv4PPCC3v/6C6rNl53k2XfCy1Jz3OFS0/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/btOFFb/btqv4PPCC3v/6C6rNl53k2XfCy1Jz3OFS0/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FbtOFFb%2Fbtqv4PPCC3v%2F6C6rNl53k2XfCy1Jz3OFS0%2Fimg.jpg' data-filename="chapter6_92.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/18roi/btqv5gfcqUu/B97W9iDVtf42NMnICsvfyK/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/18roi/btqv5gfcqUu/B97W9iDVtf42NMnICsvfyK/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2F18roi%2Fbtqv5gfcqUu%2FB97W9iDVtf42NMnICsvfyK%2Fimg.jpg' data-filename="chapter6_93.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/bFsKoI/btqv4QAZNZ0/dZUPRohvaONSBOO15CAqpK/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/bFsKoI/btqv4QAZNZ0/dZUPRohvaONSBOO15CAqpK/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FbFsKoI%2Fbtqv4QAZNZ0%2FdZUPRohvaONSBOO15CAqpK%2Fimg.jpg' data-filename="chapter6_94.jpg"></span></figure></p>
<p>Solution maual to Linear Algebra, Fourth Edition, Stephen H. Friedberg. (Chapter 6)<br /><br />Solutions to Linear Algebra, Fourth Edition, Stephen H. Friedberg. (Chapter 6)<br /><br />Linear Algebra solution manual, Fourth Edition, Stephen H. Friedberg. (Chapter 6)<br /><br />Linear Algebra solutions Friedberg. (Chapter 6)</p>
<div style="width: 100%; height: 1px; overflow: auto;"><span>1.2.Label the following statements as true or false. (a)(b)(c)(d)(e)(f )(g)(h)An inner product is a scalar-valued function on the set of ordered pairs of vectors. An inner product space must be over the field of real or complex numbers. An inner product is linear in both components. There is exactly one inner product on the vector space Rn . The triangle inequality only holds in finite-dimensional inner prod- uct spaces. Only square matrices have a conjugate-transpose. If x, y, and z are vectors in an inner product space such that x, y = x, z, then y = z. If x, y = 0 for all x in an inner product space, then y = 0 . Let x = (2, 1 + i, i) and y = (2 − i, 2, 1 + 2i) be vectors in C3 . Compute x, y, x, y, and x + y. Then verify both the Cauchy–Schwarz inequality and the triangle inequality. 3.In C([0, 1]), let f (t) = t and g(t) = et . Compute f, g (as defined in Example 3), f , g, and f + g. Then verify both the Cauchy– Schwarz inequality and the triangle inequality. 4. (a) Complete the proof in Example 5 that · , · is an inner product (the Frobenius inner product) on Mn×n (F ). (b) Use the Frobenius inner product to compute A, B, and A, B for 1 2+ i 1+ i 0 A = and B = . 3 i i −i 5.In C2 , show that x, y = xAy∗ is an inner product, where 1 i A = . −i 2 Compute x, y for x = (1 − i, 2 + 3i) and y = (2 + i, 3 − 2i). Sec. 6.1 Inner Products and Norms 337 6. Complete the proof of Theorem 6.1. 7. Complete the proof of Theorem 6.2. 8. Provide reasons why each of the following is not an inner product on the given vector spaces. (a) (a, b), (c, d) = ac − bd on R2 . (b) A, B = tr(A + B) on M2×2 (R). 1 (c) f (x), g(x) = 0 f (t)g(t) dt on P(R), where denotes differentia- tion. 9. Let β be a basis for a finite-dimensional inner product space. (a) Prove that if x, z = 0 for all z ∈ β, then x = 0 . (b) Prove that if x, z = y, z for all z ∈ β, then x = y. 10. † Let V be an inner product space, and suppose that x and y are orthog- onal vectors in V. Prove that x + y2 = x2 + y2 . Deduce the Pythagorean theorem in R2 . 11. Prove the parallelogram law on an inner product space V; that is, show that x + y2 + x − y2 = 2x2 + 2y2 for all x, y ∈ V. What does this equation state about parallelograms in R2 ? 12. † Let {v1 , v2 , . . . , vk } be an orthogonal set in V, and let a1 , a2 , . . . , ak be scalars. Prove that 5 52 5 5 k 5 5 k 5 ai vi 5 = |ai |2 vi 2 . 5 5 i=1 i=1 13.14.Suppose that · , · 1 and · , · 2 are two inner products on a vector space V. Prove that · , · = · , · 1 + · , · 2 is another inner product on V. Let A and B be n × n matrices, and let c be a scalar. Prove that (A + cB)∗ = A∗ + cB ∗ . 15.(a)Prove that if V is an inner product space, then | x, y | = x· y if and only if one of the vectors x or y is a multiple of the other. Hint: If the identity holds and y = 0 , let x, y a = , y2 338 Chap. 6 Inner Product Spaces 16.(b)(a)(b)and let z = x − ay. Prove that y and z are orthogonal and x |a| = . y Then apply Exercise 10 to x2 = ay + z2 to obtain z = 0. Derive a similar result for the equality x + y = x + y, and generalize it to the case of n vectors. Show that the vector space H with · , · defined on page 332 is an inner product space. Let V = C([0, 1]), and define 1/2 f, g = f (t)g(t) dt. 0 Is this an inner product on V? 17.Let T be a linear operator on an inner product space V, and suppose that T(x) = x for all x. Prove that T is one-to-one. 18.Let V be a vector space over F , where F = R or F = C, and let W be an inner product space over F with inner product · , · . If T : V → W is linear, prove that x, y = T(x), T(y) defines an inner product on V if and only if T is one-to-one. 19.Let V be an inner product space. Prove that (a) x ± y2 = x2 ± 2 x, y + y2 for all x, y ∈ V, where x, y denotes the real part of the complex number x, y. (b) | x − y | ≤ x − y for all x, y ∈ V. 20.21.Let V be an inner product space over F . Prove the polar identities: For all x, y ∈ V, (a) x, y = 14 x + y2 − 14 x − y2 if F = R; ,4 (b) x, y = 14 k=1 ik x + ik y2 if F = C, where i2 = −1. Let A be an n × n matrix. Define 1 1 A1 = (A + A∗ ) and A2 = (A − A∗ ). 2 2i (a) Prove that A∗ 1 = A1 , A∗ 2 = A2 , and A = A1 + iA2 . Would it be reasonable to define A1 and A2 to be the real and imaginary parts, respectively, of the matrix A? (b) Let A be an n × n matrix. Prove that the representation in (a) is ∗unique. That is, prove that if A = B1 + iB2 , where B1 = B1 and B2 ∗ = B2 , then B1 = A1 and B2 = A2 . Sec. 6.1 Inner Products and Norms 339 22.Let V be a real or complex vector space (possibly infinite-dimensional), and let β be a basis for V. For x, y ∈ V there exist v1 , v2 , . . . , vn ∈ β such that n n x = ai vi and y = bi vi . i=1 i=1 Define n x, y = ai bi . i=1 (a) Prove that · , · is an inner product on V and that β is an or- thonormal basis for V. Thus every real or complex vector space may be regarded as an inner product space. (b) Prove that if V = Rn or V = Cn and β is the standard ordered basis, then the inner product defined above is the standard inner product. 23.Let V = Fn , and let A ∈ Mn×n (F ). (a) Prove that x, Ay = A∗ x, y for all x, y ∈ V. (b) Suppose that for some B ∈ Mn×n (F ), we have x, Ay = Bx, y for all x, y ∈ V. Prove that B = A∗ . (c) Let α be the standard ordered basis for V. For any orthonormal basis β for V, let Q be the n × n matrix whose columns are the vectors in β. Prove that Q∗ = Q−1 . (d) Define linear operators T and U on V by T(x) = Ax and U(x) = A∗ x. Show that [U]β = [T]∗ β for any orthonormal basis β for V. The following definition is used in Exercises 24–27. Definition. Let V be a vector space over F , where F is either R or C. Regardless of whether V is or is not an inner product space, we may still define a norm · as a real-valued function on V satisfying the following three conditions for all x, y ∈ V and a ∈ F : (1) x ≥ 0, and x = 0 if and only if x = 0 . (2) ax = |a|· x. (3) x + y ≤ x + y. 24. Prove that the following are norms on the given vector spaces V. (a) V = Mm×n (F ); A = max |Aij | for all A ∈ V i,j (b) V = C([0, 1]); f = max |f (t)| for all f ∈ V t∈[0,1] 340 1 (c) V = C([0, 1]); f = |f (t)| dt 0 (d) V = R2 ; (a, b) = max{|a|, |b|} Chap. 6 Inner Product Spaces for all f ∈ V for all (a, b) ∈ V 25. Use Exercise 20 to show that there is no inner product · , · on R2 such that x2 = x, x for all x ∈ R2 if the norm is defined as in Exercise 24(d). 26.Let · be a norm on a vector space V, and define, for each ordered pair of vectors, the scalar d(x, y) = x − y, called the distance between x and y. Prove the following results for all x, y, z ∈ V. (a) (b) (c) (d) (e) d(x, y) ≥ 0. d(x, y) = d(y, x). d(x, y) ≤ d(x, z) + d(z, y). d(x, x) = 0. d(x, y) = 0 if x = y. 27.Let · be a norm on a real vector space V satisfying the parallelogram law given in Exercise 11. Define 1 6 7 x, y = x + y2 − x − y2 . 4 Prove that · , · defines an inner product on V such that x2 = x, x for all x ∈ V. Hints: (a)(b)(c)(d)(e)(f )(g)Prove x, 2y = 2 x, y for all x, y ∈ V. Prove x + u, y = x, y + u, y for all x, u, y ∈ V. Prove nx, y = n x, y for every positive integer n and every x, y ∈ V. 8 9 Prove m m 1 x, y = x, y for every positive integer m and every x, y ∈ V. Prove rx, y = r x, y for every rational number r and every x, y ∈ V. Prove | x, y | ≤ xy for every x, y ∈ V. Hint: Condition (3) in the definition of norm can be helpful. Prove that for every c ∈ R, every rational number r, and every x, y ∈ V, |c x, y − cx, y | = |(c−r) x, y − (c−r)x, y | ≤ 2|c−r|xy. (h) Use the fact that for any c ∈ R, |c − r| can be made arbitrarily small, where r varies over the set of rational numbers, to establish item (b) of the definition of inner product. Sec. 6.2 Gram-Schmidt Orthogonalization Process 341 28.29.Let V be a complex inner product space with an inner product · , · . Let [ · , · ] be the real-valued function such that [x, y] is the real part of the complex number x, y for all x, y ∈ V. Prove that [· , · ] is an inner product for V, where V is regarded as a vector space over R. Prove, furthermore, that [x, ix] = 0 for all x ∈ V. Let V be a vector space over C, and suppose that [· , · ] is a real inner product on V, where V is regarded as a vector space over R, such that [x, ix] = 0 for all x ∈ V. Let · , · be the complex-valued function defined by x, y = [x, y] + i[x, iy] for x, y ∈ V. 30.Prove that · , · is a complex inner product on V. Let · be a norm (as defined in Exercise 24) on a complex vector space V satisfying the parallelogram law given in Exercise 11. Prove that there is an inner product · , · on V such that x2 = x, x for all x ∈ V. Hint: Apply Exercise 27 to V regarded as a vector space over R. Then apply Exercise 29. 1. Label the following statements as true or false. (a) The Gram–Schmidt orthogonalization process allows us to con- struct an orthonormal set from an arbitrary set of vectors. Sec. 6.2 Gram-Schmidt Orthogonalization Process 353 (b)(c)(d)(e)(f )(g)Every nonzero finite-dimensional inner product space has an or- thonormal basis. The orthogonal complement of any set is a subspace. If {v1 , v2 , . . . , vn } is a basis for an inner product space V, then for any x ∈ V the scalars x, vi are the Fourier coefficients of x. An orthonormal basis must be an ordered basis. Every orthogonal set is linearly independent. Every orthonormal set is linearly independent. 2.In each part, apply the Gram–Schmidt process to the given subset S of the inner product space V to obtain an orthogonal basis for span(S). Then normalize the vectors in this basis to obtain an orthonormal basis β for span(S), and compute the Fourier coefficients of the given vector relative to β. Finally, use Theorem 6.5 to verify your result. (a) V = R3 , S = {(1, 0, 1), (0, 1, 1), (1, 3, 3)}, and x = (1, 1, 2) (b) V = R3 , S = {(1, 1, 1), (0, 1, 1), (0, 0, 1)}, and x = (1, 0, 1) 1 (c) V = P2 (R) with the inner product f (x), g(x) = 0 f (t)g(t) dt, S = {1, x, x2 }, and h(x) = 1 + x (d) V = span(S), where S = {(1, i, 0), (1 − i, 2, 4i)}, and x = (3 + i, 4i, −4) (e) V = R4 , S = {(2, −1, −2, 4), (−2, 1, −5, 5), (−1, 3, 7, 11)}, and x = (−11, 8, −4, 18) (f ) V = R4 , S = {(1, −2, −1, 3), (3, 6, 3, −1), (1, 4, 2, 8)}, and x = (−1, 2, 1, 1) 3 5 −1 9 7 −17 (g) V = M2×2 (R), S = , , , and −1 1 5 −1 2 −6 −1 27 A = −4 8 2 2 11 4 4 −12 (h) V = M2×2 (R), S = , , , and A = 2 1 2 5 3 −16 8 6 25 −13 π (i) V = span(S) with the inner product f, g = f (t)g(t) dt, 0 S = {sin t, cos t, 1, t}, and h(t) = 2t + 1 (j) V = C4, S = {(1, i, 2 − i, −1), (2 + 3i, 3i, 1 − i, 2i), (−1+7i, 6+10i, 11−4i, 3+4i)}, and x = (−2+7i, 6+9i, 9−3i, 4+4i) (k) V = C4, S = {(−4, 3 − 2i, i, 1 − 4i), (−1−5i, 5−4i, −3+5i, 7−2i), (−27−i, −7−6i, −15+25i, −7−6i)}, and x = (−13 − 7i, −12 + 3i, −39 − 11i, −26 + 5i) 354 Chap. 6 Inner Product Spaces 1 − i −2 − 3i 8i 4 (l) V = M2×2 (C), S = , , 2 + 2i 4+ i −3 − 3i −4 + 4i −25 − 38i −2 − 13i −2 + 8i −13 + i , and A = 12 − 78i −7 + 24i 10 − 10i 9 − 9i −1 + i −i −1 − 7i −9 − 8i (m) V = M2×2 (C), S = , , 2 − i 1 + 3i 1 + 10i −6 − 2i −11 − 132i −34 − 31i −7 + 5i 3 + 18i , and A = 7 − 126i −71 − 5i 9 − 6i −3 + 7i 3. In R2 , let 1 1 1 −1 β = √ , √ , √ , √ . 2 2 2 2 4.5.Find the Fourier coefficients of (3, 4) relative to β. Let S = {(1, 0, i), (1, 2, 1)} in C3 . Compute S ⊥. Let S0 = {x0 }, where x0 is a nonzero vector in R3. Describe S0 ⊥ ge- ometrically. Now suppose that S = {x1 , x2 } is a linearly independent subset of R3 . Describe S ⊥ geometrically. 6. Let V be an inner product space, and let W be a finite-dimensional subspace of V. If x ∈ / W, prove that there exists y ∈ V such that y ∈ W⊥ , but x, y = 0. Hint: Use Theorem 6.6. 7.8.9.Let β be a basis for a subspace W of an inner product space V, and let z ∈ V. Prove that z ∈ W⊥ if and only if z, v = 0 for every v ∈ β. Prove that if {w1 , w2 , . . . , wn} is an orthogonal set of nonzero vectors, then the vectors v1 , v2 , . . . , vn derived from the Gram–Schmidt process satisfy vi = wi for i = 1, 2, . . . , n. Hint: Use mathematical induction. Let W = span({(i, 0, 1)}) in C3 . Find orthonormal bases for W and W⊥ . 10. Let W be a finite-dimensional subspace of an inner product space V. Prove that there exists a projection T on W along W⊥ that satisfies N(T) = W⊥ . In addition, prove that T(x) ≤ x for all x ∈ V. Hint: Use Theorem 6.6 and Exercise 10 of Section 6.1. (Projections are defined in the exercises of Section 2.1.) 11.12.Let A be an n × n matrix with complex entries. Prove that AA∗ = I if and only if the rows of A form an orthonormal basis for Cn . Prove that for any matrix A ∈ Mm×n (F ), (R(LA∗ ))⊥ = N(LA ). Sec. 6.2 Gram-Schmidt Orthogonalization Process 355 13.Let V be an inner product space, S and S0 be subsets of V, and W be a finite-dimensional subspace of V. Prove the following results. (a) S0 ⊆ S implies that S ⊥ ⊆ S0 ⊥ . (b) S ⊆ (S ⊥ )⊥ ; so span(S) ⊆ (S ⊥)⊥ . (c) W = (W⊥ )⊥ . Hint: Use Exercise 6. (d) V = W ⊕ W⊥ . (See the exercises of Section 1.3.) 14.Let W1 and W2 be subspaces of a finite-dimensional inner product space. Prove that (W1 +W2 )⊥ = W1 ⊥ ∩W2 ⊥ and (W1 ∩W2 )⊥ = W1 ⊥ +W2 ⊥ . (See the definition of the sum of subsets of a vector space on page 22.) Hint for the second equation: Apply Exercise 13(c) to the first equation. 15.Let V be a finite-dimensional inner product space over F . (a) Parseval’s Identity. Let {v1 , v2 , . . . , vn } be an orthonormal basis for V. For any x, y ∈ V prove that n x, y = x, vi y, vi . i=1 (b)Use (a) to prove that if β is an orthonormal basis for V with inner product · , · , then for any x, y ∈ V φβ (x), φβ (y) = [x]β , [y]β = x, y , where · , · is the standard inner product on Fn . 16.(a)Bessel’s Inequality. Let V be an inner product space, and let S = {v1, v2 , . . . , vn } be an orthonormal subset of V. Prove that for any x ∈ V we have n x2 ≥ | x, vi |2 . i=1 Hint: Apply Theorem 6.6 to x ∈ V and W = span(S). Then use Exercise 10 of Section 6.1. (b) In the context of (a), prove that Bessel’s inequality is an equality if and only if x ∈ span(S). 17.Let T be a linear operator on an inner product space V. If T(x), y = 0 for all x, y ∈ V, prove that T = T0 . In fact, prove this result if the equality holds for all x and y in some basis for V. 18.Let V = C([−1, 1]). Suppose that We and Wo denote the subspaces of V consisting of the even and odd functions, respectively. (See Exercise 22 356 Chap. 6 Inner Product Spaces ⊥of Section 1.3.) Prove that We = Wo , where the inner product on V is defined by 1 f, g = f (t)g(t) dt. −1 19.In each of the following parts, find the orthogonal projection of the given vector on the given subspace W of the inner product space V. (a) V = R2 , u = (2, 6), and W = {(x, y) : y = 4x}. (b) V = R3 , u = (2, 1, 3), and W = {(x, y, z) : x + 3y − 2z = 0}. 1 (c) V = P(R) with the inner product f (x), g(x) = 0 f (t)g(t) dt, h(x) = 4 + 3x − 2x2 , and W = P1 (R). 20. In each part of Exercise 19, find the distance from the given vector to the subspace W. 1 21. Let V = C([−1, 1]) with the inner product f, g = −1 f (t)g(t) dt, and let W be the subspace P2 (R), viewed as a space of functions. Use the orthonormal basis obtained in Example 5 to compute the “best” (closest) second-degree polynomial approximation of the function h(t) = et on the interval [−1, 1]. 1 22. Let V = C([0, 1]) with the inner product f, g = 0 f (t)g(t) dt. Let W √ be the subspace spanned by the linearly independent set {t, t}. (a) Find an orthonormal basis for W. (b) Let h(t) = t2 . Use the orthonormal basis obtained in (a) to obtain the “best” (closest) approximation of h in W. 23. Let V be the vector space defined in Example 5 of Section 1.2, the space of all sequences σ in F (where F = R or F = C) such that σ(n) = 0 for only finitely many positive integers n. For σ, μ ∈ V, we ∞ define σ, μ = σ(n)μ(n). Since all but a finite number of terms of n=1 the series are zero, the series converges. (a)(b)(c)Prove that · , · is an inner product on V, and hence V is an inner product space. For each positive integer n, let en be the sequence defined by en (k) = δn,k , where δn,k is the Kronecker delta. Prove that {e1 , e2, . . .} is an orthonormal basis for V. Let σn = e1 + en and W = span({σn : n ≥ 2}. (i) Prove that e1 ∈ / W, so W = V. (ii) Prove that W ⊥ = {0 }, and conclude that W = (W⊥ )⊥ . Sec. 6.3 The Adjoint of a Linear Operator 357 Thus the assumption in Exercise 13(c) that W is finite-dimensional is essential. 1.Label the following statements as true or false. Assume that the under- lying inner product spaces are finite-dimensional. (a)(b)(c)(d)(e)Every linear operator has an adjoint. Every linear operator on V has the form x → x, y for some y ∈ V. For every linear operator T on V and every ordered basis β for V, we have [T∗ ]β = ([T]β )∗ . The adjoint of a linear operator is unique. For any linear operators T and U and scalars a and b, (aT + bU)∗ = aT∗ + bU∗ . (f )(g)For any n × n matrix A, we have (LA )∗ = LA∗ . For any linear operator T, we have (T∗ )∗ = T. 2.For each of the following inner product spaces V (over F ) and linear transformations g : V → F , find a vector y such that g(x) = x, y for all x ∈ V. 366 Chap. 6 Inner Product Spaces 3.(a) V = R3 , g(a1 , a2 , a3 ) = a1 − 2a2 + 4a3 (b) V = C2 , g(z1 , z2 ) = z1 − 2z2 1 (c) V = P2 (R) with f, h = f (t)h(t) dt, g(f ) = f (0) + f (1) 0 For each of the following inner product spaces V and linear operators T on V, evaluate T∗ at the given vector in V. (a) V = R2 , T(a, b) = (2a + b, a − 3b), x = (3, 5). (b) V = C2 , T(z1 , z2 ) = (2z1 + iz2 , (1 − i)z1 ), x = (3 − i, 1 + 2i). 1 (c) V = P1 (R) with f, g = f (t)g(t) dt, T(f ) = f + 3f , −1 f (t) = 4 − 2t 4. Complete the proof of Theorem 6.11. 5. (a) Complete the proof of the corollary to Theorem 6.11 by using Theorem 6.11, as in the proof of (c). (b) State a result for nonsquare matrices that is analogous to the corol- lary to Theorem 6.11, and prove it using a matrix argument. 6.Let T be a linear operator on an inner product space V. Let U1 = T+T∗ and U2 = TT∗ . Prove that U1 = U∗ 1 and U2 = U∗ 2 . 7.Give an example of a linear operator T on an inner product space V such that N(T) = N(T∗ ). 8.9.Let V be a finite-dimensional inner product space, and let T be a linear operator on V. Prove that if T is invertible, then T∗ is invertible and (T∗ )−1 = (T−1 )∗ . Prove that if V = W ⊕ W⊥ and T is the projection on W along W⊥ , then T = T∗ . Hint: Recall that N(T) = W⊥ . (For definitions, see the exercises of Sections 1.3 and 2.1.) 10.11.Let T be a linear operator on an inner product space V. Prove that T(x) = x for all x ∈ V if and only if T(x), T(y) = x, y for all x, y ∈ V. Hint: Use Exercise 20 of Section 6.1. For a linear operator T on an inner product space V, prove that T∗ T = T0 implies T = T0 . Is the same result true if we assume that TT∗ = T0 ? 12.Let V be an inner product space, and let T be a linear operator on V. Prove the following results. (a) R(T∗ )⊥ = N(T). (b) If V is finite-dimensional, then R(T∗ ) = N(T)⊥ . Hint: Use Exer- cise 13(c) of Section 6.2. Sec. 6.3 The Adjoint of a Linear Operator 367 13.Let T be a linear operator on a finite-dimensional vector space V. Prove the following results. (a) N(T∗ T) = N(T). Deduce that rank(T∗ T) = rank(T). (b) rank(T) = rank(T∗ ). Deduce from (a) that rank(TT∗ ) = rank(T). (c) For any n × n matrix A, rank(A∗ A) = rank(AA∗ ) = rank(A). 14.Let V be an inner product space, and let y, z ∈ V. Define T : V → V by T(x) = x, yz for all x ∈ V. First prove that T is linear. Then show that T∗ exists, and find an explicit expression for it. The following definition is used in Exercises 15–17 and is an extension of the definition of the adjoint of a linear operator. Definition. Let T : V → W be a linear transformation, where V and W are finite-dimensional inner product spaces with inner products · , · 1 and · , · 2 , respectively. A function T∗ : W → V is called an adjoint of T if T(x), y2 = x, T ∗ (y)1 for all x ∈ V and y ∈ W. 15.Let T : V → W be a linear transformation, where V and W are finite- dimensional inner product spaces with inner products · , · 1 and · , · 2 , respectively. Prove the following results. (a)(b)(c)(d)(e)There is a unique adjoint T∗ of T, and T∗ is linear. If β and γ are orthonormal bases for V and W, respectively, then [T∗ ]βγ = ([T]γβ )∗ . rank(T∗ ) = rank(T). T∗ (x), y1 = x, T(y)2 for all x ∈ W and y ∈ V. For all x ∈ V, T∗ T(x) = 0 if and only if T(x) = 0 . 16. State and prove a result that extends the first four parts of Theorem 6.11 using the preceding definition. 17. Let T : V → W be a linear transformation, where V and W are finite- dimensional inner product spaces. Prove that (R(T∗ ))⊥ = N(T), using the preceding definition. 18. † Let A be an n × n matrix. Prove that det(A∗ ) = det(A). 19.Suppose that A is an m×n matrix in which no two columns are identical. Prove that A∗ A is a diagonal matrix if and only if every pair of columns of A is orthogonal. 20.For each of the sets of data that follows, use the least squares approx- imation to find the best fits with both (i) a linear function and (ii) a quadratic function. Compute the error E in both cases. (a) {(−3, 9), (−2, 6), (0, 2), (1, 1)} 368 Chap. 6 Inner Product Spaces (b)(c){(1, 2), (3, 4), (5, 7), (7, 9), (9, 12)} {(−2, 4), (−1, 3), (0, 1), (1, −1), (2, −3)} 21. In physics, Hooke’s law states that (within certain limits) there is a linear relationship between the length x of a spring and the force y applied to (or exerted by) the spring. That is, y = cx + d, where c is called the spring constant. Use the following data to estimate the spring constant (the length is given in inches and the force is given in pounds). Length x 3.5 4.0 4.5 5.0 Force y 1.0 2.2 2.8 4.3 22.Find the minimal solution to each of the following systems of linear equations. x + 2y − z = 1 (a) x + 2y − z = 12 (b) 2x + 3y + z = 2 4x + 7y − z = 4 x + y − z =0 x + y + z − w =1 (c) 2x − y + z = 3 (d) 2x − y + w =1 x − y + z =2 23. Consider the problem of finding the least squares line y = ct + d corre- sponding to the m observations (t1 , y1 ), (t2 , y2 ), . . . , (tm , ym ). (a) Show that the equation (A∗ A)x0 = A∗ y of Theorem 6.12 takes the form of the normal equations: m m m t2 i c + ti d = tiyi i=1 i=1 i=1 and m m ti c + md = yi . i=1 i=1 These equations may also be obtained from the error E by setting the partial derivatives of E with respect to both c and d equal to zero. Sec. 6.4 Normal and Self-Adjoint Operators 369 (b) Use the second normal equation of (a) to show that the least squares line must pass through the center of mass, (t, y), where 1 m 1 m t = ti and y = yi . m mi=1 i=1 24. Let V and {e1 , e2 , . . .} be defined as in Exercise 23 of Section 6.2. Define T : V → V by ∞ T(σ)(k) = σ(i) for every positive integer k. i=k Notice that the infinite series in the definition of T converges because σ(i) = 0 for only finitely many i. (a) Prove that T is a linear operator on V. ,n (b) Prove that for any positive integer n, T(en ) = i=1 ei . (c) Prove that T has no adjoint. Hint: By way of contradiction, suppose that T∗ exists. Prove that for any positive integer n, T∗ (en )(k) = 0 for infinitely many k. 1.Label the following statements as true or false. Assume that the under- lying inner product spaces are finite-dimensional. (a)(b)(c)(d)(e)Every self-adjoint operator is normal. Operators and their adjoints have the same eigenvectors. If T is an operator on an inner product space V, then T is normal if and only if [T]β is normal, where β is any ordered basis for V. A real or complex matrix A is normal if and only if LA is normal. The eigenvalues of a self-adjoint operator must all be real. Sec. 6.4 Normal and Self-Adjoint Operators 375 (f )(g)(h)The identity and zero operators are self-adjoint. Every normal operator is diagonalizable. Every self-adjoint operator is diagonalizable. 2.For each linear operator T on an inner product space V, determine whether T is normal, self-adjoint, or neither. If possible, produce an orthonormal basis of eigenvectors of T for V and list the corresponding eigenvalues. (a) (b) (c) (d) V = R2 and T is defined by T(a, b) = (2a − 2b, −2a + 5b). V = R3 and T is defined by T(a, b, c) = (−a + b, 5b, 4a − 2b + 5c). V = C2 and T is defined by T(a, b) = (2a + ib, a + 2b). V = P2(R) and T is defined by T(f ) = f , where 1 f, g = f (t)g(t) dt. 0 (e) V = M2×2 (R) and T is defined by T(A) = At . a b c d (f ) V = M2×2 (R) and T is defined by T = . c d a b 3.Give an example of a linear operator T on R2 and an ordered basis for R2 that provides a counterexample to the statement in Exercise 1(c). 4.5.Let T and U be self-adjoint operators on an inner product space V. Prove that TU is self-adjoint if and only if TU = UT. Prove (b) of Theorem 6.15. 6.Let V be a complex inner product space, and let T be a linear operator on V. Define 1 1 T1 = (T + T∗ ) and T2 = (T − T∗ ). 2 2i (a)(b)(c)Prove that T1 and T2 are self-adjoint and that T = T1 + i T2 . Suppose also that T = U1 + iU2 , where U1 and U2 are self-adjoint. Prove that U1 = T1 and U2 = T2 . Prove that T is normal if and only if T1 T2 = T2 T1 . 7.Let T be a linear operator on an inner product space V, and let W be a T-invariant subspace of V. Prove the following results. (a) If T is self-adjoint, then TW is self-adjoint. (b) W⊥ is T∗ -invariant. (c) If W is both T- and T∗ -invariant, then (TW )∗ = (T∗ )W . (d) If W is both T- and T∗ -invariant and T is normal, then TW is normal. 376 Chap. 6 Inner Product Spaces 8. Let T be a normal operator on a finite-dimensional complex inner product space V, and let W be a subspace of V. Prove that if W is T-invariant, then W is also T∗ -invariant. Hint: Use Exercise 24 of Sec- tion 5.4. 9.Let T be a normal operator on a finite-dimensional inner product space V. Prove that N(T) = N(T∗ ) and R(T) = R(T∗). Hint: Use Theo- rem 6.15 and Exercise 12 of Section 6.3. 10.Let T be a self-adjoint operator on a finite-dimensional inner product space V. Prove that for all x ∈ V T(x) ± ix2 = T(x)2 + x2 . Deduce that T − iI is invertible and that [(T − iI)−1 ]∗ = (T + iI)−1 . 11.Assume that T is a linear operator on a complex (not necessarily finite- dimensional) inner product space V with an adjoint T∗ . Prove the following results. (a) If T is self-adjoint, then T(x), x is real for all x ∈ V. (b) If T satisfies T(x), x = 0 for all x ∈ V, then T = T0 . Hint: Replace x by x + y and then by x + iy, and expand the resulting inner products. (c) If T(x), x is real for all x ∈ V, then T = T∗ . 12. Let T be a normal operator on a finite-dimensional real inner product space V whose characteristic polynomial splits. Prove that V has an orthonormal basis of eigenvectors of T. Hence prove that T is self- adjoint. 13.An n × n real matrix A is said to be a Gramian matrix if there exists a real (square) matrix B such that A = B t B. Prove that A is a Gramian matrix if and only if A is symmetric and all of its eigenvalues are non- negative. Hint: Apply Theorem 6.17 to T = LA to obtain an orthonor- mal basis {v1 , v2 , . . . , vn } of eigenvectors with the associated eigenvalues √λ1 , λ2 , . . . , λn. Define the linear operator U by U(vi ) = λi vi . 14.Simultaneous Diagonalization. Let V be a finite-dimensional real inner product space, and let U and T be self-adjoint linear operators on V such that UT = TU. Prove that there exists an orthonormal basis for V consisting of vectors that are eigenvectors of both U and T. (The complex version of this result appears as Exercise 10 of Section 6.6.) Hint: For any eigenspace W = Eλ of T, we have that W is both T- and U-invariant. By Exercise 7, we have that W⊥ is both T- and U-invariant. Apply Theorem 6.17 and Theorem 6.6 (p. 350). Sec. 6.4 Normal and Self-Adjoint Operators 377 15.Let A and B be symmetric n × n matrices such that AB = BA. Use Exercise 14 to prove that there exists an orthogonal matrix P such that P t AP and P t BP are both diagonal matrices. 16.Prove the Cayley–Hamilton theorem for a complex n×n matrix A. That is, if f (t) is the characteristic polynomial of A, prove that f (A) = O. Hint: Use Schur’s theorem to show that A may be assumed to be upper triangular, in which case n f (t) = (Aii − t). i=1 Now if T = LA , we have (Ajj I − T)(ej ) ∈ span({e1 , e2 , . . . , ej−1 }) for j ≥ 2, where {e1 , e2 , . . . , en } is the standard ordered basis for Cn . (The general case is proved in Section 5.4.) The following definitions are used in Exercises 17 through 23. Definitions. A linear operator T on a finite-dimensional inner product space is called positive definite [positive semidefinite] if T is self-adjoint and T(x), x > 0 [T(x), x ≥ 0] for all x = 0 . An n × n matrix A with entries from R or C is called positive definite [positive semidefinite] if LA is positive definite [positive semidefinite]. 17.Let T and U be a self-adjoint linear operators on an n-dimensional inner product space V, and let A = [T]β , where β is an orthonormal basis for V. Prove the following results. (a)(b)T is positive definite [semidefinite] if and only if all of its eigenval- ues are positive [nonnegative]. T is positive definite if and only if Aij aj ai > 0 for all nonzero n-tuples (a1 , a 2 , . . . , an). i,j (c)(d)(e)(f )T is positive semidefinite if and only if A = B ∗ B for some square matrix B. If T and U are positive semidefinite operators such that T2 = U2 , then T = U. If T and U are positive definite operators such that TU = UT, then TU is positive definite. T is positive definite [semidefinite] if and only if A is positive def- inite [semidefinite]. Because of (f), results analogous to items (a) through (d) hold for ma- trices as well as operators. 378 Chap. 6 Inner Product Spaces 18. Let T : V → W be a linear transformation, where V and W are finite- dimensional inner product spaces. Prove the following results. (a) T∗ T and TT∗ are positive semidefinite. (See Exercise 15 of Sec- tion 6.3.) (b) rank(T∗ T) = rank(TT∗ ) = rank(T). 19.Let T and U be positive definite operators on an inner product space V. Prove the following results. (a)(b)(c)T + U is positive definite. If c > 0, then cT is positive definite. T−1 is positive definite. 20.Let V be an inner product space with inner product · , · , and let T be a positive definite linear operator on V. Prove that x, y = T(x), y defines another inner product on V. 21.Let V be a finite-dimensional inner product space, and let T and U be self-adjoint operators on V such that T is positive definite. Prove that both TU and UT are diagonalizable linear operators that have only real eigenvalues. Hint: Show that UT is self-adjoint with respect to the inner product x, y = T(x), y. To show that TU is self-adjoint, repeat the argument with T−1 in place of T. 22.This exercise provides a converse to Exercise 20. Let V be a finite- dimensional inner product space with inner product · , · , and let · , · be any other inner product on V. (a) Prove that there exists a unique linear operator T on V such that x, y = T(x), y for all x and y in V. Hint: Let β = {v1 , v2 , . . . , vn } be an orthonormal basis for V with respect to · , · , and define a matrix A by Aij = vj , vi for all i and j. Let T be the unique linear operator on V such that [T]β = A. (b) Prove that the operator T of (a) is positive definite with respect to both inner products. 23.Let U be a diagonalizable linear operator on a finite-dimensional inner product space V such that all of the eigenvalues of U are real. Prove that there exist positive definite linear operators T1 and T 1 and self-adjoint linear operators T2 and T 2 such that U = T2 T1 = T 1 T 2 . Hint: Let · , · be the inner product associated with V, β a basis of eigenvectors for U, · , · the inner product on V with respect to which β is orthonormal (see Exercise 22(a) of Section 6.1), and T1 the positive definite operator according to Exercise 22. Show that U is self-adjoint with respect to · , · and U = T−1 1 U∗ T1 (the adjoint is with respect to · , · ). Let = T1−1 U∗ . T2Sec. 6.5 Unitary and Orthogonal Operators and Their Matrices 379 24.This argument gives another proof of Schur’s theorem. Let T be a linear operator on a finite dimensional inner product space V. (a)(b)Suppose that β is an ordered basis for V such that [T]β is an upper triangular matrix. Let γ be the orthonormal basis for V obtained by applying the Gram–Schmidt orthogonalization process to β and then normalizing the resulting vectors. Prove that [T]γ is an upper triangular matrix. Use Exercise 32 of Section 5.4 and (a) to obtain an alternate proof of Schur’s theorem. 1.Label the following statements as true or false. Assume that the under- lying inner product spaces are finite-dimensional. (a) Every unitary operator is normal. (b) Every orthogonal operator is diagonalizable. (c) A matrix is unitary if and only if it is invertible. (d) If two matrices are unitarily equivalent, then they are also similar. (e) The sum of unitary matrices is unitary. (f ) The adjoint of a unitary operator is unitary. (g) If T is an orthogonal operator on V, then [T]β is an orthogonal matrix for any ordered basis β for V. (h) If all the eigenvalues of a linear operator are 1, then the operator must be unitary or orthogonal. (i) A linear operator may preserve the norm, but not the inner prod- uct. 2. For each of the following matrices A, find an orthogonal or unitary matrix P and a diagonal matrix D such that P ∗ AP = D. 1 2 0 −1 2 3 − 3i (a) (b) (c) 2 1 1 0 3 + 3i 5 ⎛ ⎞ ⎛ ⎞ 0 2 2 2 1 1 (d) ⎝2 0 2⎠ (e) ⎝1 2 1⎠ 2 2 0 1 1 2 3. Prove that the composite of unitary [orthogonal] operators is unitary [orthogonal]. Sec. 6.5 Unitary and Orthogonal Operators and Their Matrices 393 4.For z ∈ C, define Tz : C → C by Tz (u) = zu. Characterize those z for which Tz is normal, self-adjoint, or unitary. 5. Which of the following pairs of matrices are unitarily equivalent? ⎛ ⎞ 1 0 0 1 0 1 0 12 (a) and (b) and ⎝ 1 ⎠ 0 1 1 0 1 0 0 2 ⎛ ⎞ ⎛ ⎞ 0 1 0 2 0 0 (c) ⎝−1 0 0⎠ and ⎝0 −1 0⎠ 0 0 1 0 0 0 ⎛ ⎞ ⎛ ⎞ 0 1 0 1 0 0 (d) ⎝−1 0 0⎠ and ⎝0 i 0⎠ 0 0 1 0 0 −i ⎛ ⎞ ⎛ ⎞ 1 1 0 1 0 0 (e) ⎝0 2 2⎠ and ⎝0 2 0⎠ 0 0 3 0 0 3 6.Let V be the inner product space of complex-valued continuous func- tions on [0, 1] with the inner product 1 f, g = f (t)g(t) dt. 0 Let h ∈ V, and define T : V → V by T(f ) = hf . Prove that T is a unitary operator if and only if |h(t)| = 1 for 0 ≤ t ≤ 1. 7.Prove that if T is a unitary operator on a finite-dimensional inner prod- uct space V, then T has a unitary square root; that is, there exists a unitary operator U such that T = U2 . 8.Let T be a self-adjoint linear operator on a finite-dimensional inner product space. Prove that (T + iI)(T − iI)−1 is unitary using Exercise 10 of Section 6.4. 9.Let U be a linear operator on a finite-dimensional inner product space V. If U(x) = x for all x in some orthonormal basis for V, must U be unitary? Justify your answer with a proof or a counterexample. 10. Let A be an n × n real symmetric or complex normal matrix. Prove that n n tr(A) = λi and tr(A∗ A) = |λi |2 , i=1 i=1 where the λi ’s are the (not necessarily distinct) eigenvalues of A. 394 11.12.13.14.Chap. 6 Inner Product Spaces Find an orthogonal matrix whose first row is ( 13 , 23 , 23 ). Let A be an n × n real symmetric or complex normal matrix. Prove that n det(A) = λi , i=1 where the λi ’s are the (not necessarily distinct) eigenvalues of A. Suppose that A and B are diagonalizable matrices. Prove or disprove that A is similar to B if and only if A and B are unitarily equivalent. Prove that if A and B are unitarily equivalent matrices, then A is pos- itive definite [semidefinite] if and only if B is positive definite [semidef- inite]. (See the definitions in the exercises in Section 6.4.) 15.Let U be a unitary operator on an inner product space V, and let W be a finite-dimensional U-invariant subspace of V. Prove that (a) U(W) = W; (b) W ⊥ is U-invariant. Contrast (b) with Exercise 16. 16.Find an example of a unitary operator U on an inner product space and a U-invariant subspace W such that W⊥ is not U-invariant. 17.Prove that a matrix that is both unitary and upper triangular must be a diagonal matrix. 18. Show that “is unitarily equivalent to” is an equivalence relation on Mn×n (C). 19.Let W be a finite-dimensional subspace of an inner product space V. By Theorem 6.7 (p. 352) and the exercises of Section 1.3, V = W ⊕ W⊥ . Define U : V → V by U(v1 + v2 ) = v1 − v2 , where v1 ∈ W and v2 ∈ W⊥ . Prove that U is a self-adjoint unitary operator. 20.Let V be a finite-dimensional inner product space. A linear operator U on V is called a partial isometry if there exists a subspace W of V such that U(x) = x for all x ∈ W and U(x) = 0 for all x ∈ W⊥ . Observe that W need not be U-invariant. Suppose that U is such an operator and {v1 , v2 , . . . , vk } is an orthonormal basis for W. Prove the following results. (a)(b)U(x), U(y) = x, y for all x, y ∈ W. Hint: Use Exercise 20 of Section 6.1. {U(v1 ), U(v2 ), . . . , U(vk )} is an orthonormal basis for R(U). Sec. 6.5 Unitary and Orthogonal Operators and Their Matrices 395 (c)(d)(e)(f )There exists an orthonormal basis γ for V such that the first k columns of [U]γ form an orthonormal set and the remaining columns are zero. Let {w1 , w2 , . . . , wj } be an orthonormal basis for R(U)⊥ and β = {U(v1 ), U(v2 ), . . . , U(vk ), w1 , . . . , wj }. Then β is an orthonormal basis for V. Let T be the linear operator on V that satisfies T(U(vi )) = vi (1 ≤ i ≤ k) and T(wi ) = 0 (1 ≤ i ≤ j). Then T is well defined, and T = U∗ . Hint: Show that U(x), y = x, T(y) for all x, y ∈ β. There are four cases. U∗ is a partial isometry. This exercise is continued in Exercise 9 of Section 6.6. 21.Let A and B be n × n matrices that are unitarily equivalent. (a) Prove that tr(A∗ A) = tr(B ∗B). (b) Use (a) to prove that n n |Aij |2 = |Bij |2 . i,j=1 i,j=1 (c) Use (b) to show that the matrices 1 2 and 2 i i14 1 are not unitarily equivalent. 22.Let V be a real inner product space. (a) Prove that any translation on V is a rigid motion. (b) Prove that the composite of any two rigid motions on V is a rigid motion on V. 23.Prove the following variation of Theorem 6.22: If f : V → V is a rigid motion on a finite-dimensional real inner product space V, then there exists a unique orthogonal operator T on V and a unique translation g on V such that f = T ◦ g. 24. Let T and U be orthogonal operators on R2 . Use Theorem 6.23 to prove the following results. (a) If T and U are both reflections about lines through the origin, then UT is a rotation. (b) If T is a rotation and U is a reflection about a line through the origin, then both UT and TU are reflections about lines through the origin. 396 Chap. 6 Inner Product Spaces 25. Suppose that T and U are reflections of R2 about the respective lines L and L through the origin and that φ and ψ are the angles from the positive x-axis to L and L , respectively. By Exercise 24, UT is a rotation. Find its angle of rotation. 26.27.Suppose that T and U are orthogonal operators on R2 such that T is the rotation by the angle φ and U is the reflection about the line L through the origin. Let ψ be the angle from the positive x-axis to L. By Exercise 24, both UT and TU are reflections about lines L1 and L2 , respectively, through the origin. (a)(b)Find the angle θ from the positive x-axis to L1 . Find the angle θ from the positive x-axis to L2 . Find new coordinates x , y so that the following quadratic forms can be written as λ1 (x )2 + λ2 (y )2 . (a) (b) (c) (d) (e) x2 + 4xy + y2 2x2 + 2xy + 2y 2 x2 − 12xy − 4y 2 3x2 + 2xy + 3y 2 x2 − 2xy + y2 28.Consider the expression X t AX, where X t = (x, y, z) and A is as defined in Exercise 2(e). Find a change of coordinates x , y , z so that the preceding expression is of the form λ1 (x )2 + λ2 (y )2 + λ3 (z )2 . 29.QR-Factorization. Let w1 , w2 , . . . , wn be linearly independent vectors in Fn , and let v1 , v2 , . . . , vn be the orthogonal vectors obtained from w1 , w2 , . . . , wn by the Gram–Schmidt process. Let u1 , u2, . . . , un be the orthonormal basis obtained by normalizing the vi ’s. (a) Solving (1) in Section 6.2 for wk in terms of uk , show that k−1 wk = vk uk + wk , uj uj j=1 (1 ≤ k ≤ n). (b)Let A and Q denote the n × n matrices in which the kth columns are wk and uk , respectively. Define R ∈ Mn×n (F ) by ⎧ ⎪ ⎨vj if j = k Rjk = wk , uj if j < k ⎪ ⎩ 0 if j > k. (c)Prove A = QR. Compute Q and R as in (b) for the 3×3 matrix whose columns are the vectors w1 , w2 , w3 , respectively, in Example 4 of Section 6.2. Sec. 6.5 Unitary and Orthogonal Operators and Their Matrices 397 (d)(e)Since Q is unitary [orthogonal] and R is upper triangular in (b), we have shown that every invertible matrix is the product of a uni- tary [orthogonal] matrix and an upper triangular matrix. Suppose that A ∈ Mn×n (F ) is invertible and A = Q1 R1 = Q2 R2 , where Q1 , Q2 ∈ Mn×n (F ) are unitary and R1 , R2 ∈ Mn×n (F ) are upper −1triangular. Prove that D = R2 R1 is a unitary diagonal matrix. Hint: Use Exercise 17. The QR factorization described in (b) provides an orthogonaliza- tion method for solving a linear system Ax = b when A is in- vertible. Decompose A to QR, by the Gram–Schmidt process or other means, where Q is unitary and R is upper triangular. Then QRx = b, and hence Rx = Q∗ b. This last system can be easily solved since R is upper triangular. 1 Use the orthogonalization method and (c) to solve the system x1 + 2x2 + 2x3 = 1 x1 + 2x3 = 11 x2 + x3 = −1. 30.Suppose that β and γ are ordered bases for an n-dimensional real [com- plex] inner product space V. Prove that if Q is an orthogonal [unitary] n × n matrix that changes γ-coordinates into β-coordinates, then β is orthonormal if and only if γ is orthonormal. The following definition is used in Exercises 31 and 32. Definition. Let V be a finite-dimensional complex [real] inner product space, and let u be a unit vector in V. Define the Householder operator Hu : V → V by Hu (x) = x − 2 x, u u for all x ∈ V. 31.Let Hu be a Householder operator on a finite-dimensional inner product space V. Prove the following results. (a) Hu is linear. (b) Hu (x) = x if and only if x is orthogonal to u. (c) Hu (u) = −u. (d) H∗ u = Hu and H2 u = I, and hence Hu is a unitary [orthogonal] operator on V. (Note: If V is a real inner product space, then in the language of Sec- tion 6.11, Hu is a reflection.) 1 At one time, because of its great stability, this method for solving large sys- tems of linear equations with a computer was being advocated as a better method than Gaussian elimination even though it requires about three times as much work. (Later, however, J. H. Wilkinson showed that if Gaussian elimination is done “prop- erly,” then it is nearly as stable as the orthogonalization method.) 398 Chap. 6 Inner Product Spaces 32.Let V be a finite-dimensional inner product space over F . Let x and y be linearly independent vectors in V such that x = y. (a) If F = C, prove that there exists a unit vector u in V and a complex number θ with |θ| = 1 such that Hu (x) = θy. Hint: Choose θ so 1 that x, θy is real, and set u = (x − θy). x − θy (b) If F = R, prove that there exists a unit vector u in V such that Hu (x) = y. 1.Label the following statements as true or false. Assume that the under- lying inner product spaces are finite-dimensional. (a) All projections are self-adjoint. (b) An orthogonal projection is uniquely determined by its range. (c) Every self-adjoint operator is a linear combination of orthogonal projections. 404 Chap. 6 Inner Product Spaces (d) If T is a projection on W, then T(x) is the vector in W that is closest to x. (e) Every orthogonal projection is a unitary operator. 2.Let V = R2 , W = span({(1, 2)}), and β be the standard ordered basis for V. Compute [T]β , where T is the orthogonal projection of V on W. Do the same for V = R3 and W = span({(1, 0, 1)}). 3. For each of the matrices A in Exercise 2 of Section 6.5: (1) Verify that LA possesses a spectral decomposition. (2) For each eigenvalue of LA , explicitly define the orthogonal projec- tion on the corresponding eigenspace. (3) Verify your results using the spectral theorem. 4.Let W be a finite-dimensional subspace of an inner product space V. Show that if T is the orthogonal projection of V on W, then I − T is the orthogonal projection of V on W ⊥. 5.Let T be a linear operator on a finite-dimensional inner product space V. (a) If T is an orthogonal projection, prove that T(x) ≤ x for all x ∈ V. Give an example of a projection for which this inequality does not hold. What can be concluded about a projection for which the inequality is actually an equality for all x ∈ V? (b) Suppose that T is a projection such that T(x) ≤ x for x ∈ V. Prove that T is an orthogonal projection. 6.Let T be a normal operator on a finite-dimensional inner product space. Prove that if T is a projection, then T is also an orthogonal projection. 7.Let T be a normal operator on a finite-dimensional complex inner prod- uct space V. Use the spectral decomposition λ1 T1 + λ2 T2 + · · · + λk Tk of T to prove the following results. (a) If g is a polynomial, then k g(T) = g(λi )Ti . i=1 (b)(c)(d)(e)(f )If Tn = T0 for some n, then T = T0 . Let U be a linear operator on V. Then U commutes with T if and only if U commutes with each Ti . There exists a normal operator U on V such that U2 = T. T is invertible if and only if λi = 0 for 1 ≤ i ≤ k. T is a projection if and only if every eigenvalue of T is 1 or 0. Sec. 6.7 The Singular Value Decomposition and the Pseudoinverse 405 (g) T = −T∗ if and only if every λi is an imaginary number. 8. Use Corollary 1 of the spectral theorem to show that if T is a normal operator on a complex finite-dimensional inner product space and U is a linear operator that commutes with T, then U commutes with T∗ . 9.Referring to Exercise 20 of Section 6.5, prove the following facts about a partial isometry U. (a) U∗ U is an orthogonal projection on W. (b) UU∗ U = U. 10.Simultaneous diagonalization. Let U and T be normal operators on a finite-dimensional complex inner product space V such that TU = UT. Prove that there exists an orthonormal basis for V consisting of vectors that are eigenvectors of both T and U. Hint: Use the hint of Exercise 14 of Section 6.4 along with Exercise 8. 11. Prove (c) of the spectral theorem. 1. Label the following statements as true or false. (a) The singular values of any linear operator on a finite-dimensional vector space are also eigenvalues of the operator. (b) The singular values of any matrix A are the eigenvalues of A∗ A. (c) For any matrix A and any scalar c, if σ is a singular value of A, then |c|σ is a singular value of cA. (d) The singular values of any linear operator are nonnegative. (e) If λ is an eigenvalue of a self-adjoint matrix A, then λ is a singular value of A. (f ) For any m×n matrix A and any b ∈ Fn , the vector A† b is a solution to Ax = b. (g) The pseudoinverse of any linear operator exists even if the operator is not invertible. 2. Let T : V → W be a linear transformation of rank r, where V and W are finite-dimensional inner product spaces. In each of the following, find orthonormal bases {v1 , v2 , . . . , vn } for V and {u1 , u2 , . . . , um } for W, and the nonzero singular values σ1 ≥ σ2 ≥ · · · ≥ σr of T such that T(vi ) = σi ui for 1 ≤ i ≤ r. (a) T : R2 → R3 defined by T(x1 , x2 ) = (x1 , x1 + x2 , x1 − x2 ) (b) T : P2 (R) → P1 (R), where T(f (x)) = f (x), and the inner prod- ucts are defined as in Example 1 (c) Let V = W = span({1, sin x, cos x}) with the inner product defined 2π by f, g = 0 f (t)g(t) dt, and T is defined by T(f ) = f + 2f (d) T : C2 → C2 defined by T(z1 , z2 ) = ((1 − i)z2, (1 + i)z1 + z2 ) 3. Find a singular value decomposition for each of the following matrices. ⎛ ⎞ ⎛ ⎞ 1 1 1 1 1 0 1 ⎜0 1⎟ (a) ⎝ 1 1⎠ (b) (c) ⎜ ⎟ 1 0 −1 ⎝1 0⎠ −1 −1 1 1 ⎛ ⎞ ⎛ ⎞ 1 1 1 1 1 1 1 (d) ⎝1 −1 0⎠ (e) 1+ i 1 (f ) ⎝1 0 −2 1⎠ 1 − i −i 1 0 −1 1 −1 1 1 4. Find a polar decomposition for each of the following matrices. ⎛ ⎞ 20 4 0 (a) 1 1 (b) ⎝ 0 0 1⎠ 2 −2 4 20 0 5. Find an explicit formula for each of the following expressions. Sec. 6.7 The Singular Value Decomposition and the Pseudoinverse 419 (a) T† (x1 , x2 , x3 ), where T is the linear transformation of Exercise 2(a) (b) T† (a + bx + cx2 ), where T is the linear transformation of Exer- cise 2(b) (c) T† (a + b sin x + c cos x), where T is the linear transformation of Exercise 2(c) (d) T† (z1 , z2 ), where T is the linear transformation of Exercise 2(d) 6. Use the results of Exercise 3 to find the pseudoinverse of each of the following matrices. ⎛ ⎞ ⎛ ⎞ 1 1 1 1 1 0 1 ⎜0 1⎟ (a) ⎝ 1 1⎠ (b) (c) ⎜ ⎟ 1 0 −1 ⎝1 0⎠ −1 −1 1 1 ⎛ ⎞ ⎛ ⎞ 1 1 1 1 1 1 1 (d) ⎝1 −1 0⎠ (e) 1+ i 1 (f ) ⎝1 0 −2 1⎠ 1 − i −i 1 0 −1 1 −1 1 1 7. For each of the given linear transformations T : V → W, (i) Describe the subspace Z1 of V such that T† T is the orthogonal projection of V on Z1 . (ii) Describe the subspace Z2 of W such that TT† is the orthogonal projection of W on Z2 . (a) T is the linear transformation of Exercise 2(a) (b) T is the linear transformation of Exercise 2(b) (c) T is the linear transformation of Exercise 2(c) (d) T is the linear transformation of Exercise 2(d) 8.9.For each of the given systems of linear equations, (i) If the system is consistent, find the unique solution having mini- mum norm. (ii) If the system is inconsistent, find the “best approximation to a solution” having minimum norm, as described in Theorem 6.30(b). (Use your answers to parts (a) and (f) of Exercise 6.) x1 + x2 = 1 x1 + x 2 + x3 + x4 = 2 (a) x1 + x2 = 2 (b) x1 − 2x3 + x4 = −1 −x1 + −x2 = 0 x1 − x 2 + x3 + x4 = 2 Let V and W be finite-dimensional inner product spaces over F , and sup- pose that {v1 , v2 , . . . , vn } and {u1 , u2 , . . . , um } are orthonormal bases for V and W, respectively. Let T : V → W is a linear transformation of rank r, and suppose that σ1 ≥ σ2 ≥ · · · ≥ σr > 0 are such that σ i ui if 1 ≤ i ≤ r T(vi ) = 0 if r < i. 420 Chap. 6 Inner Product Spaces (a)Prove that {u1 , u2 , . . . , um } is a set of eigenvectors of TT∗ with corresponding eigenvalues λ1 , λ2 , . . . , λm , where σi 2 if 1 ≤ i ≤ r λi = 0 if r < i. (b)(c)(d)Let A be an m × n matrix with real or complex entries. Prove that the nonzero singular values of A are the positive square roots of the nonzero eigenvalues of AA∗ , including repetitions. Prove that TT∗ and T∗ T have the same nonzero eigenvalues, in- cluding repetitions. State and prove a result for matrices analogous to (c). 10.Use Exercise 8 of Section 2.5 to obtain another proof of Theorem 6.27, the singular value decomposition theorem for matrices. 11.This exercise relates the singular values of a well-behaved linear operator or matrix to its eigenvalues. (a)(b)Let T be a normal linear operator on an n-dimensional inner prod- uct space with eigenvalues λ1 , λ2 , . . . , λn . Prove that the singular values of T are |λ1 |, |λ2 |, . . . , |λn |. State and prove a result for matrices analogous to (a). 12.Let A be a normal matrix with an orthonormal basis of eigenvectors β = {v1 , v2 , . . . , vn } and corresponding eigenvalues λ1 , λ2 , . . . , λn . Let V be the n × n matrix whose columns are the vectors in β. Prove that for each i there is a scalar θi of absolute value 1 such that if U is the n × n matrix with θi vi as column i and Σ is the diagonal matrix such that Σii = |λi | for each i, then U ΣV ∗ is a singular value decomposition of A. 13.14.Prove that if A is a positive semidefinite matrix, then the singular values of A are the same as the eigenvalues of A. ∗Prove that if A is a positive definite matrix and A = U ΣV is a singular value decomposition of A, then U = V . 15.Let A be a square matrix with a polar decomposition A = WP . (a) Prove that A is normal if and only if WP 2 = P 2 W . (b) Use (a) to prove that A is normal if and only if WP = P W . 16. Let A be a square matrix. Prove an alternate form of the polar de- composition for A: There exists a unitary matrix W and a positive semidefinite matrix P such that A = P W . Sec. 6.7 The Singular Value Decomposition and the Pseudoinverse 421 17. Let T and U be linear operators on R2 defined for all (x1 , x2 ) ∈ R2 by 18.T(x1 , x2 ) = (x1 , 0) and U(x1, x2 ) = (x1 + x2 , 0). (a) Prove that (UT)† = T† U† . (b) Exhibit matrices A and B such that AB is defined, but (AB)† = B † A† . Let A be an m × n matrix. Prove the following results. (a) For any m × m unitary matrix G, (GA)† = A† G∗ . (b) For any n × n unitary matrix H, (AH)† = H ∗A† . 19. Let A be a matrix with real or complex entries. Prove the following results. (a)(b)(c)The nonzero singular values of A are the same as the nonzero singular values of A∗ , which are the same as the nonzero singular values of At . (A† )∗ = (A∗ )† . (A† )t = (At )† . 20. Let A be a square matrix such that A2 = O. Prove that (A† )2 = O. 21. Let V and W be finite-dimensional inner product spaces, and let T : V → W be linear. Prove the following results. (a) TT† T = T. (b) T† TT† = T† . (c) Both T† T and TT† are self-adjoint. The preceding three statements are called the Penrose conditions, and they characterize the pseudoinverse of a linear transformation as shown in Exercise 22. 22.Let V and W be finite-dimensional inner product spaces. Let T : V → W and U : W → V be linear transformations such that TUT = T, UTU = U, and both UT and TU are self-adjoint. Prove that U = T† . 23.State and prove a result for matrices that is analogous to the result of Exercise 21. 24.State and prove a result for matrices that is analogous to the result of Exercise 22. 25. Let V and W be finite-dimensional inner product spaces, and let T : V → W be linear. Prove the following results. (a) If T is one-to-one, then T∗ T is invertible and T† = (T∗ T)−1 T∗ . (b) If T is onto, then TT∗ is invertible and T† = T∗ (TT∗ )−1 . 422 Chap. 6 Inner Product Spaces 26.Let V and W be finite-dimensional inner product spaces with orthonor- mal bases β and γ, respectively, and let T : V → W be linear. Prove that ([T]γβ )† = [T† ]βγ . 27. Let V and W be finite-dimensional inner product spaces, and let T : V → W be a linear transformation. Prove part (b) of the lemma to Theorem 6.30: TT† is the orthogonal projection of W on R(T). 1.Label(a) (b) (c) (d) (e) (f ) (g) (h) (i) (j) the following statements as true or false. Every quadratic form is a bilinear form. If two matrices are congruent, they have the same eigenvalues. Symmetric bilinear forms have symmetric matrix representations. Any symmetric matrix is congruent to a diagonal matrix. The sum of two symmetric bilinear forms is a symmetric bilinear form. Two symmetric matrices with the same characteristic polynomial are matrix representations of the same bilinear form. There exists a bilinear form H such that H(x, y) = 0 for all x and y. If V is a vector space of dimension n, then dim(B(V )) = 2n. Let H be a bilinear form on a finite-dimensional vector space V with dim(V) > 1. For any x ∈ V, there exists y ∈ V such that y = 0 , but H(x, y) = 0. If H is any bilinear form on a finite-dimensional real inner product space V, then there exists an ordered basis β for V such that ψβ (H) is a diagonal matrix. 2. Prove properties 1, 2, 3, and 4 on page 423. 3.(a)(b)(c)Prove that the sum of two bilinear forms is a bilinear form. Prove that the product of a scalar and a bilinear form is a bilinear form. Prove Theorem 6.31. 4.Determine which of the mappings that follow are bilinear forms. Justify your answers. (a)Let V = C[0, 1] be the space of continuous real-valued functions on the closed interval [0, 1]. For f, g ∈ V, define 1 H(f, g) = f (t)g(t)dt. 0 (b) Let V be a vector space over F , and let J ∈ B(V) be nonzero. Define H : V × V → F by H(x, y) = [J(x, y)]2 for all x, y ∈ V. 448 Chap. 6 Inner Product Spaces (c)(d)(e)(f )Define H : R × R → R by H(t1 , t2 ) = t1 + 2t2 . Consider the vectors of R2 as column vectors, and let H : R2 → R be the function defined by H(x, y) = det(x, y), the determinant of the 2 × 2 matrix with columns x and y. Let V be a real inner product space, and let H : V × V → R be the function defined by H(x, y) = x, y for x, y ∈ V. Let V be a complex inner product space, and let H : V × V → C be the function defined by H(x, y) = x, y for x, y ∈ V. 5. Verify that each of the given mappings is a bilinear form. Then compute its matrix representation with respect to the given ordered basis β. (a) H : R3 × R3 → R, where ⎛⎛ ⎞ ⎛ ⎞⎞ a1 b1 H ⎝⎝a2 ⎠ , ⎝b2 ⎠⎠ = a1 b1 − 2a1 b2 + a2 b1 − a3 b3 a3 b3 and ⎧⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎫ ⎨ 1 1 0 ⎬ β = ⎝0⎠ , ⎝ 0⎠ , ⎝1⎠ . ⎩ ⎭ 1 −1 0 (b) Let V = M2×2 (R) and 1 0 0 1 0 0 0 0 β = , , , . 0 0 0 0 1 0 0 1 Define H : V × V → R by H(A, B) = tr(A)· tr(B). (c) Let β = {cos t, sin t, cos 2t, sin 2t}. Then β is an ordered basis for V = span(β), a four-dimensional subspace of the space of all continuous functions on R. Let H : V × V → R be the function defined by H(f, g) = f (0) · g (0). 6. Let H : R2 → R be the function defined by a1 b1 a1 b 1 H , = a1 b2 + a2 b1 for , ∈ R2 . a2 b2 a2 b2 (a) Prove that H is a bilinear form. (b) Find the 2× 2 matrix A such that H(x, y) = xtAy for all x, y ∈ R2 . For a 2 × 2 matrix M with columns x and y, the bilinear form H(M ) = H(x, y) is called the permanent of M . 7.Let V and W be vector spaces over the same field, and let T : V → W be a linear transformation. For any H ∈ B(W), define T(H) : V × V → F by T(H)(x, y) = H(T(x), T(y)) for all x, y ∈ V. Prove the following results. Sec. 6.8 Bilinear and Quadratic Forms (a) If H ∈ B(W), then T(H) ∈ B(V). (b) T : B(W) → B(V) is a linear transformation. (c) If T is an isomorphism, then so is T. 449 8.Assume the notation of Theorem 6.32. (a)(b)(c)Prove that for any ordered basis β, ψβ is linear. Let β be an ordered basis for an n-dimensional space V over F , and let φβ : V → Fn be the standard representation of V with respect to β. For A ∈ Mn×n (F ), define H : V × V → F by H(x, y) = [φβ (x)]t A[φβ (y)]. Prove that H ∈ B(V). Can you establish this as a corollary to Exercise 7? Prove the converse of (b): Let H be a bilinear form on V. If A = ψβ (H), then H(x, y) = [φβ (x)]t A[φβ (y)]. 9. (a) Prove Corollary 1 to Theorem 6.32. (b) For a finite-dimensional vector space V, describe a method for finding an ordered basis for B(V). 10. Prove Corollary 2 to Theorem 6.32. 11.12.13.14.15.16.Prove Corollary 3 to Theorem 6.32. Prove that the relation of congruence is an equivalence relation. The following outline provides an alternative proof to Theorem 6.33. (a) Suppose that β and γ are ordered bases for a finite-dimensional vector space V, and let Q be the change of coordinate matrix changing γ-coordinates to β-coordinates. Prove that φβ = LQ φγ , where φβ and φγ are the standard representations of V with respect to β and γ, respectively. (b) Apply Corollary 2 to Theorem 6.32 to (a) to obtain an alternative proof of Theorem 6.33. Let V be a finite-dimensional vector space and H ∈ B(V). Prove that, for any ordered bases β and γ of V, rank(ψβ (H)) = rank(ψγ (H)). Prove the following results. (a) Any square diagonal matrix is symmetric. (b) Any matrix congruent to a diagonal matrix is symmetric. (c) the corollary to Theorem 6.35 Let V be a vector space over a field F not of characteristic two, and let H be a symmetric bilinear form on V. Prove that if K(x) = H(x, x) is the quadratic form associated with H, then, for all x, y ∈ V, 1 H(x, y) = [K(x + y) − K(x) − K(y)]. 2 450 Chap. 6 Inner Product Spaces 17.For each of the given quadratic forms K on a real inner product space V, find a symmetric bilinear form H such that K(x) = H(x, x) for all x ∈ V. Then find an orthonormal basis β for V such that ψβ (H) is a diagonal matrix. t 1(a) K : R2 → R defined by K = −2t21 + 4t1 t2 + t22 t2 t 1(b) K : R2 → R defined by K = 7t21 − 8t1 t2 + t22 t2 ⎛ ⎞ t1 (c) K : R3 → R defined by K ⎝t2 ⎠ = 3t21 + 3t22 + 3t23 − 2t1 t3 t3 18.Let S be the set of all (t1 , t2 , t3 ) ∈ R3 for which √ 3t21 + 3t22 + 3t23 − 2t1 t3 + 2 2(t1 + t3 ) + 1 = 0. Find an orthonormal basis β for R3 for which the equation relating the coordinates of points of S relative to β is simpler. Describe S geometrically. 19.Prove the following refinement of Theorem 6.37(d). (a) If 0 < rank(A) < n and A has no negative eigenvalues, then f has no local maximum at p. (b) If 0 < rank(A) < n and A has no positive eigenvalues, then f has no local minimum at p. 20. Prove the following variation of the second-derivative test for the case n = 2: Define $ 2 %$ % $ 2 %2 ∂ f (p) ∂ 2f (p) ∂ f (p) D = − . ∂t21 ∂t22 ∂t1 ∂t2 (a) (b) (c) (d) If D > 0 and ∂ 2 f (p)/∂t21 > 0, then f has a local minimum at p. If D > 0 and ∂ 2 f (p)/∂t21 < 0, then f has a local maximum at p. If D < 0, then f has no local extremum at p. If D = 0, then the test is inconclusive. Hint: Observe that, as in Theorem 6.37, D = det(A) = λ1 λ2 , where λ1 and λ2 are the eigenvalues of A. 21.Let A and E be in Mn×n (F ), with E an elementary matrix. In Sec- tion 3.1, it was shown that AE can be obtained from A by means of an elementary column operation. Prove that E t A can be obtained by means of the same elementary operation performed on the rows rather than on the columns of A. Hint: Note that E t A = (At E)t . Sec. 6.9 Einstein’s Special Theory of Relativity 451 22. For each of the following matrices A with entries from R, find a diagonal matrix D and an invertible matrix Q such that Qt AQ = D. ⎛ ⎞ 3 1 2 (a) 1 3 (b) 0 1 (c) ⎝1 4 0⎠ 3 2 1 0 2 0 −1 Hint for (b): Use an elementary operation other than interchanging columns. 23.Prove that if the diagonal entries of a diagonal matrix are permuted, then the resulting diagonal matrix is congruent to the original one. 24. Let T be a linear operator on a real inner product space V, and define H : V × V → R by H(x, y) = x, T(y) for all x, y ∈ V. (a) Prove that H is a bilinear form. (b) Prove that H is symmetric if and only if T is self-adjoint. (c) What properties must T have for H to be an inner product on V? (d) Explain why H may fail to be a bilinear form if V is a complex inner product space. 25.26.Prove the converse to Exercise 24(a): Let V be a finite-dimensional real inner product space, and let H be a bilinear form on V. Then there exists a unique linear operator T on V such that H(x, y) = x, T(y) for all x, y ∈ V. Hint: Choose an orthonormal basis β for V, let A = ψβ (H), and let T be the linear operator on V such that [T]β = A. Apply Exercise 8(c) of this section and Exercise 15 of Section 6.2 (p. 355). Prove that the number of distinct equivalence classes of congruent n × n real symmetric matrices is (n + 1)(n + 2) . 2 1. Prove (b), (c), and (d) of Theorem 6.39. 2. Complete the proof of Theorem 6.40 for the case t < 0. 3. For ⎛ ⎞ ⎛ ⎞ 1 1 ⎜0⎟ ⎜ 0⎟ w1 = ⎜ ⎟ and w2 = ⎜ ⎟ , ⎝0⎠ ⎝ 0⎠1 −1 show that (a) {w1 , w2 } is an orthogonal basis for span({e1 , e4 }); (b) span({e1 , e 4 }) is T∗ v LA Tv -invariant. 4. Prove the corollary to Theorem 6.41. Hints: (a) Prove that ⎛ ⎞ p 0 0 q ∗ ⎜ 0 1 0 0⎟ Bv ABv = ⎜ ⎝ 0 0 1 0⎠ ⎟ , −q 0 0 −p where a + b a − b p = and q = . 2 2 462 (b)(c)Chap. 6 Inner Product Spaces ∗Show that q = 0 by using the fact that Bv ABv is self-adjoint. Apply Theorem 6.40 to ⎛ ⎞ 0 ⎜1⎟ w = ⎜ ⎟ ⎝0⎠ 1 to show that p = 1. 5.6.7.Derive (24), and prove that ⎛ ⎞ −v⎛ ⎞ √ 0 ⎜ 1 − v 2 ⎟ Tv ⎜0⎟ ⎝0⎠ ⎜ ⎟ = ⎜ ⎜ ⎜ ⎜ 0 0 ⎟ ⎟ ⎟ ⎟ . (25) 1 ⎝ 1 ⎠ √ 1 − v 2 Hint: Use a technique similar to the derivation of (22). Consider three coordinate systems S, S , and S with the corresponding axes (x,x ,x ; y,y ,y ; and z,z ,z ) parallel and such that the x-, x-, and x -axes coincide. Suppose that S is moving past S at a velocity v1 > 0 (as measured on S), S is moving past S at a velocity v2 > 0 (as measured on S ), and S is moving past S at a velocity v3 > 0 (as measured on S), and that there are three clocks C, C , and C such that C is stationary relative to S, C is stationary relative to S , and C is stationary relative to S . Suppose that when measured on any of the three clocks, all the origins of S, S , and S coincide at time 0. Assuming that Tv3 = Tv2 Tv1 (i.e., Bv3 = Bv2 Bv1 ), prove that v1 + v2 v3 = . 1 + v1 v2 Note that substituting v2 = 1 in this equation yields v3 = 1. This tells us that the speed of light as measured in S or S is the same. Why would we be surprised if this were not the case? Compute (Bv )−1 . Show (Bv )−1 = B(−v) . Conclude that if S moves at a negative velocity v relative to S, then [Tv ]β = Bv , where Bv is of the form given in Theorem 6.42. 8.Suppose that an astronaut left Earth in the year 2000 and traveled to a star 99 light years away from Earth at 99% of the speed of light and that upon reaching the star immediately turned around and returned to Earth at the same speed. Assuming Einstein’s special theory of Sec. 6.9 Einstein’s Special Theory of Relativity 463 relativity, show that if the astronaut was 20 years old at the time of departure, then he or she would return to Earth at age 48.2 in the year 2200. Explain the use of Exercise 7 in solving this problem. 9. Recall the moving space vehicle considered in the study of time contrac- tion. Suppose that the vehicle is moving toward a fixed star located on the x-axis of S at a distance b units from the origin of S. If the space vehicle moves toward the star at velocity v, Earthlings (who remain “al- most” stationary relative to S) compute the time it takes for the vehicle to reach the star as t = b/v. Due to the phenomenon √ of time contraction, √ the astronaut perceives a time span of t = t 1 − v 2 = (b/v) 1 − v 2 . A paradox appears in that the astronaut perceives a time span incon- sistent with a distance of b and a velocity of v. The paradox is resolved by observing that the distance from the solar system to the star as measured by the astronaut is less than b. Assuming that the coordinate systems S and S and clocks C and Care as in the discussion of time contraction, prove the following results. (a) At time t (as measured on C), the space–time coordinates of star relative to S and C are ⎛ ⎞ b ⎜0⎟ ⎜ ⎟ . ⎝0⎠ t (b) At time t (as measured on C), the space–time coordinates of the star relative to S and C are ⎛ ⎞ b − vt√ ⎜ 1 − v2 ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ ⎟ . ⎜ 0 ⎟ ⎝ t − bv ⎠ √ 1 − v2 (c) For b − tv t − bv x = √ and t = √ , 1 − v 2 1 − v 2 √ we have x = b 1 − v 2 − t v. This result may be interpreted to mean that at time t as measured by the astronaut, the distance from the astronaut to the star as measured by the astronaut (see Figure 6.9) is b 1 − v 2 − t v. 464 Chap. 6 Inner Product Spaces z z 6 . . . . . . . . . . . . . . . . . . . . . . . . . 7 8 . . . . . . . . . . . . . .6 . . . . . . 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 4 . 1 . . . . . . . . . . . . . . . . . . . . . 2 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . C 6 . . . . . . . . . . . . . . . . . . . . . . 8 7 . . . . . . . . . . . . . . . . .6 . . . . . . 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 4 . 1 . . . . . . . . . . . . . . . . . . . . 3 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C (x , 0, 0) > r r Z ~ Z coordinates 1 y - 1 y - relative to S S x S x *(star) (b, 0, 0) coordinates Figure 6.9 relative to S (d) Conclude from the preceding equation that (1) the speed of the space vehicle relative to the star, as measured by the astronaut, is v; (2) the distance √ from Earth to the star, as measured by the astro- naut, is b 1 − v 2 . Thus distances along the line of √ motion of the space vehicle appear to be contracted by a factor of 1 − v 2 . 1.Label the following statements as true or false. (a) If Ax = b is well-conditioned, then cond(A) is small. (b) If cond(A) is large, then Ax = b is ill-conditioned. (c) If cond(A) is small, then Ax = b is well-conditioned. (d) The norm of A equals the Rayleigh quotient. (e) The norm of A always equals the largest eigenvalue of A. Sec. 6.10 Conditioning and the Rayleigh Quotient 471 2. Compute the norms of the following matrices. ⎛ ⎞ 1 −2 √ 0 3 (a) 4 0 (b) 5 3 (c) ⎜ −2 √ ⎟ 1 3 −3 3 ⎜ ⎝0 3 1⎟ ⎠ 0 √ 2 1 3 3. Prove that if B is symmetric, then B is the largest eigenvalue of B. 4. Let A and A−1 be as follows: ⎛ ⎞ ⎛ ⎞ 6 13 −17 6 −4 1 A = ⎝ 13 29 −38⎠ and A−1 = ⎝−4 11 7⎠ . −17 −38 50 −1 7 5 The eigenvalues of A are approximately 84.74, 0.2007, and 0.0588. (a) Approximate A, A−1 , and cond(A). (Note Exercise 3.) (b) Suppose that we have vectors x and x̃ such that Ax = b and b − Ax̃ ≤ 0.001. Use (a) to determine upper bounds for x̃ − A−1 b (the absolute error) and x̃ − A−1 b/A−1 b (the rel- ative error). 5.Suppose that x is the actual solution of Ax = b and that a computer arrives at an approximate solution x̃. If cond(A) = 100, b = 1, and b − Ax̃ = 0.1, obtain upper and lower bounds for x − x̃/x. 6. Let ⎛ ⎞ 2 1 1 B = ⎝1 2 1⎠ . 1 1 2 Compute ⎛ ⎞ 1 R ⎝−2⎠ , B, and cond(B). 3 7.8.Let B be a symmetric matrix. Prove that min R(x) equals the smallest x=0 eigenvalue of B. Prove that if λ is an eigenvalue of AA∗ , then λ is an eigenvalue of A∗ A. This completes the proof of the lemma to Corollary 2 to Theorem 6.43. 9. Prove that if A is an invertible matrix and Ax = b, then 1 δb δx ≤ . A · A−1 b x 472 Chap. 6 Inner Product Spaces 10. Prove the left inequality of (a) in Theorem 6.44. 11.Prove that cond(A) = 1 if and only if A is a scalar multiple of a unitary or orthogonal matrix. 12. (a) Let A and B be square matrices that are unitarily equivalent. Prove that A = B. (b) Let T be a linear operator on a finite-dimensional inner product space V. Define (c)T(x) T = max . x=0 x Prove that T = [T]β , where β is any orthonormal basis for V. Let V be an infinite-dimensional inner product space with an or- thonormal basis {v1 , v2 , . . .}. Let T be the linear operator on V such that T(vk ) = kvk . Prove that T (defined in (b)) does not exist. The next exercise assumes the definitions of singular value and pseudoinverse and the results of Section 6.7. 13.Let A be an n × n matrix of rank r with the nonzero singular values σ1 ≥ σ2 ≥ · · · ≥ σr . Prove each of the following results. (a) A = σ1 . 1 (b) A† = . σr σ1 (c) If A is invertible (and hence r = n), then cond(A) = . σn </span></div><div style="text-align:center;margin:10px 0 10px 0;clear:both"><div style="display:inline;text-align:center;"><script async src="https://pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script>
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Linear Algebrahashnuthttps://koreanfoodie.me/94https://koreanfoodie.me/94#entry94commentSat, 15 Jun 2019 13:22:40 +0900Solutions to Linear Algebra, Stephen H. Friedberg, Fourth Edition (Chapter 5)
https://koreanfoodie.me/93
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</script></div><p><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/cFoRnZ/btqv7qHUOWX/zY6T6f11LYtauJv1ecCjhK/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/cFoRnZ/btqv7qHUOWX/zY6T6f11LYtauJv1ecCjhK/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FcFoRnZ%2Fbtqv7qHUOWX%2FzY6T6f11LYtauJv1ecCjhK%2Fimg.jpg' data-filename="chapter5_1.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/b54nJX/btqv4P3ea2s/peNIP2LztnUpKQo891uF81/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/b54nJX/btqv4P3ea2s/peNIP2LztnUpKQo891uF81/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2Fb54nJX%2Fbtqv4P3ea2s%2FpeNIP2LztnUpKQo891uF81%2Fimg.jpg' data-filename="chapter5_2.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/PyIwm/btqv4PIU8qU/JktdVItqfq7OKDX3OjSN3k/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/PyIwm/btqv4PIU8qU/JktdVItqfq7OKDX3OjSN3k/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FPyIwm%2Fbtqv4PIU8qU%2FJktdVItqfq7OKDX3OjSN3k%2Fimg.jpg' data-filename="chapter5_3.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/cuVvfM/btqv7QMSAl8/MZcUUgL2VFU2FkHYjQVzoK/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/cuVvfM/btqv7QMSAl8/MZcUUgL2VFU2FkHYjQVzoK/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FcuVvfM%2Fbtqv7QMSAl8%2FMZcUUgL2VFU2FkHYjQVzoK%2Fimg.jpg' data-filename="chapter5_4.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/5Zm2k/btqv8mxNuUo/xMsAthPq7uQgKfptVHBWl0/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/5Zm2k/btqv8mxNuUo/xMsAthPq7uQgKfptVHBWl0/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2F5Zm2k%2Fbtqv8mxNuUo%2FxMsAthPq7uQgKfptVHBWl0%2Fimg.jpg' data-filename="chapter5_5.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/W1xG7/btqv7Cae1QD/6IniBFXjbdbRJK5bWGQLvk/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/W1xG7/btqv7Cae1QD/6IniBFXjbdbRJK5bWGQLvk/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FW1xG7%2Fbtqv7Cae1QD%2F6IniBFXjbdbRJK5bWGQLvk%2Fimg.jpg' data-filename="chapter5_6.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/C0ZXz/btqv5W1PQvd/14ZBb4CkqlEosjRI71G1l0/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/C0ZXz/btqv5W1PQvd/14ZBb4CkqlEosjRI71G1l0/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FC0ZXz%2Fbtqv5W1PQvd%2F14ZBb4CkqlEosjRI71G1l0%2Fimg.jpg' data-filename="chapter5_7.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/b4Nlub/btqv7DfYMtD/WhlM4U0LFzNsC3pTkGJRS0/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/b4Nlub/btqv7DfYMtD/WhlM4U0LFzNsC3pTkGJRS0/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2Fb4Nlub%2Fbtqv7DfYMtD%2FWhlM4U0LFzNsC3pTkGJRS0%2Fimg.jpg' data-filename="chapter5_8.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/m37WH/btqv4PWnnSz/9ql64klq9k7nrthidE1Le0/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/m37WH/btqv4PWnnSz/9ql64klq9k7nrthidE1Le0/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2Fm37WH%2Fbtqv4PWnnSz%2F9ql64klq9k7nrthidE1Le0%2Fimg.jpg' data-filename="chapter5_9.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/k9RMX/btqv6EmopIk/vfBu6aJLvkrYPyk2OKguzk/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/k9RMX/btqv6EmopIk/vfBu6aJLvkrYPyk2OKguzk/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2Fk9RMX%2Fbtqv6EmopIk%2FvfBu6aJLvkrYPyk2OKguzk%2Fimg.jpg' data-filename="chapter5_10.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/cFlX6I/btqv4QHMEJa/7SDbonbU2zMTA8LMnBpxt0/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/cFlX6I/btqv4QHMEJa/7SDbonbU2zMTA8LMnBpxt0/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FcFlX6I%2Fbtqv4QHMEJa%2F7SDbonbU2zMTA8LMnBpxt0%2Fimg.jpg' data-filename="chapter5_11.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/dzBp5f/btqv7gldr8I/iyIz20HPbWKdoFMhKAhJYK/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/dzBp5f/btqv7gldr8I/iyIz20HPbWKdoFMhKAhJYK/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FdzBp5f%2Fbtqv7gldr8I%2FiyIz20HPbWKdoFMhKAhJYK%2Fimg.jpg' data-filename="chapter5_12.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/b5k0Hz/btqv7qBctSy/eO62jsF5v9oFy95fBq57H0/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/b5k0Hz/btqv7qBctSy/eO62jsF5v9oFy95fBq57H0/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2Fb5k0Hz%2Fbtqv7qBctSy%2FeO62jsF5v9oFy95fBq57H0%2Fimg.jpg' data-filename="chapter5_13.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/bspMRA/btqv8mEyOiR/LqyHXXAd3Mz7iLEU8HKSD1/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/bspMRA/btqv8mEyOiR/LqyHXXAd3Mz7iLEU8HKSD1/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FbspMRA%2Fbtqv8mEyOiR%2FLqyHXXAd3Mz7iLEU8HKSD1%2Fimg.jpg' data-filename="chapter5_14.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/bpA5xQ/btqv7CnO8cg/4L6ewqsDTNVdAFc46eBsxk/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/bpA5xQ/btqv7CnO8cg/4L6ewqsDTNVdAFc46eBsxk/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FbpA5xQ%2Fbtqv7CnO8cg%2F4L6ewqsDTNVdAFc46eBsxk%2Fimg.jpg' data-filename="chapter5_15.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/bnrWjP/btqv7ZiwjYN/w7piFvIJuRlxoNmzAwptRK/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/bnrWjP/btqv7ZiwjYN/w7piFvIJuRlxoNmzAwptRK/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FbnrWjP%2Fbtqv7ZiwjYN%2Fw7piFvIJuRlxoNmzAwptRK%2Fimg.jpg' data-filename="chapter5_16.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/bEySMb/btqv4QHMEPU/D4ygMjiz8twbUfOIKFjuAk/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/bEySMb/btqv4QHMEPU/D4ygMjiz8twbUfOIKFjuAk/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FbEySMb%2Fbtqv4QHMEPU%2FD4ygMjiz8twbUfOIKFjuAk%2Fimg.jpg' data-filename="chapter5_17.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/MJJmT/btqv7YYdHDZ/Ot6hoVwGOZYYey3caZCzK1/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/MJJmT/btqv7YYdHDZ/Ot6hoVwGOZYYey3caZCzK1/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FMJJmT%2Fbtqv7YYdHDZ%2FOt6hoVwGOZYYey3caZCzK1%2Fimg.jpg' data-filename="chapter5_18.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/p3EiD/btqv7XLMSqL/96iMJJ91NQIcbiq8TQFqP1/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/p3EiD/btqv7XLMSqL/96iMJJ91NQIcbiq8TQFqP1/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2Fp3EiD%2Fbtqv7XLMSqL%2F96iMJJ91NQIcbiq8TQFqP1%2Fimg.jpg' data-filename="chapter5_19.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/k5AbS/btqv7hqTjoW/AGmHWJvLgRl489IRH0nXJK/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/k5AbS/btqv7hqTjoW/AGmHWJvLgRl489IRH0nXJK/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2Fk5AbS%2Fbtqv7hqTjoW%2FAGmHWJvLgRl489IRH0nXJK%2Fimg.jpg' data-filename="chapter5_20.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/5i9Ja/btqv8njaBK3/guslTdnryPEBr9sbKLPtv0/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/5i9Ja/btqv8njaBK3/guslTdnryPEBr9sbKLPtv0/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2F5i9Ja%2Fbtqv8njaBK3%2FguslTdnryPEBr9sbKLPtv0%2Fimg.jpg' data-filename="chapter5_21.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/dIVSBq/btqv7Zv4lyG/F5aqxhXkDXNqUZIn0xjYo1/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/dIVSBq/btqv7Zv4lyG/F5aqxhXkDXNqUZIn0xjYo1/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FdIVSBq%2Fbtqv7Zv4lyG%2FF5aqxhXkDXNqUZIn0xjYo1%2Fimg.jpg' data-filename="chapter5_22.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/c3Hty8/btqv8ncoWol/bzAWW7BcuonILTs81uhFEk/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/c3Hty8/btqv8ncoWol/bzAWW7BcuonILTs81uhFEk/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2Fc3Hty8%2Fbtqv8ncoWol%2FbzAWW7BcuonILTs81uhFEk%2Fimg.jpg' data-filename="chapter5_23.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/rH23u/btqv7B3uZqn/KRKelZYZRYiCL2Ht4U4kN0/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/rH23u/btqv7B3uZqn/KRKelZYZRYiCL2Ht4U4kN0/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FrH23u%2Fbtqv7B3uZqn%2FKRKelZYZRYiCL2Ht4U4kN0%2Fimg.jpg' data-filename="chapter5_24.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/LxRKh/btqv7gMhktF/2kawmVDzYh5UJVLLeZkYP1/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/LxRKh/btqv7gMhktF/2kawmVDzYh5UJVLLeZkYP1/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FLxRKh%2Fbtqv7gMhktF%2F2kawmVDzYh5UJVLLeZkYP1%2Fimg.jpg' data-filename="chapter5_25.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/LRGTt/btqv7XLMSw2/n5Zsbmk2BDUeb1fkAewxz1/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/LRGTt/btqv7XLMSw2/n5Zsbmk2BDUeb1fkAewxz1/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FLRGTt%2Fbtqv7XLMSw2%2Fn5Zsbmk2BDUeb1fkAewxz1%2Fimg.jpg' data-filename="chapter5_26.jpg"></span></figure><figure class='imageblock alignCenter'><span 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<p>Solution maual to Linear Algebra, Fourth Edition, Stephen H. Friedberg. (Chapter 5)<br /><br />Solutions to Linear Algebra, Fourth Edition, Stephen H. Friedberg. (Chapter 5)<br /><br />Linear Algebra solution manual, Fourth Edition, Stephen H. Friedberg. (Chapter 5)<br /><br />Linear Algebra solutions Friedberg. (Chapter 5)</p>
<div style="width: 100%; height: 1px; overflow: auto;"><span>1. Label the following statements as true or false. (a) Every linear operator on an n-dimensional vector space has n dis- tinct eigenvalues. (b) If a real matrix has one eigenvector, then it has an infinite number of eigenvectors. (c) There exists a square matrix with no eigenvectors. (d) Eigenvalues must be nonzero scalars. (e) Any two eigenvectors are linearly independent. (f ) The sum of two eigenvalues of a linear operator T is also an eigen- value of T. (g) Linear operators on infinite-dimensional vector spaces never have eigenvalues. (h) An n × n matrix A with entries from a field F is similar to a diagonal matrix if and only if there is a basis for Fn consisting of eigenvectors of A. (i) Similar matrices always have the same eigenvalues. (j) Similar matrices always have the same eigenvectors. (k) The sum of two eigenvectors of an operator T is always an eigen- vector of T. 2. For each of the following linear operators T on a vector space V and ordered bases β, compute [T]β , and determine whether β is a basis consisting of eigenvectors of T. 2 a 10a − 6b 1 2 (a) V = R , T = , and β = , b 17a − 10b 2 3 (b) V = P1 (R), T(a + bx) = (6a − 6b) + (12a − 11b)x, and β = {3 + 4x, 2 + 3x} ⎛ ⎞ ⎛ ⎞ a 3a + 2b − 2c (c) V = R3 , T ⎝ b ⎠ = ⎝−4a − 3b + 2c⎠, and c −c ⎧⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎫ ⎨ 0 1 1 ⎬ β = ⎝1⎠ , ⎝−1⎠ , ⎝0⎠ ⎩ ⎭ 1 0 2 (d) V = P2 (R), T(a + bx + cx2 ) = (−4a + 2b − 2c) − (7a + 3b + 7c)x + (7a + b + 5c)x2 , and β = {x − x2 , −1 + x2 , −1 − x + x2 } Sec. 5.1 Eigenvalues and Eigenvectors 257 (e) V = P3(R), T(a + bx + cx2 + dx3 ) = −d + (−c + d)x + (a + b − 2c)x2 + (−b + c − 2d)x3 , and β = {1 − x + x3 , 1 + x2 , 1, x + x2 } a b −7a − 4b + 4c − 4d b (f ) V = M2×2 (R), T = , and c d −8a − 4b + 5c − 4d d 1 0 −1 2 1 0 −1 0 β = , , , 1 0 0 0 2 0 0 2 3. For each of the following matrices A ∈ Mn×n (F ), (i) Determine all the eigenvalues of A. (ii) For each eigenvalue λ of A, find the set of eigenvectors correspond- ing to λ. (iii) If possible, find a basis for Fn consisting of eigenvectors of A. (iv) If successful in finding such a basis, determine an invertible matrix Q and a diagonal matrix D such that Q−1 AQ = D. 1 2 (a) A = for F = R 3 2 ⎛ ⎞ 0 −2 −3 (b) A = ⎝−1 1 −1⎠ for F = R 2 2 5 i 1 (c) A = for F = C 2 −i ⎛ ⎞ 2 0 −1 (d) A = ⎝4 1 −4⎠ for F = R 2 0 −1 4.For each linear operator T on V, find the eigenvalues of T and an ordered basis β for V such that [T]β is a diagonal matrix. (a) V = R2 and T(a, b) = (−2a + 3b, −10a + 9b) (b) V = R3 and T(a, b, c) = (7a − 4b + 10c, 4a − 3b + 8c, −2a + b − 2c) (c) V = R3 and T(a, b, c) = (−4a + 3b − 6c, 6a − 7b + 12c, 6a − 6b + 11c) (d) V = P1(R) and T(ax + b) = (−6a + 2b)x + (−6a + b) (e) V = P2(R) and T(f (x)) = xf (x) + f (2)x + f (3) (f ) V = P3(R) and T(f (x)) = f (x) + f (2)x (g) V = P3(R) and T(f (x)) = xf (x) + f (x) − f (2) a b d b (h) V = M2×2 (R) and T = c d c a 258 Chap. 5 a b c d (i) V = M2×2 (R) and T = c d a b (j) V = M2×2 (R) and T(A) = At + 2 · tr(A) · I2 Diagonalization 5. Prove Theorem 5.4. 6.Let T be a linear operator on a finite-dimensional vector space V, and let β be an ordered basis for V. Prove that λ is an eigenvalue of T if and only if λ is an eigenvalue of [T]β . 7.Let T be a linear operator on a finite-dimensional vector space V. We define the determinant of T, denoted det(T), as follows: Choose any ordered basis β for V, and define det(T) = det([T]β ). (a)(b)(c)(d)(e)Prove that the preceding definition is independent of the choice of an ordered basis for V. That is, prove that if β and γ are two ordered bases for V, then det([T]β ) = det([T]γ ). Prove that T is invertible if and only if det(T) = 0. Prove that if T is invertible, then det(T−1 ) = [det(T)]−1 . Prove that if U is also a linear operator on V, then det(TU) = det(T)· det(U). Prove that det(T − λIV ) = det([T]β − λI) for any scalar λ and any ordered basis β for V. 8.(a)(b)(c)Prove that a linear operator T on a finite-dimensional vector space is invertible if and only if zero is not an eigenvalue of T. Let T be an invertible linear operator. Prove that a scalar λ is an eigenvalue of T if and only if λ−1 is an eigenvalue of T−1 . State and prove results analogous to (a) and (b) for matrices. 9. Prove that the eigenvalues of an upper triangular matrix M are the diagonal entries of M . 10. Let V be a finite-dimensional vector space, and let λ be any scalar. (a)(b)(c)For any ordered basis β for V, prove that [λIV ]β = λI. Compute the characteristic polynomial of λIV . Show that λIV is diagonalizable and has only one eigenvalue. 11.A scalar matrix is a square matrix of the form λI for some scalar λ; that is, a scalar matrix is a diagonal matrix in which all the diagonal entries are equal. (a) Prove that if a square matrix A is similar to a scalar matrix λI, then A = λI. (b) Show that a diagonalizable matrix having only one eigenvalue is a scalar matrix. Sec. 5.1 Eigenvalues and Eigenvectors 1 1 (c) Prove that is not diagonalizable. 0 1 259 12. (a) Prove that similar matrices have the same characteristic polyno- mial. (b) Show that the definition of the characteristic polynomial of a linear operator on a finite-dimensional vector space V is independent of the choice of basis for V. 13. Let T be a linear operator on a finite-dimensional vector space V over a field F , let β be an ordered basis for V, and let A = [T]β . In reference to Figure 5.1, prove the following. (a) If v ∈ V and φβ (v) is an eigenvector of A corresponding to the eigenvalue λ, then v is an eigenvector of T corresponding to λ. (b) If λ is an eigenvalue of A (and hence of T), then a vector y ∈ Fn −1 is an eigenvector of A corresponding to λ if and only if φ (y) is βan eigenvector of T corresponding to λ. 14. † For any square matrix A, prove that A and At have the same charac- teristic polynomial (and hence the same eigenvalues). 15. † (a)(b)Let T be a linear operator on a vector space V, and let x be an eigenvector of T corresponding to the eigenvalue λ. For any posi- tive integer m, prove that x is an eigenvector of Tm corresponding to the eigenvalue λm . State and prove the analogous result for matrices. 16. (a) Prove that similar matrices have the same trace. Hint: Use Exer- cise 13 of Section 2.3. (b) How would you define the trace of a linear operator on a finite- dimensional vector space? Justify that your definition is well- defined. 17.Let T be the linear operator on Mn×n (R) defined by T(A) = At . (a) Show that ±1 are the only eigenvalues of T. (b) Describe the eigenvectors corresponding to each eigenvalue of T. (c) Find an ordered basis β for M2×2 (R) such that [T]β is a diagonal matrix. (d) Find an ordered basis β for Mn×n(R) such that [T]β is a diagonal matrix for n > 2. 18.Let A, B ∈ Mn×n (C). (a) Prove that if B is invertible, then there exists a scalar c ∈ C such that A + cB is not invertible. Hint: Examine det(A + cB). 260 Chap. 5 Diagonalization (b) Find nonzero 2 × 2 matrices A and B such that both A and A + cB are invertible for all c ∈ C. 19. † Let A and B be similar n × n matrices. Prove that there exists an n- dimensional vector space V, a linear operator T on V, and ordered bases β and γ for V such that A = [T]β and B = [T]γ . Hint: Use Exercise 14 of Section 2.5. 20. Let A be an n × n matrix with characteristic polynomial f (t) = (−1)n tn + an−1 tn−1 + · · · + a1 t + a0 . Prove that f (0) = a0 = det(A). Deduce that A is invertible if and only if a0 = 0. 21.Let A and f (t) be as in Exercise 20. (a)(b)Prove that f (t) = (A11 − t)(A22 − t) · · · (Ann − t) + q(t), where q(t) is a polynomial of degree at most n−2. Hint: Apply mathematical induction to n. Show that tr(A) = (−1)n−1 an−1 . †22. (a) Let T be a linear operator on a vector space V over the field F , and let g(t) be a polynomial with coefficients from F . Prove that if x is an eigenvector of T with corresponding eigenvalue λ, then g(T)(x) = g(λ)x. That is, x is an eigenvector of g(T) with corre- sponding eigenvalue g(λ). (b) State and prove a comparable result for matrices. (c) Verify (b) for the matrix A in Exercise 3(a) with polynomial g(t) = 2 2 2t − t + 1, eigenvector x = , and corresponding eigenvalue 3 λ = 4. 23.24.25.26.Use Exercise 22 to prove that if f (t) is the characteristic polynomial of a diagonalizable linear operator T, then f (T) = T0 , the zero opera- tor. (In Section 5.4 we prove that this result does not depend on the diagonalizability of T.) Use Exercise 21(a) to prove Theorem 5.3. Prove Corollaries 1 and 2 of Theorem 5.3. Determine the number of distinct characteristic polynomials of matrices in M2×2 (Z2 ). 1.Label(a)(b)(c)(d)(e)(f )(g)the following statements as true or false. Any linear operator on an n-dimensional vector space that has fewer than n distinct eigenvalues is not diagonalizable. Two distinct eigenvectors corresponding to the same eigenvalue are always linearly dependent. If λ is an eigenvalue of a linear operator T, then each vector in Eλ is an eigenvector of T. If λ1 and λ2 are distinct eigenvalues of a linear operator T, then Eλ1 ∩ Eλ2 = {0 }. Let A ∈ Mn×n (F ) and β = {v1 , v2 , . . . , vn } be an ordered basis for Fn consisting of eigenvectors of A. If Q is the n × n matrix whose jth column is vj (1 ≤ j ≤ n), then Q−1 AQ is a diagonal matrix. A linear operator T on a finite-dimensional vector space is diago- nalizable if and only if the multiplicity of each eigenvalue λ equals the dimension of Eλ . Every diagonalizable linear operator on a nonzero vector space has at least one eigenvalue. The following two items relate to the optional subsection on direct sums. (h)(i)If a vector space is the direct sum of subspaces W1 , W2 , . . . , Wk , then Wi ∩ Wj = {0 } for i = j. If k V = Wi and Wi ∩ Wj = {0 } for i = j, i=1 then V = W1 ⊕ W2 ⊕ · · · ⊕ Wk . 2. For each of the following matrices A ∈ Mn×n (R), test A for diagonal- izability, and if A is diagonalizable, find an invertible matrix Q and a diagonal matrix D such that Q−1 AQ = D. 1 2 1 3 1 4 (a) (b) (c) 0 1 3 1 3 2 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 7 −4 0 0 0 1 1 1 0 (d) ⎝8 −5 0⎠ (e) ⎝1 0 −1⎠ (f ) ⎝0 1 2⎠ 6 −6 3 0 1 1 0 0 3 280 ⎛ ⎞ 3 1 1 (g) ⎝ 2 4 2⎠ −1 −1 1 Chap. 5 Diagonalization 3.For each of the following linear operators T on a vector space V, test T for diagonalizability, and if T is diagonalizable, find a basis β for V such that [T]β is a diagonal matrix. (a) V = P3 (R) and T is defined by T(f (x)) = f (x) + f (x), respec- tively. (b) V = P2 (R) and T is defined by T(ax2 + bx + c) = cx2 + bx + a. (c) V = R3 and T is defined by ⎛ ⎞ ⎛ ⎞ a1 a2 T ⎝a2 ⎠ = ⎝−a1 ⎠ . a3 2a3 4.5.6.7.8.9.(d) V = P2 (R) and T is defined by T(f (x)) = f (0) + f (1)(x + x2 ). (e) V = C2 and T is defined by T(z, w) = (z + iw, iz + w). (f ) V = M2×2 (R) and T is defined by T(A) = At . Prove the matrix version of the corollary to Theorem 5.5: If A ∈ Mn×n (F ) has n distinct eigenvalues, then A is diagonalizable. State and prove the matrix version of Theorem 5.6. (a) Justify the test for diagonalizability and the method for diagonal- ization stated in this section. (b) Formulate the results in (a) for matrices. For 1 4 A = ∈ M2×2 (R), 2 3 find an expression for An , where n is an arbitrary positive integer. Suppose that A ∈ Mn×n (F ) has two distinct eigenvalues, λ1 and λ2 , and that dim(Eλ1 ) = n − 1. Prove that A is diagonalizable. Let T be a linear operator on a finite-dimensional vector space V, and suppose there exists an ordered basis β for V such that [T]β is an upper triangular matrix. (a) Prove that the characteristic polynomial for T splits. (b) State and prove an analogous result for matrices. The converse of (a) is treated in Exercise 32 of Section 5.4. Sec. 5.2 Diagonalizability 281 10.11.Let T be a linear operator on a finite-dimensional vector space V with the distinct eigenvalues λ1 , λ2 , . . . , λk and corresponding multiplicities m1 , m2 , . . . , mk . Suppose that β is a basis for V such that [T]β is an upper triangular matrix. Prove that the diagonal entries of [T]β are λ1 , λ2 , . . . , λk and that each λi occurs mi times (1 ≤ i ≤ k). Let A be an n × n matrix that is similar to an upper triangular ma- trix and has the distinct eigenvalues λ1 , λ2, . . . , λk with corresponding multiplicities m1 , m2 , . . . , mk . Prove the following statements. k (a) tr(A) = mi λi i=1 (b) det(A) = (λ1 )m1 (λ2 )m2 · · · (λk )mk . 12. Let T be an invertible linear operator on a finite-dimensional vector space V. (a) Recall that for any eigenvalue λ of T, λ−1 is an eigenvalue of T−1 (Exercise 8 of Section 5.1). Prove that the eigenspace of T corre- sponding to λ is the same as the eigenspace of T−1 corresponding to λ−1 . (b) Prove that if T is diagonalizable, then T−1 is diagonalizable. 13.Let A ∈ Mn×n (F ). Recall from Exercise 14 of Section 5.1 that A and At have the same characteristic polynomial and hence share the same eigenvalues with the same multiplicities. For any eigenvalue λ of A and At , let Eλ and E λ denote the corresponding eigenspaces for A and At , respectively. (a)(b)(c)Show by way of example that for a given common eigenvalue, these two eigenspaces need not be the same. Prove that for any eigenvalue λ, dim(Eλ ) = dim(E λ ). Prove that if A is diagonalizable, then At is also diagonalizable. 14. Find the general solution to each system of differential equations. x = x + y x 1 = 8x1 + 10x2 (a) (b) y = 3x − y x2 = −5x1 − 7x2 x 1 = x1 + x 3 (c) x 2 = x2 + x 3 x 3 = 2x 3 15. Let ⎛ ⎞ a11 a12 ··· a1n ⎜ ⎜ a21 a22 ··· a2n ⎟ ⎟ A = ⎜ ⎝ . .. .. . .. . ⎟ ⎠ an1 an2 · · · ann 282 Chap. 5 Diagonalization be the coefficient matrix of the system of differential equations x 1 = a11 x1 + a12 x2 + · · · + a1n xn x 2 = a21 x1 + a22 x2 + · · · + a2n xn .. . x n = an1 x1 + an2 x2 + · · · + ann xn . Suppose that A is diagonalizable and that the distinct eigenvalues of A are λ1 , λ2 , . . . , λ k . Prove that a differentiable function x : R → Rn is a solution to the system if and only if x is of the form 16.x(t) = eλ1 t z1 + eλ2 t z2 + · · · + eλk t zk , where zi ∈ Eλi for i = 1, 2, . . . , k. Use this result to prove that the set of solutions to the system is an n-dimensional real vector space. Let C ∈ Mm×n (R), and let Y be an n × p matrix of differentiable functions. Prove (CY ) = CY , where (Y )ij = Yij for all i, j. Exercises 17 through 19 are concerned with simultaneous diagonalization. Definitions. Two linear operators T and U on a finite-dimensional vector space V are called simultaneously diagonalizable if there exists an ordered basis β for V such that both [T]β and [U]β are diagonal matrices. Similarly, A, B ∈ Mn×n (F ) are called simultaneously diagonalizable if there exists an invertible matrix Q ∈ Mn×n (F ) such that both Q−1 AQ and Q−1 BQ are diagonal matrices. 17. (a) Prove that if T and U are simultaneously diagonalizable linear operators on a finite-dimensional vector space V, then the matrices [T]β and [U]β are simultaneously diagonalizable for any ordered basis β. (b) Prove that if A and B are simultaneously diagonalizable matrices, then LA and LB are simultaneously diagonalizable linear operators. 18.(a) Prove that if T and U are simultaneously diagonalizable operators, then T and U commute (i.e., TU = UT). (b) Show that if A and B are simultaneously diagonalizable matrices, then A and B commute. The converses of (a) and (b) are established in Exercise 25 of Section 5.4. 19.Let T be a diagonalizable linear operator on a finite-dimensional vector space, and let m be any positive integer. Prove that T and Tm are simultaneously diagonalizable. Exercises 20 through 23 are concerned with direct sums. Sec. 5.3 Matrix Limits and Markov Chains 283 20.Let W1 , W2 , . . . , Wk be subspaces of a finite-dimensional vector space V such that k Wi = V. i=1 Prove that V is the direct sum of W1 , W2 , . . . , Wk if and only if k dim(V) = dim(Wi ). i=1 21.Let V be a finite-dimensional vector space with a basis β, and let β1 , β2 , . . . , βk be a partition of β (i.e., β1 , β2 , . . . , βk are subsets of β such that β = β1 ∪ β2 ∪ · · · ∪ βk and βi ∩ βj = ∅ if i = j). Prove that V = span(β1 ) ⊕ span(β2 ) ⊕ · · · ⊕ span(βk ). 22.Let T be a linear operator on a finite-dimensional vector space V, and suppose that the distinct eigenvalues of T are λ1 , λ2 , . . . , λk . Prove that span({x ∈ V : x is an eigenvector of T}) = Eλ1 ⊕ Eλ2 ⊕ · · · ⊕ Eλk . 23.Let W1 , W2, K1 , K2 , . . . , Kp , M1 , M2 , . . . , Mq be subspaces of a vector space V such that W1 = K1 ⊕K2 ⊕· · ·⊕Kp and W2 = M1 ⊕M2 ⊕· · ·⊕Mq . Prove that if W1 ∩ W2 = {0 }, then W1 + W2 = W1 ⊕ W2 = K1 ⊕ K2 ⊕ · · · ⊕ Kp ⊕ M1 ⊕ M2 ⊕ · · · ⊕ Mq . 1.Label the following statements as true or false. (a) If A ∈ Mn×n (C) and lim Am = L, then, for any invertible matrix m→∞ Q ∈ Mn×n (C), we have lim QAm Q−1 = QLQ−1 . m→∞ (b) If 2 is an eigenvalue of A ∈ Mn×n (C), then lim Am does not m→∞ exist. (c) Any vector ⎛ ⎞ x 1 ⎜ x 2 ⎟ ⎜ ⎟ ⎜ ⎝ .. . ⎟ ⎠ ∈ Rn xn (d)(e)such that x1 + x2 + · · · + xn = 1 is a probability vector. The sum of the entries of each row of a transition matrix equals 1. The product of a transition matrix and a probability vector is a probability vector. 308 Chap. 5 Diagonalization (f ) Let z be any complex number such that |z| < 1. Then the matrix ⎛ ⎞ 1 z −1 ⎝ z 1 1⎠ −1 1 z does not have 3 as an eigenvalue. (g) Every transition matrix has 1 as an eigenvalue. (h) No transition matrix can have −1 as an eigenvalue. (i) If A is a transition matrix, then lim Am exists. m→∞ (j) If A is a regular transition matrix, then lim Am exists and has m→∞ rank 1. 2. Determine whether lim Am exists for each of the following matrices m→∞ A, and compute the limit if it exists. 0.1 0.7 −1.4 0.8 0.4 0.7 (a) (b) (c) 0.7 0.1 −2.4 1.8 0.6 0.3 −1.8 4.8 −2 −1 2.0 −0.5 (d) (e) (f ) −0.8 2.2 4 3 3.0 −0.5 ⎛ ⎞ ⎛ ⎞ −1.8 0 −1.4 3.4 −0.2 0.8 (g) ⎝−5.6 1 −2.8⎠ (h) ⎝ 3.9 1.8 1.3⎠ 2.8 0 2.4 −16.5 −2.0 −4.5 ⎛ ⎞ − 1 2 − 2i 4i 1 2 + 5i ⎜ ⎟ (i) ⎝ 1 + 2i −3i −1 − 4i⎠ −1 − 2i 4i 1 + 5i ⎛ ⎞ −26 + i −28 − 4i ⎜ 3 3 28 ⎟ ⎜ ⎟ ⎜ ⎟ (j) ⎜ ⎜ ⎜ −7 3 + 2i −5 3 + i 7 − 2i ⎟ ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ + 6i + 6i 35 20i ⎠ −13 −5 −6 6 6 3. Prove that if A1 , A2 , . . . is a sequence of n × p matrices with complex entries such that lim Am = L, then lim (Am )t = Lt . m→∞ m→∞ 4. Prove that if A ∈ Mn×n (C) is diagonalizable and L = lim Am exists, m→∞ then either L = In or rank(L) < n. Sec. 5.3 Matrix Limits and Markov Chains 309 5.Find 2 × 2 matrices A and B having real entries such that lim Am , m→∞ lim B m , and lim (AB)m all exist, but m→∞ m→∞ lim (AB)m = ( lim Am )( lim B m ). m→∞ m→∞ m→∞ 6.A hospital trauma unit has determined that 30% of its patients are ambulatory and 70% are bedridden at the time of arrival at the hospital. A month after arrival, 60% of the ambulatory patients have recovered, 20% remain ambulatory, and 20% have become bedridden. After the same amount of time, 10% of the bedridden patients have recovered, 20% have become ambulatory, 50% remain bedridden, and 20% have died. Determine the percentages of patients who have recovered, are ambulatory, are bedridden, and have died 1 month after arrival. Also determine the eventual percentages of patients of each type. 7.A player begins a game of chance by placing a marker in box 2, marked Start. (See Figure 5.5.) A die is rolled, and the marker is moved one square to the left if a 1 or a 2 is rolled and one square to the right if a 3, 4, 5, or 6 is rolled. This process continues until the marker lands in square 1, in which case the player wins the game, or in square 4, in which case the player loses the game. What is the probability of winning this game? Hint: Instead of diagonalizing the appropriate transition matrix Win Start Lose 1 2 3 4 Figure 5.5 A, it is easier to represent e2 as a linear combination of eigenvectors of A and then apply An to the result. 8. Which of the following transition matrices are regular? ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0.2 0.3 0.5 0.5 0 1 0.5 0 0 (a) ⎝0.3 0.2 0.5⎠ (b) ⎝0.5 0 0⎠ (c) ⎝0.5 0 1⎠ 0.5 0.5 0 0 1 0 0 1 0 ⎛ 1 ⎞ ⎛ ⎞ 3 0 0 ⎛ ⎞ 0.5 0 1 ⎜ 1 ⎟ 1 0 0 (d) ⎝0.5 1 0⎠ (e) ⎜ 3 1 0⎟ (f ) ⎝0 0.7 0.2⎠ ⎝ ⎠ 0 0 0 1 0 1 0 0.3 0.8 3 310 Chap. 5 Diagonalization ⎛ 1 ⎞ ⎛ 1 1 ⎞ 0 0 0 0 0 2 4 4 ⎜ 1 ⎟ ⎜ 1 1 ⎟ ⎜ 2 0 0 0⎟ ⎜ 4 4 0 0⎟ ⎜ ⎟ ⎜ ⎟ (g) ⎜ 1 1 ⎟ (h) ⎜ 1 1 ⎟ ⎜ 4 4 1 0 ⎟ ⎜ 4 4 1 0⎟ ⎝ ⎠ ⎝ ⎠ 1 1 1 1 0 1 0 1 4 4 4 4 9. Compute lim Am if it exists, for each matrix A in Exercise 8. m→∞ 10.Each of the matrices that follow is a regular transition matrix for a three-state Markov chain. In all cases, the initial probability vector is ⎛ ⎞ 0.3 P = ⎝0.3⎠ . 0.4 For each transition matrix, compute the proportions of objects in each state after two stages and the eventual proportions of objects in each state by determining the fixed probability vector. ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0.6 0.1 0.1 0.8 0.1 0.2 0.9 0.1 0.1 (a) ⎝0.1 0.9 0.2⎠ (b) ⎝0.1 0.8 0.2⎠ (c) ⎝0.1 0.6 0.1⎠ 0.3 0 0.7 0.1 0.1 0.6 0 0.3 0.8 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0.4 0.2 0.2 0.5 0.3 0.2 0.6 0 0.4 (d) ⎝0.1 0.7 0.2⎠ (e) ⎝0.2 0.5 0.3⎠ (f ) ⎝0.2 0.8 0.2⎠ 0.5 0.1 0.6 0.3 0.2 0.5 0.2 0.2 0.4 11.In 1940, a county land-use survey showed that 10% of the county land was urban, 50% was unused, and 40% was agricultural. Five years later, a follow-up survey revealed that 70% of the urban land had remained urban, 10% had become unused, and 20% had become agricultural. Likewise, 20% of the unused land had become urban, 60% had remained unused, and 20% had become agricultural. Finally, the 1945 survey showed that 20% of the agricultural land had become unused while 80% remained agricultural. Assuming that the trends indicated by the 1945 survey continue, compute the percentages of urban, unused, and agricultural land in the county in 1950 and the corresponding eventual percentages. 12.A diaper liner is placed in each diaper worn by a baby. If, after a diaper change, the liner is soiled, then it is discarded and replaced by a new liner. Otherwise, the liner is washed with the diapers and reused, except that each liner is discarded and replaced after its third use (even if it has never been soiled). The probability that the baby will soil any diaper liner is one-third. If there are only new diaper liners at first, eventually what proportions of the diaper liners being used will be new, Sec. 5.3 Matrix Limits and Markov Chains 311 once used, and twice used? Hint: Assume that a diaper liner ready for use is in one of three states: new, once used, and twice used. After its use, it then transforms into one of the three states described. 13.In 1975, the automobile industry determined that 40% of American car owners drove large cars, 20% drove intermediate-sized cars, and 40% drove small cars. A second survey in 1985 showed that 70% of the large- car owners in 1975 still owned large cars in 1985, but 30% had changed to an intermediate-sized car. Of those who owned intermediate-sized cars in 1975, 10% had switched to large cars, 70% continued to drive intermediate-sized cars, and 20% had changed to small cars in 1985. Finally, of the small-car owners in 1975, 10% owned intermediate-sized cars and 90% owned small cars in 1985. Assuming that these trends continue, determine the percentages of Americans who own cars of each size in 1995 and the corresponding eventual percentages. 14. Show that if A and P are as in Example 5, then ⎛ ⎞ rm rm+1 rm+1 Am = ⎝rm+1 rm rm+1⎠ , rm+1 rm+1 rm where $ % 1 (−1)m rm = 1 + . 3 2 m−1Deduce that ⎛ ⎞ (−1)m ⎛300 ⎞ ⎜ 200 + 2m (100) ⎟ ⎜ ⎟ 600(Am P ) = Am ⎝200⎠ = ⎜ ⎜ 200 ⎟ ⎟ . ⎜ ⎟ 100 ⎝ (−1) m+1 ⎠ 200 + (100) 2m 15. Prove that if a 1-dimensional subspace W of Rn contains a nonzero vec- tor with all nonnegative entries, then W contains a unique probability vector. 16. Prove Theorem 5.15 and its corollary. 17. Prove the two corollaries of Theorem 5.18. 18. Prove the corollary of Theorem 5.19. 19. Suppose that M and M are n × n transition matrices. 312 Chap. 5 Diagonalization (a)(b)(c)Prove that if M is regular, N is any n × n transition matrix, and c is a real number such that 0 < c ≤ 1, then cM + (1 − c)N is a regular transition matrix. Suppose that for all i, j, we have that Mij > 0 whenever Mij > 0. Prove that there exists a transition matrix N and a real number c with 0 < c ≤ 1 such that M = cM + (1 − c)N . Deduce that if the nonzero entries of M and M occur in the same positions, then M is regular if and only if M is regular. The following definition is used in Exercises 20–24. Definition. For A ∈ Mn×n (C), define eA = lim Bm , where m→∞ A2 Am Bm = I + A + + ··· + 2! m! (see Exercise 22). Thus eA is the sum of the infinite series A2 A3 I + A + + + ··· , 2! 3! and Bm is the mth partial sum of this series. (Note the analogy with the power series a2 a3 ea = 1 + a + + + ··· , 2! 3! which is valid for all complex numbers a.) 20. Compute eO and eI , where O and I denote the n × n zero and identity matrices, respectively. 21. Let P −1 AP = D be a diagonal matrix. Prove that eA = P eD P −1. 22. Let A ∈ Mn×n (C) be diagonalizable. Use the result of Exercise 21 to show that eA exists. (Exercise 21 of Section 7.2 shows that eA exists for every A ∈ Mn×n (C).) 23. Find A, B ∈ M2×2 (R) such that eA eB = eA+B . 24. Prove that a differentiable function x : R → Rn is a solution to the system of differential equations defined in Exercise 15 of Section 5.2 if and only if x(t) = etA v for some v ∈ Rn , where A is defined in that exercise. 1. Label the following statements as true or false. (a) There exists a linear operator T with no T-invariant subspace. (b) If T is a linear operator on a finite-dimensional vector space V and W is a T-invariant subspace of V, then the characteristic polyno- mial of TW divides the characteristic polynomial of T. (c) Let T be a linear operator on a finite-dimensional vector space V, and let v and w be in V. If W is the T-cyclic subspace generated by v, W is the T-cyclic subspace generated by w, and W = W , then v = w. (d) If T is a linear operator on a finite-dimensional vector space V, then for any v ∈ V the T-cyclic subspace generated by v is the same as the T-cyclic subspace generated by T(v). (e) Let T be a linear operator on an n-dimensional vector space. Then there exists a polynomial g(t) of degree n such that g(T) = T0. (f ) Any polynomial of degree n with leading coefficient (−1)n is the characteristic polynomial of some linear operator. (g) If T is a linear operator on a finite-dimensional vector space V, and if V is the direct sum of k T-invariant subspaces, then there is an ordered basis β for V such that [T]β is a direct sum of k matrices. 322 Chap. 5 Diagonalization 2.For each of the following linear operators T on the vector space V, determine whether the given subspace W is a T-invariant subspace of V. (a) V = P3 (R), T(f (x)) = f (x), and W = P2 (R) (b) V = P(R), T(f (x)) = xf (x), and W = P2 (R) (c) V = R3 , T(a, b, c) = (a + b + c, a + b + c, a + b + c), and W = {(t, t, t) : t ∈ R}& 1 ' (d) V = C([0, 1]), T(f (t)) = f (x) dx t, and 0W = {f ∈ V : f (t) = at + b for some a and b} 0 1 (e) V = M2×2 (R), T(A) = A, and W = {A ∈ V : At = A} 1 0 3.Let T be a linear operator on a finite-dimensional vector space V. Prove that the following subspaces are T-invariant. (a) {0 } and V (b) N(T) and R(T) (c) Eλ , for any eigenvalue λ of T 4. Let T be a linear operator on a vector space V, and let W be a T- invariant subspace of V. Prove that W is g(T)-invariant for any poly- nomial g(t). 5.Let T be a linear operator on a vector space V. Prove that the inter- section of any collection of T-invariant subspaces of V is a T-invariant subspace of V. 6. For each linear operator T on the vector space V, find an ordered basis for the T-cyclic subspace generated by the vector z. (a) V = R4 , T(a, b, c, d) = (a + b, b − c, a + c, a + d), and z = e1 . (b) V = P3 (R), T(f (x)) = f (x), and z = x3 . 0 1 (c) V = M2×2 (R), T(A) = At , and z = . 1 0 1 1 0 1 (d) V = M2×2 (R), T(A) = A, and z = . 2 2 1 0 7.Prove that the restriction of a linear operator T to a T-invariant sub- space is a linear operator on that subspace. 8.Let T be a linear operator on a vector space with a T-invariant subspace W. Prove that if v is an eigenvector of TW with corresponding eigenvalue λ, then the same is true for T. 9.For each linear operator T and cyclic subspace W in Exercise 6, compute the characteristic polynomial of TW in two ways, as in Example 6. Sec. 5.4 Invariant Subspaces and the Cayley–Hamilton Theorem 323 10.For each linear operator in Exercise 6, find the characteristic polynomial f (t) of T, and verify that the characteristic polynomial of TW (computed in Exercise 9) divides f (t). 11.12.Let T be a linear operator on a vector space V, let v be a nonzero vector in V, and let W be the T-cyclic subspace of V generated by v. Prove that (a) W is T-invariant. (b) Any T-invariant subspace of V containing v also contains W. B1 B2 Prove that A = in the proof of Theorem 5.21. O B3 13.Let T be a linear operator on a vector space V, let v be a nonzero vector in V, and let W be the T-cyclic subspace of V generated by v. For any w ∈ V, prove that w ∈ W if and only if there exists a polynomial g(t) such that w = g(T)(v). 14.Prove that the polynomial g(t) of Exercise 13 can always be chosen so that its degree is less than or equal to dim(W). 15.Use the Cayley–Hamilton theorem (Theorem 5.23) to prove its corol- lary for matrices. Warning: If f (t) = det(A − tI) is the characteristic polynomial of A, it is tempting to “prove” that f (A) = O by saying “f (A) = det(A − AI) = det(O) = 0.” But this argument is nonsense. Why? 16. Let T be a linear operator on a finite-dimensional vector space V. (a) Prove that if the characteristic polynomial of T splits, then so does the characteristic polynomial of the restriction of T to any T-invariant subspace of V. (b) Deduce that if the characteristic polynomial of T splits, then any nontrivial T-invariant subspace of V contains an eigenvector of T. 17. Let A be an n × n matrix. Prove that dim(span({In , A, A2 , . . .})) ≤ n. 18.Let A be an n × n matrix with characteristic polynomial f (t) = (−1)ntn + an−1 tn−1 + · · · + a1 t + a0 . (a) Prove that A is invertible if and only if a0 = 0. (b) Prove that if A is invertible, then A−1 = (−1/a0 )[(−1)n An−1 + an−1 An−2 + · · · + a1 In ]. 324 Chap. 5 Diagonalization (c) Use (b) to compute A−1 for ⎛ ⎞ 1 2 1 A = ⎝0 2 3⎠ . 0 0 −1 19. Let A denote the k × k matrix ⎛ ⎞ 0 0 ··· 0 −a0 ⎜1 0 · · · 0 −a1 ⎟ ⎜ ⎟ ⎜0 1 · · · 0 −a2 ⎟ ⎜ ⎟ ⎜ ⎜ .. . .. . .. . .. . ⎟ ⎟ , ⎜ ⎟ ⎝0 0 · · · 0 −ak−2 ⎠ 0 0 ··· 1 −ak−1 where a0 , a1 , . . . , ak−1 are arbitrary scalars. Prove that the character- istic polynomial of A is (−1)k (a0 + a1 t + · · · + ak−1 tk−1 + tk ). Hint: Use mathematical induction on k, expanding the determinant along the first row. 20. Let T be a linear operator on a vector space V, and suppose that V is a T-cyclic subspace of itself. Prove that if U is a linear operator on V, then UT = TU if and only if U = g(T) for some polynomial g(t). Hint: Suppose that V is generated by v. Choose g(t) according to Exercise 13 so that g(T)(v) = U(v). 21.Let T be a linear operator on a two-dimensional vector space V. Prove that either V is a T-cyclic subspace of itself or T = cI for some scalar c. 22. Let T be a linear operator on a two-dimensional vector space V and suppose that T = cI for any scalar c. Show that if U is any linear operator on V such that UT = TU, then U = g(T) for some polynomial g(t). 23.Let T be a linear operator on a finite-dimensional vector space V, and let W be a T-invariant subspace of V. Suppose that v1 , v2 , . . . , vk are eigenvectors of T corresponding to distinct eigenvalues. Prove that if v1 + v2 + · · · + vk is in W, then vi ∈ W for all i. Hint: Use mathematical induction on k. 24. Prove that the restriction of a diagonalizable linear operator T to any nontrivial T-invariant subspace is also diagonalizable. Hint: Use the result of Exercise 23. Sec. 5.4 Invariant Subspaces and the Cayley–Hamilton Theorem 325 25. (a) Prove the converse to Exercise 18(a) of Section 5.2: If T and U are diagonalizable linear operators on a finite-dimensional vector space V such that UT = TU, then T and U are simultaneously diagonalizable. (See the definitions in the exercises of Section 5.2.) Hint: For any eigenvalue λ of T, show that Eλ is U-invariant, and apply Exercise 24 to obtain a basis for Eλ of eigenvectors of U. (b) State and prove a matrix version of (a). 26.Let T be a linear operator on an n-dimensional vector space V such that T has n distinct eigenvalues. Prove that V is a T-cyclic subspace of itself. Hint: Use Exercise 23 to find a vector v such that {v, T(v), . . . , Tn−1 (v)} is linearly independent. Exercises 27 through 32 require familiarity with quotient spaces as defined in Exercise 31 of Section 1.3. Before attempting these exercises, the reader should first review the other exercises treating quotient spaces: Exercise 35 of Section 1.6, Exercise 40 of Section 2.1, and Exercise 24 of Section 2.4. For the purposes of Exercises 27 through 32, T is a fixed linear operator on a finite-dimensional vector space V, and W is a nonzero T-invariant subspace of V. We require the following definition. Definition. Let T be a linear operator on a vector space V, and let W be a T-invariant subspace of V. Define T : V/W → V/W by T(v + W) = T(v) + W for any v + W ∈ V/W. 27. (a) Prove that T is well defined. That is, show that T(v + W) = T(v + W) whenever v + W = v + W. (b) Prove that T is a linear operator on V/W. (c) Let η : V → V/W be the linear transformation defined in Exer- cise 40 of Section 2.1 by η(v) = v + W. Show that the diagram of Figure 5.6 commutes; that is, prove that ηT = Tη. (This exercise does not require the assumption that V is finite-dimensional.) V −−−−→ T V ⏐ ⏐ ⏐ ⏐η η! ! V/W −−−−→ T V/W Figure 5.6 28.Let f (t), g(t), and h(t) be the characteristic polynomials of T, TW , and T, respectively. Prove that f (t) = g(t)h(t). Hint: Extend an ordered basis γ = {v1 , v 2 , . . . , vk } for W to an ordered basis β = {v1 , v2 , . . . , vk , vk+1 , . . . , vn } for V. Then show that the collection of 326 Chap. 5 Diagonalization cosets α = {vk+1 + W, vk+2 + W, . . . , vn + W} is an ordered basis for V/W, and prove that B1 B2 [T]β = , O B3 29.30.where B1 = [T]γ and B3 = [T]α . Use the hint in Exercise 28 to prove that if T is diagonalizable, then so is T. Prove that if both TW and T are diagonalizable and have no common eigenvalues, then T is diagonalizable. The results of Theorem 5.22 and Exercise 28 are useful in devising methods for computing characteristic polynomials without the use of determinants. This is illustrated in the next exercise. ⎛ ⎞ 1 1 −3 31. Let A = ⎝2 3 4⎠, let T = LA , and let W be the cyclic subspace 1 2 1 of R3 generated by e1 . (a)(b)(c)Use Theorem 5.22 to compute the characteristic polynomial of TW . Show that {e2 + W} is a basis for R3 /W, and use this fact to compute the characteristic polynomial of T. Use the results of (a) and (b) to find the characteristic polynomial of A. 32. Prove the converse to Exercise 9(a) of Section 5.2: If the characteristic polynomial of T splits, then there is an ordered basis β for V such that [T]β is an upper triangular matrix. Hints: Apply mathematical induction to dim(V). First prove that T has an eigenvector v, let W = span({v}), and apply the induction hypothesis to T : V/W → V/W. Exercise 35(b) of Section 1.6 is helpful here. Exercises 33 through 40 are concerned with direct sums. 33.Let T be a linear operator on a vector space V, and let W1 , W2 , . . . , Wk be T-invariant subspaces of V. Prove that W1 + W2 + · · · + Wk is also a T-invariant subspace of V. 34.Give a direct proof of Theorem 5.25 for the case k = 2. (This result is used in the proof of Theorem 5.24.) 35.Prove Theorem 5.25. Hint: Begin with Exercise 34 and extend it using mathematical induction on k, the number of subspaces. Sec. 5.4 Invariant Subspaces and the Cayley–Hamilton Theorem 327 36. Let T be a linear operator on a finite-dimensional vector space V. Prove that T is diagonalizable if and only if V is the direct sum of one-dimensional T-invariant subspaces. 37.Let T be a linear operator on a finite-dimensional vector space V, and let W1 , W2 , . . . , Wk be T-invariant subspaces of V such that V = W1 ⊕ W2 ⊕ · · · ⊕ Wk . Prove that det(T) = det(TW1 ) det(TW2 ) · · · det(TWk ). 38.Let T be a linear operator on a finite-dimensional vector space V, and let W1 , W2 , . . . , Wk be T-invariant subspaces of V such that V = W1 ⊕ W2 ⊕ · · · ⊕ Wk . Prove that T is diagonalizable if and only if TWi is diagonalizable for all i. 39. Let C be a collection of diagonalizable linear operators on a finite- dimensional vector space V. Prove that there is an ordered basis β such that [T]β is a diagonal matrix for all T ∈ C if and only if the operators of C commute under composition. (This is an extension of Exercise 25.) Hints for the case that the operators commute: The result is trivial if each operator has only one eigenvalue. Otherwise, establish the general result by mathematical induction on dim(V), using the fact that V is the direct sum of the eigenspaces of some operator in C that has more than one eigenvalue. 40.Let B1 , B2 , . . . , Bk be square matrices with entries in the same field, and let A = B1 ⊕ B2 ⊕ · · · ⊕ Bk . Prove that the characteristic polynomial of A is the product of the characteristic polynomials of the Bi ’s. 41. Let ⎛ ⎞ 1 2 ··· n ⎜ n +1 n +2 · · · 2n⎟ ⎜ ⎟ A = ⎜ ⎝ .. . .. . .. . ⎟ ⎠ . n2 − n + 1 n2 − n + 2 · · · n2 Find the characteristic polynomial of A. Hint: First prove that A has rank 2 and that span({(1, 1, . . . , 1), (1, 2, . . . , n)}) is LA -invariant. 42. Let A ∈ Mn×n (R) be the matrix defined by Aij = 1 for all i and j. Find the characteristic polynomial of A. </span></div><div style="text-align:center;margin:10px 0 10px 0;clear:both"><div style="display:inline;text-align:center;"><script async src="https://pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script>
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Linear Algebrahashnuthttps://koreanfoodie.me/93https://koreanfoodie.me/93#entry93commentSat, 15 Jun 2019 13:18:56 +0900Solutions to Linear Algebra, Stephen H. Friedberg, Fourth Edition (Chapter 4)
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data-url='https://k.kakaocdn.net/dn/t2ny9/btqv5gfcjY2/6sCmlwzd7emAFmPJ1SZgkK/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/t2ny9/btqv5gfcjY2/6sCmlwzd7emAFmPJ1SZgkK/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2Ft2ny9%2Fbtqv5gfcjY2%2F6sCmlwzd7emAFmPJ1SZgkK%2Fimg.jpg' data-filename="chapter4_3.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/bR6uLb/btqv8lMqfaB/6a4LXRglpx8PHMj0Z2AAS0/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/bR6uLb/btqv8lMqfaB/6a4LXRglpx8PHMj0Z2AAS0/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FbR6uLb%2Fbtqv8lMqfaB%2F6a4LXRglpx8PHMj0Z2AAS0%2Fimg.jpg' data-filename="chapter4_4.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/s2zSf/btqv4RmoLCT/kazZJA55pFh7OEu9xfbtKK/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/s2zSf/btqv4RmoLCT/kazZJA55pFh7OEu9xfbtKK/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2Fs2zSf%2Fbtqv4RmoLCT%2FkazZJA55pFh7OEu9xfbtKK%2Fimg.jpg' data-filename="chapter4_5.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/AWdrd/btqv6ENqy3n/o492kbSnw5lunwZ36ZkvuK/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/AWdrd/btqv6ENqy3n/o492kbSnw5lunwZ36ZkvuK/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FAWdrd%2Fbtqv6ENqy3n%2Fo492kbSnw5lunwZ36ZkvuK%2Fimg.jpg' data-filename="chapter4_6.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/8Bsws/btqv5gGfjq3/BiRnlXtRWBCeHi1xlMDcXK/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/8Bsws/btqv5gGfjq3/BiRnlXtRWBCeHi1xlMDcXK/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2F8Bsws%2Fbtqv5gGfjq3%2FBiRnlXtRWBCeHi1xlMDcXK%2Fimg.jpg' data-filename="chapter4_7.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/pfKt6/btqv5WHxrUW/Ts4nwcKoCx5EFdsEOqk3ck/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/pfKt6/btqv5WHxrUW/Ts4nwcKoCx5EFdsEOqk3ck/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FpfKt6%2Fbtqv5WHxrUW%2FTs4nwcKoCx5EFdsEOqk3ck%2Fimg.jpg' data-filename="chapter4_8.jpg"></span></figure><figure class='imageblock alignCenter'><span 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data-url='https://k.kakaocdn.net/dn/taKx0/btqv4QueTMX/WRmBstyaoiK96lkB82cKS1/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/taKx0/btqv4QueTMX/WRmBstyaoiK96lkB82cKS1/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FtaKx0%2Fbtqv4QueTMX%2FWRmBstyaoiK96lkB82cKS1%2Fimg.jpg' data-filename="chapter4_11.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/cI6QC4/btqv4P3d2LI/snLZaRbFpWJgR4G9abT1KK/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/cI6QC4/btqv4P3d2LI/snLZaRbFpWJgR4G9abT1KK/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FcI6QC4%2Fbtqv4P3d2LI%2FsnLZaRbFpWJgR4G9abT1KK%2Fimg.jpg' data-filename="chapter4_12.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/VlYRu/btqv7o4opG2/QtAnKyIpeKFZ90VQNAHKPk/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/VlYRu/btqv7o4opG2/QtAnKyIpeKFZ90VQNAHKPk/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FVlYRu%2Fbtqv7o4opG2%2FQtAnKyIpeKFZ90VQNAHKPk%2Fimg.jpg' data-filename="chapter4_13.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/brYo3s/btqv5VhAmMd/KKKRWjASdcKaaMki8b4R41/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/brYo3s/btqv5VhAmMd/KKKRWjASdcKaaMki8b4R41/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FbrYo3s%2Fbtqv5VhAmMd%2FKKKRWjASdcKaaMki8b4R41%2Fimg.jpg' data-filename="chapter4_14.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/l9upH/btqv4QueTTb/quK0ypqnGZUPRw382dQnJ1/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/l9upH/btqv4QueTTb/quK0ypqnGZUPRw382dQnJ1/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2Fl9upH%2Fbtqv4QueTTb%2FquK0ypqnGZUPRw382dQnJ1%2Fimg.jpg' data-filename="chapter4_15.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/qv0dO/btqv7hEszQi/OaaVRo4orrgI6g9XLFau9k/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/qv0dO/btqv7hEszQi/OaaVRo4orrgI6g9XLFau9k/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2Fqv0dO%2Fbtqv7hEszQi%2FOaaVRo4orrgI6g9XLFau9k%2Fimg.jpg' data-filename="chapter4_16.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/bRK69s/btqv5fUSsV6/sySf6FsOUFYlxMvnYpN061/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/bRK69s/btqv5fUSsV6/sySf6FsOUFYlxMvnYpN061/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FbRK69s%2Fbtqv5fUSsV6%2FsySf6FsOUFYlxMvnYpN061%2Fimg.jpg' data-filename="chapter4_17.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/blmzcD/btqv7YcRl7K/wUPePtLA9WGak74hojvHKk/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/blmzcD/btqv7YcRl7K/wUPePtLA9WGak74hojvHKk/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FblmzcD%2Fbtqv7YcRl7K%2FwUPePtLA9WGak74hojvHKk%2Fimg.jpg' data-filename="chapter4_18.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/xSriM/btqv7qHUJMQ/trCcnrdAAK7Eb11RdYGKr0/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/xSriM/btqv7qHUJMQ/trCcnrdAAK7Eb11RdYGKr0/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FxSriM%2Fbtqv7qHUJMQ%2FtrCcnrdAAK7Eb11RdYGKr0%2Fimg.jpg' data-filename="chapter4_19.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/Jeppz/btqv5gsKQh9/kGreE9uqHscKsI1VKdD9s0/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/Jeppz/btqv5gsKQh9/kGreE9uqHscKsI1VKdD9s0/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FJeppz%2Fbtqv5gsKQh9%2FkGreE9uqHscKsI1VKdD9s0%2Fimg.jpg' data-filename="chapter4_20.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/b09mho/btqv4rVIkiI/MAKkY86G7F9KeL98tZxLAK/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/b09mho/btqv4rVIkiI/MAKkY86G7F9KeL98tZxLAK/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2Fb09mho%2Fbtqv4rVIkiI%2FMAKkY86G7F9KeL98tZxLAK%2Fimg.jpg' data-filename="chapter4_21.jpg"></span></figure><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/cdMsQr/btqv5gsKQoF/BtD1CvNKK8kwnYXUXqpet1/img.jpg' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/cdMsQr/btqv5gsKQoF/BtD1CvNKK8kwnYXUXqpet1/img.jpg' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FcdMsQr%2Fbtqv5gsKQoF%2FBtD1CvNKK8kwnYXUXqpet1%2Fimg.jpg' data-filename="chapter4_22.jpg"></span></figure></p>
<p>Solution maual to Linear Algebra, Fourth Edition, Stephen H. Friedberg. (Chapter 4)<br /><br />Solutions to Linear Algebra, Fourth Edition, Stephen H. Friedberg. (Chapter 4)<br /><br />Linear Algebra solution manual, Fourth Edition, Stephen H. Friedberg. (Chapter 4)<br /><br />Linear Algebra solutions Friedberg. (Chapter 4)</p>
<div style="width: 100%; height: 1px; overflow: auto;"><span>1.Label the following statements as true or false. (a) The function det : M2×2 (F ) → F is a linear transformation. (b) The determinant of a 2 × 2 matrix is a linear function of each row of the matrix when the other row is held fixed. (c) If A ∈ M2×2 (F ) and det(A) = 0, then A is invertible. (d) If u and v are vectors in R2 emanating from the origin, then the area of the parallelogram having u and v as adjacent sides is u det . v 208 Chap. 4 Determinants (e) A coordinate system is right-handed if and only if its orientation equals 1. 2. Compute the determinants of the following matrices in M2×2 (R). 6 −3 −5 2 8 0 (a) (b) (c) 2 4 6 1 3 −1 3. Compute the determinants of the following matrices in M2×2 (C). −1 + i 1 − 4i 5 − 2i 6 + 4i 2i 3 (a) (b) (c) 3 + 2i 2 − 3i −3 + i 7i 4 6i 4. For each of the following pairs of vectors u and v in R2 , compute the area of the parallelogram determined by u and v. (a) u = (3, −2) and v = (2, 5) (b) u = (1, 3) and v = (−3, 1) (c) u = (4, −1) and v = (−6, −2) (d) u = (3, 4) and v = (2, −6) 5. Prove that if B is the matrix obtained by interchanging the rows of a 2 × 2 matrix A, then det(B) = − det(A). 6. Prove that if the two columns of A ∈ M2×2 (F ) are identical, then det(A) = 0. 7. Prove that det(At ) = det(A) for any A ∈ M2×2 (F ). 8. Prove that if A ∈ M2×2 (F ) is upper triangular, then det(A) equals the product of the diagonal entries of A. 9. Prove that det(AB) = det(A)· det(B) for any A, B ∈ M2×2 (F ). 10. The classical adjoint of a 2 × 2 matrix A ∈ M2×2 (F ) is the matrix A22 −A12 C = . −A21 A11 Prove that (a) CA = AC = [det(A)]I. (b) det(C) = det(A). (c) The classical adjoint of At is C t . (d) If A is invertible, then A−1 = [det(A)]−1 C. 11. Let δ : M2×2 (F ) → F be a function with the following three properties. (i) δ is a linear function of each row of the matrix when the other row is held fixed. (ii) If the two rows of A ∈ M2×2 (F ) are identical, then δ(A) = 0. Sec. 4.2 Determinants of Order n 209 (iii) If I is the 2 × 2 identity matrix, then δ(I) = 1. Prove that δ(A) = det(A) for all A ∈ M2×2 (F ). (This result is general- ized in Section 4.5.) 12.Let {u, v} be an ordered basis for R2 . Prove that u O =1 v if and only if {u, v} forms a right-handed coordinate system. Hint: Recall the definition of a rotation given in Example 2 of Section 2.1. 1.Label(a)(b)(c)(d)(e)(f )(g)(h)the following statements as true or false. The function det : Mn×n (F ) → F is a linear transformation. The determinant of a square matrix can be evaluated by cofactor expansion along any row. If two rows of a square matrix A are identical, then det(A) = 0. If B is a matrix obtained from a square matrix A by interchanging any two rows, then det(B) = − det(A). If B is a matrix obtained from a square matrix A by multiplying a row of A by a scalar, then det(B) = det(A). If B is a matrix obtained from a square matrix A by adding k times row i to row j, then det(B) = k det(A). If A ∈ Mn×n (F ) has rank n, then det(A) = 0. The determinant of an upper triangular matrix equals the product of its diagonal entries. Sec. 4.2 Determinants of Order n 2. Find the value of k that satisfies the following equation: ⎛ ⎞ ⎛ ⎞ 3a1 3a2 3a3 a1 a2 a3 det ⎝ 3b1 3b2 3b3 ⎠ = k det ⎝ b1 b2 b3 ⎠ . 3c1 3c2 3c3 c1 c2 c3 221 3. Find the value of k that satisfies the following equation: ⎛ ⎞ ⎛ ⎞ 2a1 2a2 2a3 a1 a2 a3 det ⎝3b1 + 5c1 3b2 + 5c2 3b3 + 5c3 ⎠ = k det ⎝ b1 b2 b3 ⎠ . 7c1 7c2 7c3 c1 c2 c3 4. Find the value of k that satisfies the following equation: ⎛ ⎞ ⎛ ⎞ b1 + c1 b2 + c2 b3 + c3 a1 a2 a3 det ⎝a1 + c1 a2 + c2 a3 + c3 ⎠ = k det ⎝ b1 b2 b3⎠ . a1 + b1 a2 + b2 a3 + b3 c1 c2 c3 In Exercises 5–12, evaluate the determinant of the given matrix by cofactor expansion along the indicated row. ⎛ ⎞ ⎛ ⎞ 0 1 2 1 0 2 ⎝−1 0 −3⎠ ⎝ 0 1 5⎠ 5. 6. 2 3 0 −1 3 0 along the first row along the first row ⎛ ⎞ ⎛ ⎞ 0 1 2 1 0 2 ⎝−1 0 −3⎠ ⎝ 0 1 5⎠ 7. 8. 2 3 0 −1 3 0 along the second row along the third row ⎛ ⎞ ⎛ ⎞ 0 1+ i 2 i 2+ i 0 ⎝−2i 0 1 − i⎠ ⎝−1 3 2i ⎠ 9. 10. 3 4i 0 0 −1 1 − i along the third row along the second row ⎛ ⎞ ⎛ ⎞ 0 2 1 3 1 −1 2 −1 ⎜ ⎜ 1 0 −2 2⎟ ⎟ ⎜−3 ⎜ 4 1 −1⎟ ⎟ 11. ⎝ 3 −1 0 1⎠ 12. ⎝ 2 −5 −3 8⎠ −1 1 2 0 −2 6 −4 1 along the fourth row along the fourth row In Exercises 13–22, evaluate the determinant of the given matrix by any le- gitimate method. 222 Chap. 4 Determinants ⎛ ⎞ ⎛ ⎞ 0 0 1 2 3 4 13. ⎝0 2 3⎠ 14. ⎝5 6 0⎠ 4 5 6 7 0 0 ⎛ ⎞ ⎛ ⎞ 1 2 3 −1 3 2 15. ⎝4 5 6⎠ 16. ⎝ 4 −8 1⎠ 7 8 9 2 2 5 ⎛ ⎞ ⎛ ⎞ 0 1 1 1 −2 3 17. ⎝ 1 2 −5⎠ 18. ⎝−1 2 −5⎠ 6 −4 3 3 −1 2 ⎛ ⎞ ⎛ ⎞ i 2 −1 −1 2 + i 3 19. ⎝ 3 1+ i 2 ⎠ 20. ⎝1 − i i 1 ⎠ −2i 1 4 − i 3i 2 −1 + i ⎛ ⎞ ⎛ ⎞ 1 0 −2 3 1 −2 3 −12 21. ⎜−3 ⎜ 1 1 2⎟ ⎟ 22. ⎜−5 ⎜ 12 −14 19⎟ ⎟ ⎝ 0 4 −1 1⎠ ⎝−9 22 −20 31⎠ 2 3 0 1 −4 9 −14 15 23. Prove that the determinant of an upper triangular matrix is the product of its diagonal entries. 24. Prove the corollary to Theorem 4.3. 25. Prove that det(kA) = kn det(A) for any A ∈ Mn×n (F ). 26. Let A ∈ Mn×n (F ). Under what conditions is det(−A) = det(A)? 27. Prove that if A ∈ Mn×n (F ) has two identical columns, then det(A) = 0. 28. Compute det(Ei ) if Ei is an elementary matrix of type i. 29. † Prove that if E is an elementary matrix, then det(E t ) = det(E). 30. Let the rows of A ∈ Mn×n (F ) be a1 , a2 , . . . , an , and let B be the matrix in which the rows are an , an−1 , . . . , a1 . Calculate det(B) in terms of det(A). 1.Label the following statements as true or false. (a) (b) (c) (d) (e) (f ) (g)(h)If E is an elementary matrix, then det(E) = ±1. For any A, B ∈ Mn×n(F ), det(AB) = det(A)· det(B). A matrix M ∈ Mn×n (F ) is invertible if and only if det(M ) = 0. A matrix M ∈ Mn×n (F ) has rank n if and only if det(M ) = 0. For any A ∈ Mn×n (F ), det(At ) = − det(A). The determinant of a square matrix can be evaluated by cofactor expansion along any column. Every system of n linear equations in n unknowns can be solved by Cramer’s rule. Let Ax = b be the matrix form of a system of n linear equations in n unknowns, where x = (x1 , x2 , . . . , xn )t . If det(A) = 0 and if Mk is the n × n matrix obtained from A by replacing row k of A by bt , then the unique solution of Ax = b is xk = det(Mk ) det(A) for k = 1, 2, . . . , n. In Exercises 2–7, use Cramer’s rule to solve the given system of linear equa- tions. 2. a11 x1 + a12 x2 = b1 a21 x1 + a22 x2 = b2 where a11 a22 − a12 a21 = 0 2x1 + x2 − 3x3 = 5 3. x1 − 2x2 + x3 = 10 3x1 + 4x2 − 2x3 = 0 2x1 + x2 − 3x3 = 1 4. x1 − 2x2 + x3 = 0 3x1 + 4x2 − 2x3 = −5 x1 − x2 + 4x3 = −2 6. −8x1 + 3x2 + x3 = 0 2x1 − x2 + x3 = 6 x1 − x2 + 4x3 = −4 5. −8x1 + 3x2 + x3 = 8 2x1 − x2 + x3 = 0 3x1 + x2 + x3 = 4 7. −2x1 − x2 = 12 x1 + 2x2 + x3 = −8 8.Use Theorem 4.8 to prove a result analogous to Theorem 4.3 (p. 212), but for columns. 9.Prove that an upper triangular n × n matrix is invertible if and only if all its diagonal entries are nonzero. Sec. 4.3 Properties of Determinants 229 10.11.12.13.14.15.16.17.†A matrix M ∈ Mn×n (C) is called nilpotent if, for some positive integer k, M k = O, where O is the n × n zero matrix. Prove that if M is nilpotent, then det(M ) = 0. A matrix M ∈ Mn×n (C) is called skew-symmetric if M t = −M . Prove that if M is skew-symmetric and n is odd, then M is not invert- ible. What happens if n is even? A matrix Q ∈ Mn×n (R) is called orthogonal if QQt = I. Prove that if Q is orthogonal, then det(Q) = ±1. For M ∈ Mn×n (C), let M be the matrix such that (M )ij = Mij for all i, j, where Mij is the complex conjugate of Mij . (a) Prove that det(M ) = det(M ). (b) A matrix Q ∈ Mn×n (C) is called unitary if QQ ∗ = I, where Q∗ = Qt. Prove that if Q is a unitary matrix, then | det(Q)| = 1. Let β = {u1 , u2 , . . . , un } be a subset of Fn containing n distinct vectors, and let B be the matrix in Mn×n (F ) having uj as column j. Prove that β is a basis for Fn if and only if det(B) = 0. Prove that if A, B ∈ Mn×n (F ) are similar, then det(A) = det(B). Use determinants to prove that if A, B ∈ Mn×n (F ) are such that AB = I, then A is invertible (and hence B = A−1 ). Let A, B ∈ Mn×n (F ) be such that AB = −BA. Prove that if n is odd and F is not a field of characteristic two, then A or B is not invertible. 18.19.20.Complete the proof of Theorem 4.7 by showing that if A is an elementary matrix of type 2 or type 3, then det(AB) = det(A)· det(B). A matrix A ∈ Mn×n (F ) is called lower triangular if Aij = 0 for 1 ≤ i < j ≤ n. Suppose that A is a lower triangular matrix. Describe det(A) in terms of the entries of A. Suppose that M ∈ Mn×n (F ) can be written in the form A B M = , O I where A is a square matrix. Prove that det(M ) = det(A). 21. † Prove that if M ∈ Mn×n (F ) can be written in the form A B M = , O C where A and C are square matrices, then det(M ) = det(A)· det(C). 230 Chap. 4 Determinants 22. Let T : Pn (F ) → Fn+1 be the linear transformation defined in Exer- cise 22 of Section 2.4 by T(f ) = (f (c0 ), f (c1 ), . . . , f (cn )), where c0 , c1 , . . . , cn are distinct scalars in an infinite field F . Let β be the standard ordered basis for Pn (F ) and γ be the standard ordered basis for Fn+1 . (a) Show that M = [T]γβ has the form ⎛ ⎞ 1 c0 c20 · · · cn 0 ⎜1 ⎜ c1 c21 · · · cn 1 ⎟ ⎟ ⎜ . ⎟ . ⎝ .. .. . .. . .. . ⎠ 1 cn c2 n · · · cnn (b)(c)A matrix with this form is called a Vandermonde matrix. Use Exercise 22 of Section 2.4 to prove that det(M ) = 0. Prove that det(M ) = (cj − ci ), 0≤i<j≤n the product of all terms of the form cj − ci for 0 ≤ i < j ≤ n. 23. Let A ∈ Mn×n (F ) be nonzero. For any m (1 ≤ m ≤ n), an m × m submatrix is obtained by deleting any n − m rows and any n − m columns of A. (a) Let k (1 ≤ k ≤ n) denote the largest integer such that some k × k submatrix has a nonzero determinant. Prove that rank(A) = k. (b) Conversely, suppose that rank(A) = k. Prove that there exists a k × k submatrix with a nonzero determinant. 24. Let A ∈ Mn×n (F ) have the form ⎛ ⎞ 0 0 0 ··· 0 a0 ⎜−1 ⎜ 0 0 ··· 0 a1 ⎟ ⎟ A = ⎜ ⎜ 0 −1 0 ··· 0 a2 ⎟ ⎟ . ⎜ ⎝ .. . .. . .. . .. . .. . ⎟ ⎠ 0 0 0 ··· −1 an−1 Compute det(A + tI), where I is the n × n identity matrix. 25.Let cjk denote the cofactor of the row j, column k entry of the matrix A ∈ Mn×n (F ). (a) Prove that if B is the matrix obtained from A by replacing column k by ej , then det(B) = cjk . Sec. 4.3 Properties of Determinants 231 (b) Show that for 1 ≤ j ≤ n, we have ⎛ ⎞ cj1 ⎜ cj2 ⎟ ⎜ ⎟ A ⎜ . ⎟ = det(A)· ej . ⎝ .. ⎠ cjn Hint: Apply Cramer’s rule to Ax = ej . (c) Deduce that if C is the n × n matrix such that Cij = cji , then AC = [det(A)]I. (d) Show that if det(A) = 0, then A−1 = [det(A)]−1 C. The following definition is used in Exercises 26–27. Definition. The classical adjoint of a square matrix A is the transpose of the matrix whose ij-entry is the ij-cofactor of A. 26. Find the classical adjoint of each of the following matrices. ⎛ ⎞ 4 0 0 (a) A11 A12 (b) ⎝0 4 0⎠ A21 A22 0 0 4 ⎛ ⎞ ⎛ ⎞ −4 0 0 3 6 7 (c) ⎝ 0 2 0⎠ (d) ⎝0 4 8⎠ 0 0 5 0 0 5 ⎛ ⎞ ⎛ ⎞ 1 − i 0 0 7 1 4 (e) ⎝ 4 3i 0 ⎠ (f ) ⎝ 6 −3 0⎠ 2i 1 + 4i −1 −3 5 −2 ⎛ ⎞ ⎛ ⎞ −1 2 5 3 2+ i 0 (g) ⎝ 8 0 −3⎠ (h) ⎝−1 + i 0 i ⎠ 4 6 1 0 1 3 − 2i 27. Let C be the classical adjoint of A ∈ Mn×n (F ). Prove the following statements. (a) det(C) = [det(A)]n−1 . (b) C t is the classical adjoint of At . (c) If A is an invertible upper triangular matrix, then C and A−1 are both upper triangular matrices. 28. Let y1 , y2 , . . . , yn be linearly independent functions in C∞ . For each y ∈ C∞ , define T(y) ∈ C∞ by ⎛ ⎞ y(t) y1 (t) y2 (t) · · · yn (t) ⎜ y (t) y1 (t) y2 (t) · · · yn (t) ⎟ ⎜ ⎟ [T(y)](t) = det ⎜ .. .. .. .. ⎟ . ⎝ . . . . ⎠ y (n) (t) y1 (n) (t) y2 (n) (t) · · · yn (n) (t) 232 Chap. 4 Determinants The preceding determinant is called the Wronskian of y, y1 , . . . , yn . (a) Prove that T : C∞ → C∞ is a linear transformation. (b) Prove that N(T) = span({y1 , y2 , . . . , yn }). 1. Label the following statements as true or false. (a)(b)(c)(d)(e)(f )(g)(h)(i)(j)(k)The determinant of a square matrix may be computed by expand- ing the matrix along any row or column. In evaluating the determinant of a matrix, it is wise to expand along a row or column containing the largest number of zero en- tries. If two rows or columns of A are identical, then det(A) = 0. If B is a matrix obtained by interchanging two rows or two columns of A, then det(B) = det(A). If B is a matrix obtained by multiplying each entry of some row or column of A by a scalar, then det(B) = det(A). If B is a matrix obtained from A by adding a multiple of some row to a different row, then det(B) = det(A). The determinant of an upper triangular n×n matrix is the product of its diagonal entries. For every A ∈ Mn×n (F ), det(At ) = − det(A). If A, B ∈ Mn×n (F ), then det(AB) = det(A)· det(B). If Q is an invertible matrix, then det(Q−1 ) = [det(Q)]−1 . A matrix Q is invertible if and only if det(Q) = 0. 2. Evaluate the determinant of the following 2 × 2 matrices. 4 −5 −1 7 (a) (b) 2 3 3 8 2 + i −1 + 3i 3 4i (c) (d) 1 − 2i 3 − i −6i 2i 3.Evaluate the determinant of the following matrices in the manner indi- cated. Sec. 4.4 Summary—Important Facts about Determinants 237 ⎛ ⎞ ⎛ ⎞ 0 1 2 1 0 2 ⎝−1 0 −3⎠ ⎝ 0 1 5⎠ (a) (b) 2 3 0 −1 3 0 along the first row along the first column ⎛ ⎞ ⎛ ⎞ 0 1 2 1 0 2 ⎝−1 0 −3⎠ ⎝ 0 1 5⎠ (c) (d) 2 3 0 −1 3 0 along the second column along the third row ⎛ ⎞ ⎛ ⎞ 0 1+ i 2 i 2+ i 0 ⎝−2i 0 1 − i⎠ ⎝−1 3 2i ⎠ (e) (f ) 3 4i 0 0 −1 1 − i along the third row along the third column ⎛ ⎞ ⎛ ⎞ 0 2 1 3 1 −1 2 −1 ⎜ ⎜ 1 0 −2 2⎟ ⎟ ⎜−3 ⎜ 4 1 −1⎟ ⎟ (g) ⎝ 3 −1 0 1⎠ (h) ⎝ 2 −5 −3 8⎠ −1 1 2 0 −2 6 −4 1 along the fourth column along the fourth row 4. Evaluate the determinant of the following matrices by any legitimate method. ⎛ ⎞ ⎛ ⎞ 1 2 3 −1 3 2 (a) ⎝4 5 6⎠ (b) ⎝ 4 −8 1⎠ 7 8 9 2 2 5 ⎛ ⎞ ⎛ ⎞ 0 1 1 1 −2 3 (c) ⎝1 2 −5⎠ (d) ⎝−1 2 −5⎠ 6 −4 3 3 −1 2 ⎛ ⎞ ⎛ ⎞ i 2 −1 −1 2+ i 3 (e) ⎝ 3 1+ i 2 ⎠ (f ) ⎝1 − i i 1 ⎠ −2i 1 4 − i 3i 2 −1 + i ⎛ ⎞ ⎛ ⎞ 1 0 −2 3 1 −2 3 −12 (g) ⎜−3 ⎜ 1 1 2⎟ ⎟ (h) ⎜−5 ⎜ 12 −14 19⎟ ⎟ ⎝ 0 4 −1 1⎠ ⎝−9 22 −20 31⎠ 2 3 0 1 −4 9 −14 15 5. Suppose that M ∈ M n×n (F ) can be written in the form A B M = , O I where A is a square matrix. Prove that det(M ) = det(A). 238 Chap. 4 Determinants 6. † Prove that if M ∈ Mn×n (F ) can be written in the form A B M = , O C where A and C are square matrices, then det(M ) = det(A)· det(C). 1.Label the following statements as true or false. (a)(b)(c)(d)(e)(f )Any n-linear function δ : Mn×n (F ) → F is a linear transformation. Any n-linear function δ : Mn×n (F ) → F is a linear function of each row of an n × n matrix when the other n − 1 rows are held fixed. If δ : Mn×n (F ) → F is an alternating n-linear function and the matrix A ∈ Mn×n (F ) has two identical rows, then δ(A) = 0. If δ : Mn×n (F ) → F is an alternating n-linear function and B is obtained from A ∈ Mn×n (F ) by interchanging two rows of A, then δ(B) = δ(A). There is a unique alternating n-linear function δ : Mn×n (F ) → F . The function δ : Mn×n (F ) → F defined by δ(A) = 0 for every A ∈ Mn×n (F ) is an alternating n-linear function. 2. Determine all the 1-linear functions δ : M1×1 (F ) → F . Determine which of the functions δ : M3×3 (F ) → F in Exercises 3–10 are 3-linear functions. Justify each answer. Sec. 4.5 A Characterization of the Determinant 243 3.4.5.6.7.8.9.10.11.12.13.14.15.16.17.18.19.20.δ(A) = k, where k is any nonzero scalar δ(A) = A22 δ(A) = A11 A23 A32 δ(A) = A11 + A23 + A32 δ(A) = A11 A21 A32 δ(A) = A11 A31 A32 δ(A) = A211 A222 A233 δ(A) = A11 A22 A33 − A11 A21 A32 Prove Corollaries 2 and 3 of Theorem 4.10. Prove Theorem 4.11. Prove that det : M2×2 (F ) → F is a 2-linear function of the columns of a matrix. Let a, b, c, d ∈ F . Prove that the function δ : M2×2 (F ) → F defined by δ(A) = A11 A22 a + A11 A21 b + A12 A22 c + A12 A21 d is a 2-linear function. Prove that δ : M2×2 (F ) → F is a 2-linear function if and only if it has the form δ(A) = A11 A22 a + A11 A21 b + A12 A22 c + A12 A21 d for some scalars a, b, c, d ∈ F . Prove that if δ : Mn×n (F ) → F is an alternating n-linear function, then there exists a scalar k such that δ(A) = k det(A) for all A ∈ Mn×n (F ). Prove that a linear combination of two n-linear functions is an n-linear function, where the sum and scalar product of n-linear functions are as defined in Example 3 of Section 1.2 (p. 9). Prove that the set of all n-linear functions over a field F is a vector space over F under the operations of function addition and scalar mul- tiplication as defined in Example 3 of Section 1.2 (p. 9). Let δ : Mn×n (F ) → F be an n-linear function and F a field that does not have characteristic two. Prove that if δ(B) = −δ(A) whenever B is obtained from A ∈ Mn×n (F ) by interchanging any two rows of A, then δ(M ) = 0 whenever M ∈ Mn×n (F ) has two identical rows. Give an example to show that the implication in Exercise 19 need not hold if F has characteristic two.
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<p><br />Solution maual to Linear Algebra, Fourth Edition, Stephen H. Friedberg. (Chapter 3)<br /><br />Solutions to Linear Algebra, Fourth Edition, Stephen H. Friedberg. (Chapter 3)<br /><br />Linear Algebra solution manual, Fourth Edition, Stephen H. Friedberg. (Chapter 3)<br /><br />Linear Algebra solutions Friedberg. (Chapter 3)</p>
<div style="width: 100%; height: 1px; overflow: auto;"><span>1. Label the following statements as true or false. (a) An elementary matrix is always square. (b) The only entries of an elementary matrix are zeros and ones. (c) The n × n identity matrix is an elementary matrix. (d) The product of two n × n elementary matrices is an elementary matrix. (e) The inverse of an elementary matrix is an elementary matrix. (f ) The sum of two n×n elementary matrices is an elementary matrix. (g) The transpose of an elementary matrix is an elementary matrix. (h) If B is a matrix that can be obtained by performing an elementary row operation on a matrix A, then B can also be obtained by performing an elementary column operation on A. (i) If B is a matrix that can be obtained by performing an elemen- tary row operation on a matrix A, then A can be obtained by performing an elementary row operation on B. 2. Let ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 2 3 1 0 3 1 0 3 A = ⎝1 0 1⎠ , B = ⎝1 −2 1⎠ , and C = ⎝0 −2 −2⎠ . 1 −1 1 1 −3 1 1 −3 1 Find an elementary operation that transforms A into B and an elemen- tary operation that transforms B into C. By means of several additional operations, transform C into I3 . 3. Use the proof of Theorem 3.2 to obtain the inverse of each of the fol- lowing elementary matrices. ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0 0 1 1 0 0 1 0 0 (a) ⎝0 1 0⎠ (b) ⎝0 3 0⎠ (c) ⎝ 0 1 0⎠ 1 0 0 0 0 1 −2 0 1 4.Prove the assertion made on page 149: Any elementary n×n matrix can be obtained in at least two ways—either by performing an elementary row operation on In or by performing an elementary column operation on In . 5.6.Prove that E is an elementary matrix if and only if E t is. Let A be an m × n matrix. Prove that if B can be obtained from A by an elementary row [column] operation, then B t can be obtained from At by the corresponding elementary column [row] operation. 7. Prove Theorem 3.1. 152 Chap. 3 Elementary Matrix Operations and Systems of Linear Equations 8.Prove that if a matrix Q can be obtained from a matrix P by an elemen- tary row operation, then P can be obtained from Q by an elementary matrix of the same type. Hint: Treat each type of elementary row operation separately. 9.Prove that any elementary row [column] operation of type 1 can be obtained by a succession of three elementary row [column] operations of type 3 followed by one elementary row [column] operation of type 2. 10. Prove that any elementary row [column] operation of type 2 can be obtained by dividing some row [column] by a nonzero scalar. 11. Prove that any elementary row [column] operation of type 3 can be obtained by subtracting a multiple of some row [column] from another row [column]. 12. Let A be an m × n matrix. Prove that there exists a sequence of elementary row operations of types 1 and 3 that transforms A into an upper triangular matrix. 1. Label the following statements as true or false. (a)(b)(c)(d)(e)(f )(g)(h)(i)The rank of a matrix is equal to the number of its nonzero columns. The product of two matrices always has rank equal to the lesser of the ranks of the two matrices. The m × n zero matrix is the only m × n matrix having rank 0. Elementary row operations preserve rank. Elementary column operations do not necessarily preserve rank. The rank of a matrix is equal to the maximum number of linearly independent rows in the matrix. The inverse of a matrix can be computed exclusively by means of elementary row operations. The rank of an n × n matrix is at most n. An n × n matrix having rank n is invertible. 2. Find the rank of the following matrices. ⎛ ⎞ ⎛ ⎞ 1 1 0 1 1 0 (a) ⎝0 1 1⎠ (b) ⎝2 1 1⎠ (c) 1 0 2 1 1 4 1 1 0 1 1 1 166 Chap. 3 Elementary Matrix Operations and Systems of Linear Equations ⎛ ⎞ 1 2 3 1 1 1 2 1 ⎜1 4 0 1 2⎟ (d) (e) ⎜ ⎟ 2 4 2 ⎝0 2 −3 0 1⎠ 1 0 0 0 0 ⎛ ⎞ ⎛ ⎞ 1 2 0 1 1 1 1 0 1 ⎜ 2 4 1 3 0⎟ ⎜2 2 0 2⎟ (f ) ⎜ ⎟ (g) ⎜ ⎟ ⎝ 3 6 2 5 1⎠ ⎝1 1 0 1⎠ −4 −8 1 −3 1 1 1 0 1 3.Prove that for any m × n matrix A, rank(A) = 0 if and only if A is the zero matrix. 4. Use elementary row and column operations to transform each of the following matrices into a matrix D satisfying the conditions of Theo- rem 3.6, and then determine the rank of each matrix. ⎛ ⎞ ⎛ ⎞ 1 1 1 2 2 1 (a) ⎝2 0 −1 2⎠ (b) ⎝−1 2⎠ 1 1 1 2 2 1 5. For each of the following matrices, compute the rank and the inverse if it exists. ⎛ ⎞ 1 2 1 (a) 1 2 (b) 1 2 (c) ⎝1 3 4⎠ 1 1 2 4 2 3 −1 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0 −2 4 1 2 1 1 2 1 (d) ⎝1 1 −1⎠ (e) ⎝−1 1 2⎠ (f ) ⎝1 0 1⎠ 2 4 −5 1 0 1 1 1 1 ⎛ ⎞ ⎛ ⎞ 1 2 1 0 1 0 1 1 (g) ⎜ ⎜ 2 5 5 1⎟ ⎟ (h) ⎜1 ⎜ 1 −1 2⎟ ⎟ ⎝−2 −3 0 3⎠ ⎝2 0 1 0⎠ 3 4 −2 −3 0 −1 1 −3 6.For each of the following linear transformations T, determine whether T is invertible, and compute T−1 if it exists. (a) T : P2 (R) → P2 (R) defined by T(f (x)) = f (x) + 2f (x) − f (x). (b) T : P2 (R) → P2 (R) defined by T(f (x)) = (x + 1)f (x). (c) T : R3 → R3 defined by T(a1 , a2 , a3 ) = (a 1 + 2a2 + a3 , −a1 + a2 + 2a3 , a1 + a3 ). Sec. 3.2 The Rank of a Matrix and Matrix Inverses 167 (d) T : R3 → P2 (R) defined by T(a1 , a2 , a3 ) = (a1 + a2 + a3 ) + (a1 − a2 + a3 )x + a1 x2 . (e) T : P2 (R) → R3 defined by T(f (x)) = (f (−1), f (0), f (1)). (f ) T : M2×2 (R) → R4 defined by T(A) = (tr(A), tr(At ), tr(EA), tr(AE)), where 0 1 E = . 1 0 7. Express the invertible matrix ⎛ ⎞ 1 2 1 ⎝1 0 1⎠ 1 1 2 as a product of elementary matrices. 8. Let A be an m × n matrix. Prove that if c is any nonzero scalar, then rank(cA) = rank(A). 9. Complete the proof of the corollary to Theorem 3.4 by showing that elementary column operations preserve rank. 10. Prove Theorem 3.6 for the case that A is an m × 1 matrix. 11. Let ⎛ ⎞ 1 0 ··· 0 ⎜ 0 ⎟ ⎜ ⎟ B = ⎜ ⎝ .. . B ⎟ ⎠ , 0 where B is an m × n submatrix of B. Prove that if rank(B) = r, then rank(B ) = r − 1. 12. Let B and D be m × n matrices, and let B and D be (m + 1) × (n + 1) matrices respectively defined by ⎛ ⎞ ⎛ ⎞ 1 0 ··· 0 1 0 ··· 0 ⎜ 0 ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ ⎟ B = ⎜ . ⎟ and D = ⎜ . ⎟ . ⎝ .. B ⎠ ⎝ .. D ⎠ 0 0 Prove that if B can be transformed into D by an elementary row [column] operation, then B can be transformed into D by an elementary row [column] operation. 168 Chap. 3 Elementary Matrix Operations and Systems of Linear Equations 13.14.15.16.17.18.19.Prove (b) and (c) of Corollary 2 to Theorem 3.6. Let T, U : V → W be linear transformations. (a) Prove that R(T + U) ⊆ R(T) + R(U). (See the definition of the sum of subsets of a vector space on page 22.) (b) Prove that if W is finite-dimensional, then rank(T+U) ≤ rank(T)+ rank(U). (c) Deduce from (b) that rank(A + B) ≤ rank(A) + rank(B) for any m × n matrices A and B. Suppose that A and B are matrices having n rows. M (A|B) = (M A|M B) for any m × n matrix M . Prove that Supply the details to the proof of (b) of Theorem 3.4. Prove that if B is a 3 × 1 matrix and C is a 1 × 3 matrix, then the 3 × 3 matrix BC has rank at most 1. Conversely, show that if A is any 3 × 3 matrix having rank 1, then there exist a 3 × 1 matrix B and a 1 × 3 matrix C such that A = BC. Let A be an m × n matrix and B be an n × p matrix. Prove that AB can be written as a sum of n matrices of rank one. Let A be an m × n matrix with rank m and B be an n × p matrix with rank n. Determine the rank of AB. Justify your answer. 20. Let ⎛ ⎞ 1 0 −1 2 1 A = ⎜−1 ⎜ 1 3 −1 0⎟ ⎟ . ⎝−2 1 4 −1 3⎠ 3 −1 −5 1 −6 (a) Find a 5 × 5 matrix M with rank 2 such that AM = O, where O is the 4 × 5 zero matrix. (b) Suppose that B is a 5 × 5 matrix such that AB = O. Prove that rank(B) ≤ 2. 21. Let A be an m × n matrix with rank m. Prove that there exists an n × m matrix B such that AB = Im . 22. Let B be an n × m matrix with rank m. Prove that there exists an m × n matrix A such that AB = Im . 1.Label(a)(b)(c)(d)(e)(f )(g)(h)the following statements as true or false. Any system of linear equations has at least one solution. Any system of linear equations has at most one solution. Any homogeneous system of linear equations has at least one so- lution. Any system of n linear equations in n unknowns has at most one solution. Any system of n linear equations in n unknowns has at least one solution. If the homogeneous system corresponding to a given system of lin- ear equations has a solution, then the given system has a solution. If the coefficient matrix of a homogeneous system of n linear equa- tions in n unknowns is invertible, then the system has no nonzero solutions. The solution set of any system of m linear equations in n unknowns is a subspace of Fn . 2.For each of the following homogeneous systems of linear equations, find the dimension of and a basis for the solution set. 180 3.4.5.6.7.Chap. 3 Elementary Matrix Operations and Systems of Linear Equations x1 + 3x2 = 0 x1 + x2 − x3 = 0 (a) (b) 2x1 + 6x2 = 0 4x1 + x2 − 2x3 = 0 2x1 + x2 − x3 = 0 x1 + 2x2 − x3 = 0 (c) (d) x1 − x2 + x3 = 0 2x1 + x2 + x3 = 0 x1 + 2x2 − 2x3 = 0 x1 + 2x2 = 0 (e) x1 + 2x2 − 3x3 + x4 = 0 (f ) x1 − x2 = 0 x1 + 2x2 + x3 + x4 = 0 (g) x2 − x3 + x4 = 0 Using the results of Exercise 2, find all solutions to the following sys- tems. x1 + 3x2 = 5 x1 + x2 − x3 = 1 (a) (b) 2x1 + 6x2 = 10 4x1 + x2 − 2x3 = 3 2x1 + x2 − x3 = 5 x1 + 2x2 − x3 = 3 (c) (d) x1 − x2 + x3 = 1 2x1 + x2 + x3 = 6 x1 + 2x2 − 2x3 = 4 x1 + 2x2 = 5 (e) x1 + 2x2 − 3x3 + x4 = 1 (f ) x1 − x2 = −1 x1 + 2x2 + x3 + x4 = 1 (g) x2 − x3 + x4 = 1 For each system of linear equations with the invertible coefficient matrix A, (1) Compute A−1 . (2) Use A−1 to solve the system. x1 + 2x2 − x3 = 5 x1 + 3x2 = 4 (a) (b) x1 + x2 + x3 = 1 2x1 + 5x2 = 3 2x1 − 2x2 + x3 = 4 Give an example of a system of n linear equations in n unknowns with infinitely many solutions. Let T : R3 → R2 be defined by T(a, b, c) = (a + b, 2a − c). Determine T−1 (1, 11). Determine which of the following systems of linear equations has a so- lution. Sec. 3.3 8.9.10.11.Systems of Linear Equations—Theoretical Aspects 181 x1 + x2 − x3 + 2x4 = 2 x1 + x2 − x3 = 1 (a) x1 + x2 + 2x3 =1 (b) 2x1 + x2 + 3x3 = 2 2x1 + 2x2 + x3 + 2x4 = 4 x1 + x2 + 3x3 − x4 =0 x1 + 2x2 + 3x3 = 1 x1 + x2 + x3 + x4 =1 (c) x1 + x2 − x3 = 0 (d) x1 − 2x2 + x3 − x4 =1 x1 + 2x2 + x3 = 3 4x1 + x2 + 8x3 − x4 =0 x1 + 2x2 − x3 = 1 (e) 2x1 + x2 + 2x3 = 3 x1 − 4x2 + 7x3 = 4 Let T : R3 → R3 be defined by T(a, b, c) = (a + b, b − 2c, a + 2c). For each vector v in R3 , determine whether v ∈ R(T). (a) v = (1, 3, −2) (b) v = (2, 1, 1) Prove that the system of linear equations Ax = b has a solution if and only if b ∈ R(LA ). Prove or give a counterexample to the following statement: If the co- efficient matrix of a system of m linear equations in n unknowns has rank m, then the system has a solution. In the closed model of Leontief with food, clothing, and housing as the basic industries, suppose that the input–output matrix is ⎛ 7 1 3 ⎞ 16 2 16 ⎜ 5 1 5 ⎟ A = ⎜ 16 6 16 ⎟ . ⎝ ⎠1 1 1 4 3 2 At what ratio must the farmer, tailor, and carpenter produce in order for equilibrium to be attained? 12.A certain economy consists of two sectors: goods and services. Suppose that 60% of all goods and 30% of all services are used in the production of goods. What proportion of the total economic output is used in the production of goods? 13. In the notation of the open model of Leontief, suppose that ⎛ ⎞ 1 1 A = ⎝ 2 1 5 1 ⎠ and d = 2 5 3 5 are the input–output matrix and the demand vector, respectively. How much of each commodity must be produced to satisfy this demand? 182 Chap. 3 Elementary Matrix Operations and Systems of Linear Equations 14.A certain economy consisting of the two sectors of goods and services supports a defense system that consumes $90 billion worth of goods and $20 billion worth of services from the economy but does not contribute to economic production. Suppose that 50 cents worth of goods and 20 cents worth of services are required to produce $1 worth of goods and that 30 cents worth of of goods and 60 cents worth of services are required to produce $1 worth of services. What must the total output of the economic system be to support this defense system? 1.Label the following statements as true or false. (a) If (A |b ) is obtained from (A|b) by a finite sequence of elementary column operations, then the systems Ax = b and A x = b are equivalent. Sec. 3.4 Systems of Linear Equations—Computational Aspects 195 (b)(c)(d)(e)(f )(g)If (A |b ) is obtained from (A|b) by a finite sequence of elemen- tary row operations, then the systems Ax = b and A x = b are equivalent. If A is an n × n matrix with rank n, then the reduced row echelon form of A is In . Any matrix can be put in reduced row echelon form by means of a finite sequence of elementary row operations. If (A|b) is in reduced row echelon form, then the system Ax = b is consistent. Let Ax = b be a system of m linear equations in n unknowns for which the augmented matrix is in reduced row echelon form. If this system is consistent, then the dimension of the solution set of Ax = 0 is n − r, where r equals the number of nonzero rows in A. If a matrix A is transformed by elementary row operations into a matrix A in reduced row echelon form, then the number of nonzero rows in A equals the rank of A. 2. Use Gaussian elimination to solve the following systems of linear equa- tions. x1 − 2x2 − x3 =1 x1 + 2x2 − x3 = −1 2x1 − 3x2 + x3 =6 (a) 2x1 + 2x2 + x3 = 1 (b) 3x1 − 5x2 =7 3x1 + 5x2 − 2x3 = −1 x1 + 5x3 =9 x1 + 2x2 + 2x4 = 6 3x1 + 5x2 − x3 + 6x4 = 17 (c) 2x1 + 4x2 + x3 + 2x4 = 12 2x1 − 7x3 + 11x4 = 7 x1 − x2 − 2x3 + 3x4 = −7 2x1 − x2 + 6x3 + 6x4 = −2 (d) −2x1 + x2 − 4x3 − 3x4 = 0 3x1 − 2x2 + 9x3 + 10x4 = −5 x1 − 4x2 − x3 + x4 = 3 x1 + 2x2 − x3 + 3x4 = 2 (e) 2x1 − 8x2 + x3 − 4x4 = 9 (f ) 2x1 + 4x2 − x3 + 6x4 = 5 −x1 + 4x2 − 2x3 + 5x4 = −6 x2 + 2x4 = 3 2x1 − 2x2 − x3 + 6x4 − 2x5 = 1 (g) x1 − x2 + x3 + 2x4 − x5 = 2 4x1 − 4x2 + 5x3 + 7x4 − x5 = 6 3x1 − x2 + x3 − x4 + 2x5 = 5 x1 − x2 − x3 − 2x4 − x5 = 2 (h) 5x1 − 2x2 + x3 − 3x4 + 3x5 = 10 2x1 − x2 − 2x4 + x5 = 5 196 Chap. 3 Elementary Matrix Operations and Systems of Linear Equations (i) (j) 3x1 − x2 + 2x3 + 4x4 + x5 = 2 x1 − x2 + 2x3 + 3x4 + x5 = −1 2x1 − 3x2 + 6x3 + 9x4 + 4x5 = −5 7x1 − 2x2 + 4x3 + 8x4 + x5 = 6 2x1 + 3x3 − 4x5 = 5 3x1 − 4x2 + 8x3 + 3x4 = 8 x1 − x2 + 2x3 + x4 − x5 = 2 −2x1 + 5x2 − 9x3 − 3x4 − 5x5 = −8 3.Suppose that the augmented matrix of a system Ax = b is transformed into a matrix (A |b ) in reduced row echelon form by a finite sequence of elementary row operations. (a)(b)Prove that rank(A ) = rank(A |b ) if and only if (A |b ) contains a row in which the only nonzero entry lies in the last column. Deduce that Ax = b is consistent if and only if (A |b ) contains no row in which the only nonzero entry lies in the last column. 4. For each of the systems that follow, apply Exercise 3 to determine whether the system is consistent. If the system is consistent, find all solutions. Finally, find a basis for the solution set of the corresponding homogeneous system. x1 + 2x2 − x3 + x4 = 2 x1 + x2 − 3x3 + x4 = −2 (a) 2x1 + x2 + x3 − x4 = 3 (b) x1 + x2 + x3 − x4 = 2 x1 + 2x2 − 3x3 + 2x4 = 2 x1 + x2 − x3 = 0 x1 + x2 − 3x3 + x4 = 1 (c) x1 + x2 + x3 − x4 = 2 x1 + x2 − x3 =0 5. Let the reduced row echelon form of A be ⎛ ⎞ 1 0 2 0 −2 ⎝0 1 −5 0 −3⎠ . 0 0 0 1 6 Determine A if the first, second, and fourth columns of A are ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 0 1 ⎝−1⎠ , ⎝−1⎠ , and ⎝−2⎠ , 3 1 0 respectively. 6. Let the reduced row echelon form of A be ⎛ ⎞ 1 −3 0 4 0 5 ⎜0 0 1 3 0 2⎟ ⎜ ⎟ . ⎝0 0 0 0 1 −1⎠ 0 0 0 0 0 0 Sec. 3.4 Systems of Linear Equations—Computational Aspects 197 Determine A if the first, third, and sixth columns of A are ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 −1 3 ⎜−2⎟ ⎜ ⎟ , ⎜ ⎜ 1⎟ ⎟ , and ⎜−9⎟ ⎜ ⎟ , ⎝−1⎠ ⎝ 2⎠ ⎝ 2⎠ 3 −4 5 respectively. 7. It can be shown that the vectors u1 = (2, −3, 1), u2 = (1, 4, −2), u3 = (−8, 12, −4), u4 = (1, 37, −17), and u5 = (−3, −5, 8) generate R3 . Find a subset of {u1 , u2 , u3 , u4 , u5 } that is a basis for R3 . 8. Let W denote the subspace of R5 consisting of all vectors having coor- dinates that sum to zero. The vectors u1 = (2, −3, 4, −5, 2), u2 = (−6, 9, −12, 15, −6), u3 = (3, −2, 7, −9, 1), u4 = (2, −8, 2, −2, 6), u5 = (−1, 1, 2, 1, −3), u6 = (0, −3, −18, 9, 12), u7 = (1, 0, −2, 3, −2), and u8 = (2, −1, 1, −9, 7) generate W. Find a subset of {u1 , u2 , . . . , u8} that is a basis for W. 9. Let W be the subspace of M2×2 (R) consisting of the symmetric 2 × 2 matrices. The set 0 −1 1 2 2 1 1 −2 −1 2 S = , , , , −1 1 2 3 1 9 −2 4 2 −1 generates W. Find a subset of S that is a basis for W. 10.Let V = {(x1 , x2 , x3 , x4 , x5 ) ∈ R5 : x1 − 2x2 + 3x3 − x4 + 2x5 = 0}. (a) Show that S = {(0, 1, 1, 1, 0)} is a linearly independent subset of V. (b) Extend S to a basis for V. 11.Let V be as in Exercise 10. (a) Show that S = {(1, 2, 1, 0, 0)} is a linearly independent subset of V. (b) Extend S to a basis for V. 12.Let V denote the set of all solutions to the system of linear equations x1 − x2 + 2x4 − 3x5 + x6 = 0 2x1 − x2 − x3 + 3x4 − 4x5 + 4x6 = 0. 198 13.14.15.Chap. 3 Elementary Matrix Operations and Systems of Linear Equations (a) Show that S = {(0, −1, 0, 1, 1, 0), (1, 0, 1, 1, 1, 0)} is a linearly inde- pendent subset of V. (b) Extend S to a basis for V. Let V be as in Exercise 12. (a) Show that S = {(1, 0, 1, 1, 1, 0), (0, 2, 1, 1, 0, 0)} is a linearly inde- pendent subset of V. (b) Extend S to a basis for V. If (A|b) is in reduced row echelon form, prove that A is also in reduced row echelon form. Prove the corollary to Theorem 3.16: The reduced row echelon form of a matrix is unique. </span></div><div style="text-align:center;margin:10px 0 10px 0;clear:both"><div style="display:inline;text-align:center;"><script async src="https://pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script>
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Linear Algebrahashnuthttps://koreanfoodie.me/91https://koreanfoodie.me/91#entry91commentSat, 15 Jun 2019 13:12:39 +0900Solutions to Linear Algebra, Stephen H. Friedberg, Fourth Edition (Chapter 2)
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<p>Solution maual to Linear Algebra, Fourth Edition, Stephen H. Friedberg. (Chapter 2)<br /><br />Solutions to Linear Algebra, Fourth Edition, Stephen H. Friedberg. (Chapter 2)<br /><br />Linear Algebra solution manual, Fourth Edition, Stephen H. Friedberg. (Chapter 2)<br /><br />Linear Algebra solutions Friedberg. (Chapter 2)</p>
<div style="width:100%; height:1px; overflow:auto"><font size="1px" color=#FFFFFF>1.Label the following statements as true or false. In each part, V and W
are finite-dimensional vector spaces (over F ), and T is a function from
V to W.
(a)(b)(c)(d)(e)(f )(g)(h)If T is linear, then T preserves sums and scalar products.
If T(x + y) = T(x) + T(y), then T is linear.
T is one-to-one if and only if the only vector x such that T(x) = 0
is x = 0 .
If T is linear, then T(0 V ) = 0 W .
If T is linear, then nullity(T) + rank(T) = dim(W).
If T is linear, then T carries linearly independent subsets of V onto
linearly independent subsets of W.
If T, U : V → W are both linear and agree on a basis for V, then
T = U.
Given x1 , x2 ∈ V and y1 , y2 ∈ W, there exists a linear transforma-
tion T : V → W such that T(x1 ) = y1 and T(x2 ) = y2 .
For Exercises 2 through 6, prove that T is a linear transformation, and find
bases for both N(T) and R(T). Then compute the nullity and rank of T, and
verify the dimension theorem. Finally, use the appropriate theorems in this
section to determine whether T is one-to-one or onto.
2. T : R3 → R2 defined by T(a1 , a2 , a3 ) = (a1 − a2 , 2a3 ).
3. T : R2 → R3 defined by T(a1 , a2 ) = (a1 + a2 , 0, 2a1 − a2 ).
4. T : M2×3 (F ) → M2×2 (F ) defined by
a11
a12
a13
2a11 − a12
a13 + 2a12
T
=
.
a21
a22
a23
0
0
5. T : P2 (R) → P3 (R) defined by T(f (x)) = xf (x) + f (x).
Sec. 2.1
Linear Transformations, Null Spaces, and Ranges
75
6.T : Mn×n (F ) → F defined by T(A) = tr(A). Recall (Example 4, Sec-
tion 1.3) that
n
tr(A) =
Aii .
i=1
7. Prove properties 1, 2, 3, and 4 on page 65.
8. Prove that the transformations in Examples 2 and 3 are linear.
9.10.In this exercise, T : R2 → R2 is a function. For each of the following
parts, state why T is not linear.
(a)
(b)
(c)
(d)
(e)
T(a1 , a2 ) =T(a1 , a2 ) =T(a1 , a2 ) =T(a1 , a2 ) =T(a1 , a2 ) =(1, a2 )
(a1 , a21 )
(sin a1, 0)
(|a1 |, a2 )
(a1 + 1, a2 )
Suppose that T : R2 → R2 is linear, T(1, 0) = (1, 4), and T(1, 1) = (2, 5).
What is T(2, 3)? Is T one-to-one?
11.12.13.14.Prove that there exists a linear transformation T : R2 → R3 such that
T(1, 1) = (1, 0, 2) and T(2, 3) = (1, −1, 4). What is T(8, 11)?
Is there a linear transformation T : R3 → R2 such that T(1, 0, 3) = (1, 1)
and T(−2, 0, −6) = (2, 1)?
Let V and W be vector spaces, let T : V → W be linear, and let
{w1 , w2 , . . . , wk } be a linearly independent subset of R(T). Prove that
if S = {v1 , v2 , . . . , vk } is chosen so that T(vi) = wi for i = 1, 2, . . . , k,
then S is linearly independent.
Let V and W be vector spaces and T : V → W be linear.
(a) Prove that T is one-to-one if and only if T carries linearly inde-
pendent subsets of V onto linearly independent subsets of W.
(b) Suppose that T is one-to-one and that S is a subset of V. Prove
that S is linearly independent if and only if T(S) is linearly inde-
pendent.
(c) Suppose β = {v1 , v2 , . . . , vn } is a basis for V and T is one-to-one
and onto. Prove that T(β) = {T(v1 ), T(v2 ), . . . , T(vn )} is a basis
for W.
15.Recall the definition of P(R) on page 10. Define
x
T : P(R) → P(R) by T(f (x)) =
f (t) dt.
0
Prove that T linear and one-to-one, but not onto.
76
Chap. 2
Linear Transformations and Matrices
16. Let T : P(R) → P(R) be defined by T(f (x)) = f (x). Recall that T is
linear. Prove that T is onto, but not one-to-one.
17. Let V and W be finite-dimensional vector spaces and T : V → W be
linear.
(a) Prove that if dim(V) < dim(W), then T cannot be onto.
(b) Prove that if dim(V) > dim(W), then T cannot be one-to-one.
18. Give an example of a linear transformation T : R2 → R2 such that
N(T) = R(T).
19.Give an example of distinct linear transformations T and U such that
N(T) = N(U) and R(T) = R(U).
20.Let V and W be vector spaces with subspaces V1 and W1 , respectively.
If T : V → W is linear, prove that T(V1) is a subspace of W and that
{x ∈ V : T(x) ∈ W1 } is a subspace of V.
21.Let V be the vector space of sequences described in Example 5 of Sec-
tion 1.2. Define the functions T, U : V → V by
T(a1 , a2, . . .) = (a2 , a3 , . . .) and U(a1 , a2 , . . .) = (0, a1 , a2 , . . .).
T and U are called the left shift and right shift operators on V,
respectively.
(a) Prove that T and U are linear.
(b) Prove that T is onto, but not one-to-one.
(c) Prove that U is one-to-one, but not onto.
22.23.Let T : R3 → R be linear. Show that there exist scalars a, b, and c such
that T(x, y, z) = ax + by + cz for all (x, y, z) ∈ R3 . Can you generalize
this result for T : Fn → F ? State and prove an analogous result for
T : Fn → Fm .
Let T : R3 → R be linear. Describe geometrically the possibilities for
the null space of T. Hint: Use Exercise 22.
The following definition is used in Exercises 24–27 and in Exercise 30.
Definition. Let V be a vector space and W1 and W2 be subspaces of
V such that V = W1 ⊕ W2 . (Recall the definition of direct sum given in the
exercises of Section 1.3.) A function T : V → V is called the projection on
W1 along W2 if, for x = x1 + x2 with x1 ∈ W1 and x2 ∈ W2 , we have
T(x) = x1 .
24. Let T : R2 → R2 . Include figures for each of the following parts.
Sec. 2.1
Linear Transformations, Null Spaces, and Ranges
77
25.26.(a) Find a formula for T(a, b), where T represents the projection on
the y-axis along the x-axis.
(b) Find a formula for T(a, b), where T represents the projection on
the y-axis along the line L = {(s, s) : s ∈ R}.
Let T : R3 → R3 .
(a)(b)(c)If T(a, b, c) = (a, b, 0), show that T is the projection on the xy-
plane along the z-axis.
Find a formula for T(a, b, c), where T represents the projection on
the z-axis along the xy-plane.
If T(a, b, c) = (a − c, b, 0), show that T is the projection on the
xy-plane along the line L = {(a, 0, a) : a ∈ R}.
Using the notation in the definition above, assume that T : V → V is
the projection on W1 along W2 .
(a)
(b)
(c)
(d)
Prove that T is linear and W1 = {x ∈ V : T(x) = x}.
Prove that W1 = R(T) and W2 = N(T).
Describe T if W1 = V.
Describe T if W1 is the zero subspace.
27. Suppose that W is a subspace of a finite-dimensional vector space V.
(a) Prove that there exists a subspace W and a function T : V → V
such that T is a projection on W along W .
(b) Give an example of a subspace W of a vector space V such that
there are two projections on W along two (distinct) subspaces.
The following definitions are used in Exercises 28–32.
Definitions. Let V be a vector space, and let T : V → V be linear. A
subspace W of V is said to be T-invariant if T(x) ∈ W for every x ∈ W, that
is, T(W) ⊆ W. If W is T-invariant, we define the restriction of T on W to
be the function TW : W → W defined by TW (x) = T(x) for all x ∈ W.
Exercises 28–32 assume that W is a subspace of a vector space V and that
T : V → V is linear. Warning: Do not assume that W is T-invariant or that
T is a projection unless explicitly stated.
28. Prove that the subspaces {0 }, V, R(T), and N(T) are all T-invariant.
29. If W is T-invariant, prove that TW is linear.
30. Suppose that T is the projection on W along some subspace W . Prove
that W is T-invariant and that TW = IW .
31. Suppose that V = R(T)⊕W and W is T-invariant. (Recall the definition
of direct sum given in the exercises of Section 1.3.)
78
Chap. 2
Linear Transformations and Matrices
(a)(b)(c)Prove that W ⊆ N(T).
Show that if V is finite-dimensional, then W = N(T).
Show by example that the conclusion of (b) is not necessarily true
if V is not finite-dimensional.
32.Suppose that W is T-invariant. Prove that N(TW ) = N(T) ∩ W and
R(TW ) = T(W).
33. Prove Theorem 2.2 for the case that β is infinite, that is, R(T) =
span({T(v) : v ∈ β}).
34.Prove the following generalization of Theorem 2.6: Let V and W be
vector spaces over a common field, and let β be a basis for V. Then for
any function f : β → W there exists exactly one linear transformation
T : V → W such that T(x) = f (x) for all x ∈ β.
Exercises 35 and 36 assume the definition of direct sum given in the exercises
of Section 1.3.
35.Let V be a finite-dimensional vector space and T : V → V be linear.
(a) Suppose that V = R(T) + N(T). Prove that V = R(T) ⊕ N(T).
(b) Suppose that R(T) ∩ N(T) = {0 }. Prove that V = R(T) ⊕ N(T).
Be careful to say in each part where finite-dimensionality is used.
36.Let V and T be as defined in Exercise 21.
(a)(b)Prove that V = R(T)+N(T), but V is not a direct sum of these two
spaces. Thus the result of Exercise 35(a) above cannot be proved
without assuming that V is finite-dimensional.
Find a linear operator T1 on V such that R(T1 ) ∩ N(T1 ) = {0 } but
V is not a direct sum of R(T1 ) and N(T1 ). Conclude that V being
finite-dimensional is also essential in Exercise 35(b).
37.A function T : V → W between vector spaces V and W is called additive
if T(x + y) = T(x) + T(y) for all x, y ∈ V. Prove that if V and W
are vector spaces over the field of rational numbers, then any additive
function from V into W is a linear transformation.
38.Let T : C → C be the function defined by T(z) = z. Prove that T is
additive (as defined in Exercise 37) but not linear.
39.Prove that there is an additive function T : R → R (as defined in Ex-
ercise 37) that is not linear. Hint: Let V be the set of real numbers
regarded as a vector space over the field of rational numbers. By the
corollary to Theorem 1.13 (p. 60), V has a basis β. Let x and y be two
distinct vectors in β, and define f : β → V by f (x) = y, f (y) = x, and
f (z) = z otherwise. By Exercise 34, there exists a linear transformation
Sec. 2.2
The Matrix Representation of a Linear Transformation
79
T : V → V such that T(u) = f (u) for all u ∈ β. Then T is additive, but
for c = y/x, T(cx) = cT(x).
The following exercise requires familiarity with the definition of quotient space
given in Exercise 31 of Section 1.3.
40.Let V be a vector space and W be a subspace of V. Define the mapping
η : V → V/W by η(v) = v + W for v ∈ V.
(a) Prove that η is a linear transformation from V onto V/W and that
N(η) = W.
(b) Suppose that V is finite-dimensional. Use (a) and the dimen-
sion theorem to derive a formula relating dim(V), dim(W), and
dim(V/W).
(c) Read the proof of the dimension theorem. Compare the method of
solving (b) with the method of deriving the same result as outlined
in Exercise 35 of Section 1.6.
1.2.3.4.Label the following statements as true or false. Assume that V and
W are finite-dimensional vector spaces with ordered bases β and γ,
respectively, and T, U : V → W are linear transformations.
(a)
(b)
(c)
(d)
(e)
(f )
For any scalar a, aT + U is a linear transformation from V to W.
[T]γβ = [U]γβ implies that T = U.
If m = dim(V) and n = dim(W), then [T]γβ is an m × n matrix.
[T + U]γβ = [T]γβ + [U]γβ .
L(V, W) is a vector space.
L(V, W) = L(W, V).
Let β and γ be the standard ordered bases for Rn and Rm , respectively.
For each linear transformation T : Rn → Rm , compute [T]γβ .
(a)
T :
R2
→ R3 defined by T(a1 , a2 ) = (2a1 − a2 , 3a1 + 4a2 , a1 ).
(b)
T :
R3
→ R2 defined by T(a1 , a2 , a3 ) = (2a1 + 3a2 − a3 , a1 + a3 ).
(c)
T :
R3
→ R defined by T(a1 , a2 , a3 ) = 2a1 + a2 − 3a3 .
(d)
T :
R3
→ R3 defined by
T(a1 , a2 , a3 ) = (2a2 + a3 , −a1 + 4a2 + 5a3 , a1 + a3 ).
(e) T : Rn → Rn defined by T(a1 , a2 , . . . , an ) = (a1 , a1 , . . . , a1 ).
(f ) T : Rn → Rn defined by T(a1 , a2 , . . . , an ) = (an , an−1, . . . , a1 ).
(g) T : Rn → R defined by T(a1 , a2 , . . . , an ) = a1 + an .
Let T : R2 → R3 be defined by T(a1 , a2) = (a1 − a2 , a1 , 2a1 + a2 ). Let β
be the standard ordered basis for R2 and γ = {(1, 1, 0), (0, 1, 1), (2, 2, 3)}.
Compute [T]γβ . If α = {(1, 2), (2, 3)}, compute [T]γα .
Define
T : M2×2 (R) → P2 (R)
by
T
a
c
b
d
= (a + b) + (2d)x + bx2 .
Let
1 0
0
1
0
0
0
0
β =
,
,
,
and γ = {1, x, x2 }.
0 0
0
0
1
0
0
1
Compute [T]γβ .
5. Let
1
0
0
1
0
0
0
0
α =
,
,
,
,
0
0
0
0
1
0
0
1
β = {1, x, x2 },
and
γ = {1}.
Sec. 2.2
The Matrix Representation of a Linear Transformation
85
(a) Define T : M2×2 (F ) → M2×2 (F ) by T(A) = At . Compute [T]α .
(b) Define
f (0) 2f (1)
T : P2 (R) → M2×2 (R)
by
T(f (x)) =
0
f (3)
,
(c)(d)(e)where denotes differentiation. Compute [T]α
.
βDefine T : M2×2 (F ) → F by T(A) = tr(A). Compute [T]γα .
Define T : P2 (R) → R by T(f (x)) = f (2). Compute [T]γβ .
If
1
−2
A =
,
0
4
(f )(g)compute [A]α .
If f (x) = 3 − 6x + x2 , compute [f (x)]β .
For a ∈ F , compute [a]γ .
6. Complete the proof of part (b) of Theorem 2.7.
7. Prove part (b) of Theorem 2.8.
8. † Let V be an n-dimensional vector space with an ordered basis β. Define
T : V → Fn by T(x) = [x]β . Prove that T is linear.
9.Let V be the vector space of complex numbers over the field R. Define
T : V → V by T(z) = z, where z is the complex conjugate of z. Prove
that T is linear, and compute [T]β , where β = {1, i}. (Recall by Exer-
cise 38 of Section 2.1 that T is not linear if V is regarded as a vector
space over the field C.)
10.Let V be a vector space with the ordered basis β = {v1 , v2 , . . . , vn }.
Define v0 = 0 . By Theorem 2.6 (p. 72), there exists a linear trans-
formation T : V → V such that T(vj ) = vj + vj−1 for j = 1, 2, . . . , n.
Compute [T]β .
11.Let V be an n-dimensional vector space, and let T : V → V be a linear
transformation. Suppose that W is a T-invariant subspace of V (see the
exercises of Section 2.1) having dimension k. Show that there is a basis
β for V such that [T]β has the form
A
B
,
O
C
where A is a k × k matrix and O is the (n − k) × k zero matrix.
86
Chap. 2
Linear Transformations and Matrices
12.Let V be a finite-dimensional vector space and T be the projection on
W along W , where W and W are subspaces of V. (See the definition
in the exercises of Section 2.1 on page 76.) Find an ordered basis β for
V such that [T]β is a diagonal matrix.
13.14.Let V and W be vector spaces, and let T and U be nonzero linear
transformations from V into W. If R(T) ∩ R(U) = {0 }, prove that
{T, U} is a linearly independent subset of L(V, W).
Let V = P(R), and for j ≥ 1 define Tj (f (x)) = f (j) (x), where f (j) (x)
is the jth derivative of f (x). Prove that the set {T1 , T2 , . . . , Tn } is a
linearly independent subset of L(V) for any positive integer n.
15. Let V and W be vector spaces, and let S be a subset of V. Define
S 0 = {T ∈ L(V, W) : T(x) = 0 for all x ∈ S}. Prove the following
statements.
(a) S 0 is a subspace of L(V, W).
(b) If S1 and S2 are subsets of V and S1 ⊆ S2 , then S2 0 ⊆ S1 0 .
(c) If V1 and V2 are subspaces of V, then (V1 + V2 )0 = V1 0 ∩ V2 0 .
16.Let V and W be vector spaces such that dim(V) = dim(W), and let
T : V → W be linear. Show that there exist ordered bases β and γ for
V and W, respectively, such that [T]γβ is a diagonal matrix.
1.Label the following statements as true or false. In each part, V, W,
and Z denote vector spaces with ordered (finite) bases α, β, and γ,
respectively; T : V → W and U : W → Z denote linear transformations;
and A and B denote matrices.
(a) [UT]γα = [T]βα [U]γβ .
(b)
[T(v)]β = [T]βα[v]α for all v ∈ V.
(c)
[U(w)]β = [U]βα [w]β for all w ∈ W.
(d)
(e)
(f )
(g)
(h)
(i)
(j)
[IV ]α = I.
[T2 ]βα = ([T]βα )2 .
A2 = I implies that A = I or A = −I.
T = LA for some matrix A.
A2 = O implies that A = O, where O denotes the zero matrix.
LA+B = LA + LB .
If A is square and Aij = δij for all i and j, then A = I.
2. (a) Let
1
3
1
0
−3
A =
,
B =
,
2
−1
4
1
2
⎛ ⎞
2
C =
1
1 4
,
and D = ⎝−2⎠ .
−1
−2 0
3
Compute A(2B + 3C), (AB)D, and A(BD).
(b) Let
⎛
⎞
⎛
⎞
2 5
3 −2 0
A = ⎝−3 1⎠ , B = ⎝1 −1 4⎠ , and C = 4
0
3 .
4 2
5
5 3
Compute At , At B, BC t , CB, and CA.
3. Let g(x) = 3 + x. Let T : P2 (R) → P2 (R) and U : P2 (R) → R3 be the
linear transformations respectively defined by
T(f (x)) = f (x)g(x) + 2f (x) and U (a + bx + cx2 ) = (a + b, c, a − b).
Let β and γ be the standard ordered bases of P2 (R) and R3 , respectively.
Sec. 2.3
Composition of Linear Transformations and Matrix Multiplication
97
(a)(b)Compute [U]γβ , [T]β , and [UT]γβ directly. Then use Theorem 2.11
to verify your result.
Let h(x) = 3 − 2x + x2 . Compute [h(x)]β and [U(h(x))]γ . Then
use [U]γβ from (a) and Theorem 2.14 to verify your result.
4.5.For each of the following parts, let T be the linear transformation defined
in the corresponding part of Exercise 5 of Section 2.2. Use Theorem 2.14
to compute the following vectors:
1 4
(a) [T(A)]α , where A =
.
−1 6
(b) [T(f (x))]α , where f (x)
= 4 − 6x + 3x2
.
1 3
(c) [T(A)]γ , where A =
.
2 4
(d) [T(f (x))]γ , where f (x) = 6 − x + 2x2 .
Complete the proof of Theorem 2.12 and its corollary.
6. Prove (b) of Theorem 2.13.
7. Prove (c) and (f) of Theorem 2.15.
8.9.10.11.12.13.Prove Theorem 2.10. Now state and prove a more general result involv-
ing linear transformations with domains unequal to their codomains.
Find linear transformations U, T : F2 → F2 such that UT = T0 (the zero
transformation) but TU = T0 . Use your answer to find matrices A and
B such that AB = O but BA = O.
Let A be an n × n matrix. Prove that A is a diagonal matrix if and
only if Aij = δij Aij for all i and j.
Let V be a vector space, and let T : V → V be linear. Prove that T2 = T0
if and only if R(T) ⊆ N(T).
Let V, W, and Z be vector spaces, and let T : V → W and U : W → Z
be linear.
(a) Prove that if UT is one-to-one, then T is one-to-one. Must U also
be one-to-one?
(b) Prove that if UT is onto, then U is onto. Must T also be onto?
(c) Prove that if U and T are one-to-one and onto, then UT is also.
Let A and B be n × n matrices. Recall that the trace of A is defined
by
n
tr(A) =
Aii .
i=1
Prove that tr(AB) = tr(BA) and tr(A) = tr(At ).
98
Chap. 2
Linear Transformations and Matrices
14.Assume the notation in Theorem 2.13.
(a) Suppose that z is a (column) vector in Fp . Use Theorem 2.13(b)
to prove that Bz is a linear combination of the columns of B. In
particular, if z = (a1 , a2 , . . . , ap )t , then show that
p
Bz =
aj vj .
j=1
(b) Extend (a) to prove that column j of AB is a linear combination
of the columns of A with the coefficients in the linear combination
being the entries of column j of B.
(c) For any row vector w ∈ Fm , prove that wA is a linear combination
of the rows of A with the coefficients in the linear combination
being the coordinates of w. Hint: Use properties of the transpose
operation applied to (a).
(d) Prove the analogous result to (b) about rows: Row i of AB is a
linear combination of the rows of B with the coefficients in the
linear combination being the entries of row i of A.
†15. Let M and A be matrices for which the product matrix M A is defined.
If the jth column of A is a linear combination of a set of columns
of A, prove that the jth column of M A is a linear combination of the
corresponding columns of M A with the same corresponding coefficients.
16.Let V be a finite-dimensional vector space, and let T : V → V be linear.
(a) If rank(T) = rank(T2 ), prove that R(T) ∩ N(T) = {0 }. Deduce
that V = R(T) ⊕ N(T) (see the exercises of Section 1.3).
(b) Prove that V = R(Tk ) ⊕ N(Tk ) for some positive integer k.
17.Let V be a vector space. Determine all linear transformations T : V → V
such that T = T2 . Hint: Note that x = T(x) + (x − T(x)) for every
x in V, and show that V = {y : T(y) = y} ⊕ N(T) (see the exercises of
Section 1.3).
18.Using only the definition of matrix multiplication, prove that multipli-
cation of matrices is associative.
19.For an incidence matrix A with related matrix B defined by Bij = 1 if
i is related to j and j is related to i, and Bij = 0 otherwise, prove that
i belongs to a clique if and only if (B 3 )ii > 0.
20.Use Exercise 19 to determine the cliques in the relations corresponding
to the following incidence matrices.
Sec. 2.4
Invertibility and Isomorphisms
⎛
⎞
⎛
⎞
0
1
0
1
0
0
1
1
⎜1
0
0
0⎟
⎜1
0
0
1⎟
(a) ⎜
⎟
(b)
⎜
⎟
⎝0
1
0
1⎠
⎝1
0
0
1⎠
1
0
1
0
1
0
1
0
99
21.Let A be an incidence matrix that is associated with a dominance rela-
tion. Prove that the matrix A + A2 has a row [column] in which each
entry is positive except for the diagonal entry.
22. Prove that the matrix
⎛
⎞
0
1
0
A = ⎝0
0
1⎠
1
0
0
corresponds to a dominance relation. Use Exercise 21 to determine
which persons dominate [are dominated by] each of the others within
two stages.
23.Let A be an n × n incidence matrix that corresponds to a dominance
relation. Determine the number of nonzero entries of A.
1.Label the following statements as true or false. In each part, V and
W are vector spaces with ordered (finite) bases α and β, respectively,
T : V → W is linear, and A and B are matrices.
−1
(a) [T]βα
= [T−1 ]βα .
(b) T is invertible if and only if T is one-to-one and onto.
(c) T = LA , where A = [T]βα .
(d) M2×3 (F ) is isomorphic to F5 .
(e) Pn (F ) is isomorphic to Pm (F ) if and only if n = m.
(f ) AB = I implies that A and B are invertible.
(g) If A is invertible, then (A−1 )−1 = A.
(h) A is invertible if and only if LA is invertible.
(i) A must be square in order to possess an inverse.
2.For each of the following linear transformations T, determine whether
T is invertible and justify your answer.
(a)
T :
R2 → R3 defined by T(a1 , a2 ) = (a1 − 2a2 , a2 , 3a1 + 4a2 ).
(b)
T :
R2 → R3 defined by T(a1 , a2 ) = (3a1 − a2 , a2 , 4a1 ).
(c)
T :
R3 → R3 defined by T(a1 , a2 , a3 ) = (3a1 − 2a3, a2 , 3a1 + 4a2 ).
(d)
T :
P3 (R) → P2 (R) defined by T(p(x)) = p (x).
a b
(e) T : M2×2 (R) → P2 (R) defined by T
= a + 2bx + (c + d)x2 .
c d
a b
a + b
a
(f ) T : M2×2 (R) → M2×2 (R) defined by T
=
.
c d
c
c + d
Sec. 2.4
Invertibility and Isomorphisms
107
3.Which of the following pairs of vector spaces are isomorphic?your answers.
(a)
(b)
(c)
(d)
F3 and P3 (F ).
F4 and P3 (F ).
M2×2 (R) and P3 (R).
V = {A ∈ M2×2 (R) : tr(A) = 0} and R4 .
Justify
4. † Let A and B be n × n invertible matrices. Prove that AB is invertible
and (AB)−1 = B −1 A−1 .
5. † Let A be invertible. Prove that At is invertible and (At )−1 = (A−1 )t .
6. Prove that if A is invertible and AB = O, then B = O.
7.Let A be an n × n matrix.
(a)(b)Suppose that A2 = O. Prove that A is not invertible.
Suppose that AB = O for some nonzero n × n matrix B. Could A
be invertible? Explain.
8. Prove Corollaries 1 and 2 of Theorem 2.18.
9.Let A and B be n × n matrices such that AB is invertible. Prove that A
and B are invertible. Give an example to show that arbitrary matrices
A and B need not be invertible if AB is invertible.
10. † Let A and B be n × n matrices such that AB = In .
(a) Use Exercise 9 to conclude that A and B are invertible.
(b) Prove A = B −1 (and hence B = A−1 ). (We are, in effect, saying
that for square matrices, a “one-sided” inverse is a “two-sided”
inverse.)
(c) State and prove analogous results for linear transformations de-
fined on finite-dimensional vector spaces.
11.12.13.14.Verify that the transformation in Example 5 is one-to-one.
Prove Theorem 2.21.
Let ∼ mean “is isomorphic to.” Prove that ∼ is an equivalence relation
on the class of vector spaces over F .
Let
a
a + b
V =
: a, b, c ∈ F
.
0
c
Construct an isomorphism from V to F3 .
108
Chap. 2
Linear Transformations and Matrices
15.Let V and W be finite-dimensional vector spaces, and let T : V → W be
a linear transformation. Suppose that β is a basis for V. Prove that T
is an isomorphism if and only if T(β) is a basis for W.
16.Let B be an n × n invertible matrix. Define Φ : Mn×n (F ) → Mn×n (F )
by Φ(A) = B −1 AB. Prove that Φ is an isomorphism.
17. † Let V and W be finite-dimensional vector spaces and T : V → W be an
isomorphism. Let V0 be a subspace of V.
(a)(b)Prove that T(V0 ) is a subspace of W.
Prove that dim(V0 ) = dim(T(V0 )).
18. Repeat Example 7 with the polynomial p(x) = 1 + x + 2x2 + x3 .
19.In Example 5 of Section 2.1, the mapping T : M2×2 (R) → M2×2 (R) de-
fined by T(M ) = M t for each M ∈ M2×2 (R) is a linear transformation.
Let β = {E 11 , E 12 , E 21 , E 22 }, which is a basis for M2×2 (R), as noted in
Example 3 of Section 1.6.
(a)(b)Compute [T]β .
Verify that LA φβ (M ) = φβ T(M ) for A = [T]β and
1
2
M =
.
3
4
20. † Let T : V → W be a linear transformation from an n-dimensional vector
space V to an m-dimensional vector space W. Let β and γ be ordered
bases for V and W, respectively. Prove that rank(T) = rank(LA ) and
that nullity(T) = nullity(LA ), where A = [T]γβ . Hint: Apply Exercise 17
to Figure 2.2.
21.Let V and W be finite-dimensional vector spaces with ordered bases
β = {v1 , v2 , . . . , vn } and γ = {w1 , w2 , . . . , wm }, respectively. By The-
orem 2.6 (p. 72), there exist linear transformations Tij : V → W such
that
wi if k = j
Tij (vk ) =
0
if k = j.
First prove that {Tij : 1 ≤ i ≤ m, 1 ≤ j ≤ n} is a basis for L(V, W).
Then let M ij be the m × n matrix with 1 in the ith row and jth column
and 0 elsewhere, and prove that [Tij ]γβ = M ij . Again by Theorem 2.6,
there exists a linear transformation Φ : L(V, W) → Mm×n (F ) such that
Φ(Tij ) = M ij . Prove that Φ is an isomorphism.
Sec. 2.4
Invertibility and Isomorphisms
109
22.Let c0 , c1 , . . . , cn be distinct scalars from an infinite field F . Define
T : Pn (F ) → Fn+1 by T(f ) = (f (c0 ), f (c1 ), . . . , f (cn )). Prove that T is
an isomorphism. Hint: Use the Lagrange polynomials associated with
c0 , c1 , . . . , cn.
23.Let V denote the vector space defined in Example 5 of Section 1.2, and
let W = P(F ). Define
n
T : V → W
by
T(σ) =
σ(i)xi ,
i=0
where n is the largest integer such that σ(n) = 0. Prove that T is an
isomorphism.
The following exercise requires familiarity with the concept of quotient space
defined in Exercise 31 of Section 1.3 and with Exercise 40 of Section 2.1.
24. Let T : V → Z be a linear transformation of a vector space V onto a
vector space Z. Define the mapping
T : V/N(T) → Z
by
T(v + N(T)) = T(v)
for any coset v + N(T) in V/N(T).
(a) Prove that T is well-defined; that is, prove that if v + N(T) =
v + N(T), then T(v) = T(v ).
(b) Prove that T is linear.
(c) Prove that T is an isomorphism.
(d) Prove that the diagram shown in Figure 2.3 commutes; that is,
prove that T = Tη.
V
T
- Z
η
T
U
V/N(T)
Figure 2.3
25.Let V be a nonzero vector space over a field F , and suppose that S is
a basis for V. (By the corollary to Theorem 1.13 (p. 60) in Section 1.7,
every vector space has a basis). Let C(S, F ) denote the vector space of
all functions f ∈ F(S, F ) such that f (s) = 0 for all but a finite number
110
Chap. 2
Linear Transformations and Matrices
of vectors in S. (See Exercise 14 of Section 1.3.) Let Ψ : C(S, F ) → V
be the function defined by
Ψ(f ) =
f (s)s.
s∈S,f (s)=0
Prove that Ψ is an isomorphism. Thus every nonzero vector space can
be viewed as a space of functions.
1.Label(a)(b)(c)(d)(e)the following statements as true or false.
Suppose that β = {x1 , x2 , . . . , xn } and β = {x 1 , x 2 , . . . , x n } are
ordered bases for a vector space and Q is the change of coordinate
matrix that changes β -coordinates into β-coordinates. Then the
jth column of Q is [xj ]β .
Every change of coordinate matrix is invertible.
Let T be a linear operator on a finite-dimensional vector space V,
let β and β be ordered bases V,for and let Q be the change of
coordinate matrix that changes β -coordinates into β-coordinates.
Then [T]β = Q[T]β Q−1 .
The matrices A, B ∈ Mn×n (F ) are called similar if B = Qt AQ for
some Q ∈ Mn×n (F ).
Let T be a linear operator on a finite-dimensional vector space V.
Then for any ordered bases β and γ for V, [T]β is similar to [T]γ .
2.For each of the following pairs of ordered bases β and β for R2 , find
the change of coordinate matrix that changes β -coordinates into β-
coordinates.
(a)
β
= {e1 , e2 } and β = {(a1 , a2 ), (b1 , b2 )}
(b)
β
= {(−1, 3), (2, −1)} and β = {(0, 10), (5, 0)}
(c)
β
= {(2, 5), (−1, −3)} and β = {e1 , e2 }
(d)
β
= {(−4, 3), (2, −1)} and β = {(2, 1), (−4, 1)}
3.For each of the following pairs of ordered bases β and β for P2 (R),
find the change of coordinate matrix that changes β -coordinates into
β-coordinates.
(a) β = {x2 , x, 1} and
β = {a2 x2 + a1 x + a0 , b2 x2 + b1 x + b0 , c2 x2 + c1 x + c0 }
(b) β = {1, x, x2} and
β = {a2 x2 + a1 x + a0 , b2 x2 + b1 x + b0 , c2 x2 + c1 x + c0 }
(c) β = {2x2 − x, 3x2 + 1, x2 } and β = {1, x, x2 }
(d) β = {x2 − x + 1, x + 1, x2 + 1} and
β = {x2 + x + 4, 4x2 − 3x + 2, 2x2 + 3}
(e) β = {x2 − x, x2 + 1, x − 1} and
β = {5x2 − 2x − 3, −2x2 + 5x + 5, 2x2 − x − 3}
(f ) β = {2x2 − x + 1, x2 + 3x − 2, −x2 + 2x + 1} and
β = {9x − 9, x2 + 21x − 2, 3x2 + 5x + 2}
4. Let T be the linear operator on R2 defined by
a
2a + b
T
=
,
b
a − 3b
Sec. 2.5
The Change of Coordinate Matrix
let β be the standard ordered basis for R2 , and let
1
1
β =
,
.
1
2
117
Use Theorem 2.23 and the fact that
−1
1 1
2 −1
=
1 2
−1
1
to find [T]β .
5. Let T be the linear operator on P1 (R) defined by T(p(x)) = p (x),
the derivative of p(x). Let β = {1, x} and β = {1 + x, 1 − x}. Use
Theorem 2.23 and the fact that
⎛
⎞
−1
1
1
1
1
= ⎝ 2
1
2
⎠
1 −1
2
− 12
to find [T]β .
6. For each matrix A and ordered basis β, find [LA ]β . Also, find an invert-
ible matrix Q such that [L A ]β = Q−1 AQ.
1 3
1
1
(a) A =
and β =
,
1 1
1
2
1 2
1
1
(b) A =
and β =
,
2 1
1
−1
⎛
⎞
⎧⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎫
1 1 −1
⎨ 1
1
1 ⎬
(c) A = ⎝2 0
1⎠ and β = ⎝1⎠ , ⎝0⎠ , ⎝1⎠
⎩
⎭
1 1
0
1
1
2
⎛
⎞
⎧⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎫
13
1
4
⎨
1
1
1 ⎬
(d) A = ⎝ 1 13 4⎠ and β = ⎝ 1⎠ , ⎝ −1⎠ , ⎝1⎠
⎩
⎭
4
4 10
−2
0
1
7. In R2 , let L be the line y = mx, where m = 0. Find an expression for
T(x, y), where
(a) T is the reflection of R2 about L.
(b) T is the projection on L along the line perpendicular to L. (See
the definition of projection in the exercises of Section 2.1.)
8. Prove the following generalization of Theorem 2.23. Let T : V → W be
a linear transformation from a finite-dimensional vector space V to a
finite-dimensional vector space W. Let β and β be ordered bases for
118
Chap. 2
Linear Transformations and Matrices
V, and let γ and γ be ordered bases for W. Then [T]γ β = P −1 [T]γβ Q,
where Q is the matrix that changes β -coordinates into β-coordinates
and P is the matrix that changes γ -coordinates into γ-coordinates.
9.10.Prove that “is similar to” is an equivalence relation on Mn×n (F ).
Prove that if A and B are similar n × n matrices, then tr(A) = tr(B).
Hint: Use Exercise 13 of Section 2.3.
11. Let V be a finite-dimensional vector space with ordered bases α, β,
and γ.
(a) Prove that if Q and R are the change of coordinate matrices that
change α-coordinates into β-coordinates and β-coordinates into
γ-coordinates, respectively, then RQ is the change of coordinate
matrix that changes α-coordinates into γ-coordinates.
(b) Prove that if Q changes α-coordinates into β-coordinates, then
Q−1 changes β-coordinates into α-coordinates.
12. Prove the corollary to Theorem 2.23.
13. † Let V be a finite-dimensional vector space over a field F , and let β =
{x1 , x2 , . . . , xn } be an ordered basis for V. Let Q be an n × n invertible
matrix with entries from F . Define
n
x j =
Qij xi
for 1 ≤ j ≤ n,
i=1
and set β = {x 1 , x 2 , . . . , x n }. Prove that β is a basis for V and hence
that Q is the change of coordinate matrix changing β -coordinates into
β-coordinates.
14.Prove the converse of Exercise 8: If A and B are each m × n matrices
with entries from a field F , and if there exist invertible m × m and n × n
−1matrices P and Q, respectively, such that B = P AQ, then there exist
an n-dimensional vector space V and an m-dimensional vector space W
(both over F ), ordered bases β and β for V and γ and γ for W, and a
linear transformation T : V → W such that
A = [T]γβ
and B = [T]γ β .
Hints: Let V = Fn , W = Fm , T = LA , and β and γ be the standard
ordered bases for Fn and Fm , respectively. Now apply the results of
Exercise 13 to obtain ordered bases β and γ from β and γ via Q and
P , respectively.
1.Label the following statements as true or false. Assume that all vector
spaces are finite-dimensional.
(a)(b)(c)(d)(e)(f )(g)Every linear transformation is a linear functional.
A linear functional defined on a field may be represented as a 1 × 1
matrix.
Every vector space is isomorphic to its dual space.
Every vector space is the dual of some other vector space.
If T is an isomorphism from V onto V∗ and β is a finite ordered
basis for V, then T(β) = β ∗ .
If T is a linear transformation from V to W, then the domain of
(Tt )t is V∗∗ .
If V is isomorphic to W, then V∗ is isomorphic to W ∗ .
124
Chap. 2
Linear Transformations and Matrices
(h) The derivative of a function may be considered as a linear func-
tional on the vector space of differentiable functions.
2.For the following functions f on a vector space V, determine which are
linear functionals.
(a)
V = P(R); f(p(x)) = 2p (0) + p (1), where denotes differentiation
(b)
V = R2 ; f(x, y) = (2x, 4y)
(c)
V = M2×2 (F ); f(A) = tr(A)
(d)
V = R3 ; f(x, y, z) = x2 + y2 + z 2
1
(e)
V = P(R); f(p(x)) = p(t) dt
0(f )
V = M2×2 (F ); f(A) = A11
3.4.For each of the following vector spaces V and bases β, find explicit
formulas for vectors of the dual basis β ∗ for V ∗ , as in Example 4.
(a) V = R3 ; β = {(1, 0, 1), (1, 2, 1), (0, 0, 1)}
(b) V = P2 (R); β = {1, x, x2 }
Let V = R3 , and define f1 , f2 , f3 ∈ V∗ as follows:
5.f1 (x, y, z) = x − 2y,
f2 (x, y, z) = x + y + z,
f3 (x, y, z) = y − 3z.
Prove that {f1 , f2 , f3 } is a basis for V∗ , and then find a basis for V for
which it is the dual basis.
Let V = P1 (R), and, for p(x) ∈ V, define f1 , f2 ∈ V∗ by
1
2
f1 (p(x)) =
p(t) dt
and f2 (p(x)) =
p(t) dt.
0
0
Prove that {f1 , f2 } is a basis for V ∗ , and find a basis for V for which it
is the dual basis.
6.Define f ∈ (R2 )∗ by f(x, y) = 2x + y and T : R2 → R2 by T(x, y) =
(3x + 2y, x).
(a)(b)(c)Compute Tt (f).
Compute [Tt ]β ∗ , where β is the standard ordered basis for R2 and
β ∗ = {f1 , f2 } is the dual basis, by finding scalars a, b, c, and d such
that Tt (f1 ) = af1 + cf2 and Tt (f2 ) = bf1 + df2 .
Compute [T]β and ([T]β )t , and compare your results with (b).
7.Let V = P1 (R) and W = R2 with respective standard ordered bases β
and γ. Define T : V → W by
T(p(x)) = (p(0) − 2p(1), p(0) + p (0)),
where p (x) is the derivative of p(x).
Sec. 2.6
Dual Spaces
125
(a)(b)(c)For f ∈ W∗ defined by f(a, b) = a − 2b, compute Tt (f).
∗
Compute [Tt ]β γ∗ without appealing to Theorem 2.25.
Compute [T]γβ and its transpose, and compare your results with
(b).
8.9.Show that every plane through the origin in R3 may be identified with
the null space of a vector in (R3 )∗ . State an analogous result for R2 .
Prove that a function T : Fn → Fm is linear if and only if there exist
f1 , f2 , . . . , fm ∈ (Fn )∗ such that T(x) = (f1(x), f2 (x), . . . , fm (x)) for all
x ∈ Fn . Hint: If T is linear, define fi (x) = (gi T)(x) for x ∈ Fn ; that is,
fi = Tt (gi ) for 1 ≤ i ≤ m, where {g1 , g2 , . . . , gm } is the dual basis of
the standard ordered basis for Fm .
10.Let(a)(b)(c)V= Pn(F ), and let c0 , c1 , . . . , cn be distinct scalars in F .
For 0 ≤ i ≤ n, define fi ∈ V∗ by fi(p(x)) = p(ci ). Prove that
{f0 , f1 , . . . , fn } is a basis for V∗ . Hint: Apply any linear combi-
nation of this set that equals the zero transformation to p(x) =
(x − c1 )(x − c2 ) · · · (x − cn ), and deduce that the first coefficient is
zero.
Use the corollary to Theorem 2.26 and (a) to show that there exist
unique polynomials p0 (x), p1 (x), . . . , pn (x) such that pi (cj ) = δij
for 0 ≤ i ≤ n. These polynomials are the Lagrange polynomials
defined in Section 1.6.
For any scalars a0 , a1 , . . . , an (not necessarily distinct), deduce that
there exists a unique polynomial q(x) of degree at most n such that
q(ci ) = ai for 0 ≤ i ≤ n. In fact,
n
q(x) =
ai pi (x).
i=0
(d)(e)Deduce the Lagrange interpolation formula:
n
p(x) =
p(ci )pi (x)
i=0
for any p(x) ∈ V.
Prove that
b
n
p(t) dt =
p(ci )di ,
a
i=0
where
b
di =
pi (t) dt.
a
126
Chap. 2
Linear Transformations and Matrices
Suppose now that
i(b − a)
ci = a +
for i = 0, 1, . . . , n.
n
For n = 1, the preceding result yields the trapezoidal rule for
evaluating the definite integral of a polynomial. For n = 2, this
result yields Simpson’s rule for evaluating the definite integral of
a polynomial.
11.Let V and W be finite-dimensional vector spaces over F , and let ψ1 and
ψ2 be the isomorphisms between V and V∗∗ and W and W∗∗ , respec-
tively, as defined in Theorem 2.26. Let T : V → W be linear, and define
Ttt = (Tt )t . Prove that the diagram depicted in Figure 2.6 commutes
(i.e., prove that ψ2 T = Ttt ψ1).
T
V −−−−→ W
⏐
⏐
⏐
⏐ψ1 !
!ψ
2
V∗∗ −−−−→ Ttt
W ∗∗
Figure 2.6
12.Let V be a finite-dimensional vector space with the ordered basis β.
∗∗Prove that ψ(β) = β , where ψ is defined in Theorem 2.26.
In Exercises 13 through 17, V denotes a finite-dimensional vector space over
F . For every subset S of V, define the annihilator S 0 of S as
13.14.S 0 = {f ∈ V∗ : f(x) = 0 for all x ∈ S}.
(a)(b)(c)(d)(e)Prove that S 0 is a subspace of V∗ .
If W is a subspace of V and x ∈ W, prove that there exists f ∈ W 0
such that f(x) = 0.
Prove that (S 0 )0 = span(ψ(S)), where ψ is defined as in Theo-
rem 2.26.
For subspaces W1 and W2 , prove that W1 = W2 if and only if
W1 0 = W2 0 .
For subspaces W1 and W2 , show that (W1 + W2 )0 = W1 0 ∩ W2 0 .
Prove that if W is a subspace of V, then dim(W) + dim(W0 ) = dim(V).
Hint: Extend an ordered basis {x1 , x2 , . . . , xk } of W to an ordered ba-
sis β = {x1 , x2 , . . . , xn } of V. Let β ∗ = {f1 , f2 , . . . , fn }. Prove that
{fk+1 , fk+2 , . . . , fn } is a basis for W 0 .
Sec. 2.7
Homogeneous Linear Differential Equations with Constant Coefficients 127
15. Suppose that W is a finite-dimensional vector space and that T : V → W
is linear. Prove that N(Tt ) = (R(T))0 .
16.Use Exercises 14 and 15 to deduce that rank(LAt ) = rank(LA ) for any
A ∈ Mm×n (F ).
17.Let T be a linear operator on V, and let W be a subspace of V. Prove
that W is T-invariant (as defined in the exercises of Section 2.1) if and
only if W0 is Tt -invariant.
18.Let V be a nonzero vector space over a field F , and let S be a basis
for V. (By the corollary to Theorem 1.13 (p. 60) in Section 1.7, every
vector space has a basis.) Let Φ : V ∗ → L(S, F ) be the mapping defined
by Φ(f) = fS , the restriction of f to S. Prove that Φ is an isomorphism.
Hint: Apply Exercise 34 of Section 2.1.
19.Let V be a nonzero vector space, and let W be a proper subspace of V
(i.e., W = V). Prove that there exists a nonzero linear functional f ∈ V∗
such that f(x) = 0 for all x ∈ W. Hint: For the infinite-dimensional
case, use Exercise 34 of Section 2.1 as well as results about extending
linearly independent sets to bases in Section 1.7.
20. Let V and W be nonzero vector spaces over the same field, and let
T : V → W be a linear transformation.
(a)(b)Prove that T is onto if and only if Tt is one-to-one.
Prove that Tt is onto if and only if T is one-to-one.
Hint: Parts of the proof require the result of Exercise 19 for the infinite-
dimensional case.
1.Label(a)(b)(c)(d)(e)(f )(g)the following statements as true or false.
The set of solutions to an nth-order homogeneous linear differential
equation with constant coefficients is an n-dimensional subspace of
C∞ .
The solution space of a homogeneous linear differential equation
with constant coefficients is the null space of a differential operator.
The auxiliary polynomial of a homogeneous linear differential
equation with constant coefficients is a solution to the differential
equation.
Any solution to a homogeneous linear differential equation with
constant coefficients is of the form aect or atk ect , where a and c
are complex numbers and k is a positive integer.
Any linear combination of solutions to a given homogeneous linear
differential equation with constant coefficients is also a solution to
the given equation.
For any homogeneous linear differential equation with constant
coefficients having auxiliary polynomial p(t), if c1 , c2 , . . . , ck are
the distinct zeros of p(t), then {ec1 t , ec2 t , . . . , eck t } is a basis for
the solution space of the given differential equation.
Given any polynomial p(t) ∈ P(C), there exists a homogeneous lin-
ear differential equation with constant coefficients whose auxiliary
polynomial is p(t).
Sec. 2.7
Homogeneous Linear Differential Equations with Constant Coefficients 141
2. For each of the following parts, determine whether the statement is true
or false. Justify your claim with either a proof or a counterexample,
whichever is appropriate.
(a) Any finite-dimensional subspace of C∞ is the solution space of a
homogeneous linear differential equation with constant coefficients.
(b) There exists a homogeneous linear differential equation with con-
stant coefficients whose solution space has the basis {t, t2 }.
(c) For any homogeneous linear differential equation with constant
coefficients, if x is a solution to the equation, so is its derivative
x .
Given two polynomials p(t) and q(t) in P(C), if x ∈ N(p(D)) and y ∈
N(q(D)), then
(d) x + y ∈ N(p(D)q(D)).
(e) xy ∈ N(p(D)q(D)).
3.Find a basis for the solution space of each of the following differential
equations.
(a)
y + 2y + y = 0
(b)
y = y(c)
y (4) − 2y (2) + y = 0
(d)
y + 2y + y = 0
(e)
y (3) − y(2) + 3y (1) + 5y = 0
4.Find a basis for each of the following subspaces of C∞ .
(a)(b)(c)N(D2 − D − I)
N(D3 − 3D2 + 3D − I)
N(D3 + 6D2 + 8D)
5.6.Show that C∞ is a subspace of F(R, C).
(a) Show that D : C∞ → C∞ is a linear operator.
(b) Show that any differential operator is a linear operator on C∞ .
7.Prove that if {x, y} is a basis for a vector space over C, then so is
1
1
(x + y), (x − y) .
2
2i
8.Consider a second-order homogeneous linear differential equation with
constant coefficients in which the auxiliary polynomial has distinct con-
jugate complex roots a + ib and a − ib, where a, b ∈ R. Show that
{eat cos bt, eat sin bt} is a basis for the solution space.
142
9.Chap. 2
Linear Transformations and Matrices
Suppose that {U1 , U2 , . . . , Un } is a collection of pairwise commutative
linear operators on a vector space V (i.e., operators such that UiUj =
Uj Ui for all i, j). Prove that, for any i (1 ≤ i ≤ n),
N(Ui ) ⊆ N(U1 U2 · · · Un ).
10.Prove Theorem 2.33 and its corollary. Hint: Suppose that
b1 e c1 t + b2 ec2 t + · · · + bn ecn t = 0
(where the ci ’s are distinct).
To show the bi’s are zero, apply mathematical induction on n as follows.
Verify the theorem for n = 1. Assuming that the theorem is true for
n − 1 functions, apply the operator D − cn I to both sides of the given
equation to establish the theorem for n distinct exponential functions.
11.Prove Theorem 2.34. Hint: First verify that the alleged basis lies in
the solution space. Then verify that this set is linearly independent by
mathematical induction on k as follows. The case k = 1 is the lemma
to Theorem 2.34. Assuming that the theorem holds for k − 1 distinct
ci ’s, apply the operator (D − ck I)nk to any linear combination of the
alleged basis that equals 0 .
12.Let V be the solution space of an nth-order homogeneous linear differ-
ential equation with constant coefficients having auxiliary polynomial
p(t). Prove that if p(t) = g(t)h(t), where g(t) and h(t) are polynomials
of positive degree, then
N(h(D)) = R(g(DV )) = g(D)(V),
where DV : V → V is defined by DV (x) = x for x ∈ V. Hint: First prove
g(D)(V) ⊆ N(h(D)). Then prove that the two spaces have the same
finite dimension.
13. A differential equation
y (n) + an−1 y (n−1) + · · · + a1 y (1) + a0 y = x
is called a nonhomogeneous linear differential equation with constant
coefficients if the ai ’s are constant and x is a function that is not iden-
tically zero.
(a) Prove that for any x ∈ C∞ there exists y ∈ C∞ such that y is
a solution to the differential equation. Hint: Use Lemma 1 to
Theorem 2.32 to show that for any polynomial p(t), the linear
operator p(D) : C∞ → C∞ is onto.
Sec. 2.7
Homogeneous Linear Differential Equations with Constant Coefficients 143
(b)Let V be the solution space for the homogeneous linear equation
y(n) + an−1 y(n−1) + · · · + a1y (1) + a0 y = 0 .
Prove that if z is any solution to the associated nonhomogeneous
linear differential equation, then the set of all solutions to the
nonhomogeneous linear differential equation is
{z + y : y ∈ V}.
14.Given any nth-order homogeneous linear differential equation with con-
stant coefficients, prove that, for any solution x and any t0 ∈ R, if
x(t0 ) = x (t0 ) = · · · = x(n−1) (t0 ) = 0, then x = 0 (the zero function).
Hint: Use mathematical induction on n as follows. First prove the con-
clusion for the case n = 1. Next suppose that it is true for equations of
order n − 1, and consider an nth-order differential equation with aux-
iliary polynomial p(t). Factor p(t) = q(t)(t − c), and let z = q((D))x.
Show that z(t0 ) = 0 and z − cz = 0 to conclude that z = 0 . Now apply
the induction hypothesis.
15. Let V be the solution space of an nth-order homogeneous linear dif-
ferential equation with constant coefficients. Fix t0 ∈ R, and define a
mapping Φ : V → Cn by
⎛
⎞
x(t0 )
⎜ x (t0 ) ⎟
⎜
⎟
Φ(x) = ⎜
⎝
..
.
⎟ ⎠
for each x in V.
x(n−1) (t0 )
(a)(b)Prove that Φ is linear and its null space is the zero subspace of V.
Deduce that Φ is an isomorphism. Hint: Use Exercise 14.
Prove the following: For any nth-order homogeneous linear dif-
ferential equation with constant coefficients, any t0 ∈ R, and any
complex numbers c0 , c1 , . . . , cn−1 (not necessarily distinct), there
exists exactly one solution, x, to the given differential equation
such that x(t0 ) = c0 and x(k) (t0 ) = ck for k = 1, 2, . . . n − 1.
16.Pendular Motion. It is well known that the motion of a pendulum is
approximated by the differential equation
g
θ + θ = 0 ,
l
where θ(t) is the angle in radians that the pendulum makes with a
vertical line at time t (see Figure 2.8), interpreted so that θ is positive
if the pendulum is to the right and negative if the pendulum is to the
144
Chap. 2
Linear Transformations and Matrices
S
. . . . . . . . . . . . S
. . . .
. . .
. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . θ(t) . . . . . >
. . . . . . . . . . . . .
.
.
S .
.
.
.
.
.
.
.
. . . . . . . .
. . l
. . . . .
. . . . . .
. . . .
. . .
Sq
Figure 2.8
left of the vertical line as viewed by the reader. Here l is the length
of the pendulum and g is the magnitude of acceleration due to gravity.
The variable t and constants l and g must be in compatible units (e.g.,
t in seconds, l in meters, and g in meters per second per second).
(a) Express an arbitrary solution to this equation as a linear combi-
nation of two real-valued solutions.
(b) Find the unique solution to the equation that satisfies the condi-
tions
θ(0) = θ0 > 0
andθ (0) = 0.
(The significance of these conditions is that at time t = 0 the
pendulum is released from a position displaced from the vertical
by θ0 .)
(c) Prove that it takes 2π l/g units of time for the pendulum to make
one circuit back and forth. (This time is called the period of the
pendulum.)
17. Periodic Motion of a Spring without Damping. Find the general solu-
tion to (3), which describes the periodic motion of a spring, ignoring
frictional forces.
18.Periodic Motion of a Spring with Damping. The ideal periodic motion
described by solutions to (3) is due to the ignoring of frictional forces.
In reality, however, there is a frictional force acting on the motion that
is proportional to the speed of motion, but that acts in the opposite
direction. The modification of (3) to account for the frictional force,
called the damping force, is given by
my + ry + ky = 0 ,
where r > 0 is the proportionality constant.
(a) Find the general solution to this equation.
Chap. 2
Index of Definitions
145
(b)(c)Find the unique solution in (a) that satisfies the initial conditions
y(0) = 0 and y (0) = v0 , the initial velocity.
For y(t) as in (b), show that the amplitude of the oscillation de-
creases to zero; that is, prove that lim y(t) = 0.
t→∞
19. In our study of differential equations, we have regarded solutions as
complex-valued functions even though functions that are useful in de-
scribing physical motion are real-valued. Justify this approach.
20.The following parts, which do not involve linear algebra, are included
for the sake of completeness.
(a) Prove Theorem 2.27. Hint: Use mathematical induction on the
number of derivatives possessed by a solution.
(b) For any c, d ∈ C, prove that
1
ec+d = cced
and e−c =
.
ec
(c) Prove Theorem 2.28.
(d) Prove Theorem 2.29.
(e) Prove the product rule for differentiating complex-valued func-
tions of a real variable: For any differentiable functions x and
y in F(R, C), the product xy is differentiable and
(xy) = x y + xy .
(f )Hint: Apply the rules of differentiation to the real and imaginary
parts of xy.
Prove that if x ∈ F(R, C) and x = 0 , then x is a constant func-
tion.
</font></div>
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Linear Algebrahashnuthttps://koreanfoodie.me/90https://koreanfoodie.me/90#entry90commentSat, 15 Jun 2019 13:09:10 +0900Solutions to Linear Algebra, Stephen H. Friedberg, Fourth Edition (Chapter 1)
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<p> </p>
<p><span style="color: #333333;">Solution maual to Linear Algebra, Fourth Edition, Stephen H. Friedberg. (Chapter 1)</span></p>
<p>Solutions to Linear Algebra, Fourth Edition, Stephen H. Friedberg. (Chapter 1)</p>
<p><span style="color: #333333;">Linear Algebra solution manual, Fourth Edition, Stephen H. Friedberg. (Chapter 1)</span><span style="color: #333333;"></span></p>
<p><span style="color: #333333;"><span style="color: #333333;">Linear Algebra solutions Friedberg. (Chapter 1)</span></span></p>
<div style="width: 100%; height: 1px; overflow: auto;"><span>1.EXERCISES Determine whether the vectors emanating from the origin and termi- nating at the following pairs of points are parallel. 6 Chap. 1 Vector Spaces (a) (b) (c) (d) (3, 1, 2) and (6, 4, 2) (−3, 1, 7) and (9, −3, −21) (5, −6, 7) and (−5, 6, −7) (2, 0, −5) and (5, 0, −2) 2.Find the equations of the lines through the following pairs of points in space. (a) (b) (c) (d) (3, −2, 4) and (−5, 7, 1) (2, 4, 0) and (−3, −6, 0) (3, 7, 2) and (3, 7, −8) (−2, −1, 5) and (3, 9, 7) 3.Find the equations of the planes containing the following points in space. (a) (b) (c) (d) (2, −5, −1), (0, 4, 6), and (−3, 7, 1) (3, −6, 7), (−2, 0, −4), and (5, −9, −2) (−8, 2, 0), (1, 3, 0), and (6, −5, 0) (1, 1, 1), (5, 5, 5), and (−6, 4, 2) 4.What are the coordinates of the vector 0 in the Euclidean plane that satisfies property 3 on page 3? Justify your answer. 5.Prove that if the vector x emanates from the origin of the Euclidean plane and terminates at the point with coordinates (a1 , a2 ), then the vector tx that emanates from the origin terminates at the point with coordinates (ta1 , ta2 ). 6.Show that the midpoint of the line segment joining the points (a, b) and (c, d) is ((a + c)/2, (b + d)/2). 7. Prove that the diagonals of a parallelogram bisect each other. 1.Label the following statements as true or false. (a)(b)(c)(d)(e)(f )(g)(h)(i)Every vector space contains a zero vector. A vector space may have more than one zero vector. In any vector space, ax = bx implies that a = b. In any vector space, ax = ay implies that x = y. A vector in Fn may be regarded as a matrix in Mn×1 (F ). An m × n matrix has m columns and n rows. In P(F ), only polynomials of the same degree may be added. If f and g are polynomials of degree n, then f + g is a polynomial of degree n. If f is a polynomial of degree n and c is a nonzero scalar, then cf is a polynomial of degree n. Sec. 1.2 Vector Spaces 13 (j) A nonzero scalar of F may be considered to be a polynomial in P(F ) having degree zero. (k) Two functions in F(S, F ) are equal if and only if they have the same value at each element of S. 2. Write the zero vector of M3×4 (F ). 3. If 1 2 3 M = , 4 5 6 what are M13 , M21 , and M22 ? 4. Perform the indicated operations. 2 5 −3 4 −2 5 (a) + 1 0 7 −5 3 2 ⎛ ⎞ ⎛ ⎞ −6 4 7 −5 (b) ⎝ 3 −2⎠ + ⎝0 −3⎠ 1 8 2 0 2 5 −3 (c) 4 1 0 7 ⎛ ⎞ −6 4 (d) −5 ⎝ 3 −2⎠ 1 8 (e) (2x4 − 7x3 + 4x + 3) + (8x3 + 2x2 − 6x + 7) (f ) (−3x3 + 7x2 + 8x − 6) + (2x3 − 8x + 10) (g) 5(2x7 − 6x4 + 8x2 − 3x) (h) 3(x5 − 2x3 + 4x + 2) Exercises 5 and 6 show why the definitions of matrix addition and scalar multiplication (as defined in Example 2) are the appropriate ones. 5. Richard Gard (“Effects of Beaver on Trout in Sagehen Creek, Cali- fornia,” J. Wildlife Management, 25, 221-242) reports the following number of trout having crossed beaver dams in Sagehen Creek. Upstream Crossings Fall Spring Summer Brook trout 8 3 1 Rainbow trout 3 0 0 Brown trout 3 0 0 14 Downstream Crossings Chap. 1 Vector Spaces Fall Spring Summer Brook trout 9 1 4 Rainbow trout 3 0 0 Brown trout 1 1 0 Record the upstream and downstream crossings in two 3 × 3 matrices, and verify that the sum of these matrices gives the total number of crossings (both upstream and downstream) categorized by trout species and season. 6. At the end of May, a furniture store had the following inventory. Early Mediter- American Spanish ranean Danish Living room suites 4 2 1 3 Bedroom suites 5 1 1 4 Dining room suites 3 1 2 6 Record these data as a 3 × 4 matrix M . To prepare for its June sale, the store decided to double its inventory on each of the items listed in the preceding table. Assuming that none of the present stock is sold until the additional furniture arrives, verify that the inventory on hand after the order is filled is described by the matrix 2M . If the inventory at the end of June is described by the matrix ⎛ ⎞ 5 3 1 2 A = ⎝6 2 1 5⎠ , 1 0 3 3 7.8.9.interpret 2M − A. How many suites were sold during the June sale? Let S = {0, 1} and F = R. In F(S, R), show that f = g and f + g = h, where f (t) = 2t + 1, g(t) = 1 + 4t − 2t2 , and h(t) = 5t + 1. In any vector space V, show that (a + b)(x + y) = ax + ay + bx + by for any x, y ∈ V and any a, b ∈ F . Prove Corollaries 1 and 2 of Theorem 1.1 and Theorem 1.2(c). 10.Let V denote the set of all differentiable real-valued functions defined on the real line. Prove that V is a vector space with the operations of addition and scalar multiplication defined in Example 3. Sec. 1.2 Vector Spaces 15 11.Let V = {0 } consist of a single vector 0 and define 0 + 0 = 0 and c0 = 0 for each scalar c in F . Prove that V is a vector space over F . (V is called the zero vector space.) 12.A real-valued function f defined on the real line is called an even func- tion if f (−t) = f (t) for each real number t. Prove that the set of even functions defined on the real line with the operations of addition and scalar multiplication defined in Example 3 is a vector space. 13.Let V denote the set of ordered pairs of real numbers. If (a1 , a2 ) and (b1 , b2 ) are elements of V and c ∈ R, define (a1 , a2 ) + (b1 , b2 ) = (a1 + b1 , a2 b2 ) and c(a1 , a2 ) = (ca1 , a2 ). Is V a vector space over R with these operations? Justify your answer. 14.Let V = {(a1 , a2 , . . . , an ) : ai ∈ C for i = 1, 2, . . . n}; so V is a vector space over C by Example 1. Is V a vector space over the field of real numbers with the operations of coordinatewise addition and multipli- cation? 15.Let V = {(a1 , a2 , . . . , an) : ai ∈ R for i = 1, 2, . . . n}; so V is a vec- tor space over R by Example 1. Is V a vector space over the field of complex numbers with the operations of coordinatewise addition and multiplication? 16.Let V denote the set of all m × n matrices with real entries; so V is a vector space over R by Example 2. Let F be the field of rational numbers. Is V a vector space over F with the usual definitions of matrix addition and scalar multiplication? 17.Let V = {(a1 , a2 ) : a1 , a2 ∈ F }, where F is a field. Define addition of elements of V coordinatewise, and for c ∈ F and (a1 , a2 ) ∈ V, define c(a1 , a2 ) = (a1 , 0). 18.Is V a vector space over F with these operations? Justify your answer. Let V = {(a1 , a2 ) : a1 , a2 ∈ R}. For (a1 , a2 ), (b1 , b2 ) ∈ V and c ∈ R, define (a1 , a2 ) + (b1 , b2 ) = (a1 + 2b1 , a2 + 3b2 ) and c(a1 , a2 ) = (ca1 , ca2 ). Is V a vector space over R with these operations? Justify your answer. 16 19.20.21.Chap. 1 Vector Spaces Let V = {(a1 , a2 ) : a1 , a2 ∈ R}. Define addition of elements of V coor- dinatewise, and for (a1 , a2 ) in V and c ∈ R, define ⎧ ⎪ ⎨(0, 0) if c = 0 c(a1 , a2 ) = ⎪ ⎩ ca1 , a 2 if c = 0. c Is V a vector space over R with these operations? Justify your answer. Let V be the set of sequences {an } of real numbers. (See Example 5 for the definition of a sequence.) For {an }, {bn } ∈ V and any real number t, define {an } + {bn } = {an + bn } and t{an} = {tan }. Prove that, with these operations, V is a vector space over R. Let V and W be vector spaces over a field F . Let Z = {(v, w) : v ∈ V and w ∈ W}. Prove that Z is a vector space over F with the operations (v1 , w 1 ) + (v2 , w2 ) = (v1 + v2 , w1 + w2 ) and c(v1 , w1 ) = (cv1, cw1 ). 22. How many matrices are there in the vector space Mm×n (Z2 )? (See Appendix C.) 1. Label the following statements as true or false. (a) If V is a vector space and W is a subset of V that is a vector space, then W is a subspace of V. (b) The empty set is a subspace of every vector space. (c) If V is a vector space other than the zero vector space, then V contains a subspace W such that W = V. (d) The intersection of any two subsets of V is a subspace of V. 20 Chap. 1 Vector Spaces (e)(f )(g)An n × n diagonal matrix can never have more than n nonzero entries. The trace of a square matrix is the product of its diagonal entries. Let W be the xy-plane in R3 ; that is, W = {(a1 , a2 , 0) : a1 , a2 ∈ R}. Then W = R2 . 2. Determine the transpose of each of the matrices that follow. In addition, if the matrix is square, compute its trace. −4 2 0 8 −6 (a) (b) 5 −1 3 4 7 ⎛ ⎞ ⎛ ⎞ −3 9 10 0 −8 (c) ⎝ 0 −2⎠ (d) ⎝ 2 −4 3⎠ 6 1 −5 7 6 −2 5 1 4 (e) 1 −1 3 5 (f ) 7 0 1 −6 ⎛ ⎞ ⎛ ⎞ 5 −4 0 6 (g) ⎝6⎠ (h) ⎝ 0 1 −3⎠ 7 6 −3 5 3.4.5.6.Prove that (aA + bB)t = aAt + bB t for any A, B ∈ Mm×n (F ) and any a, b ∈ F . Prove that (At )t = A for each A ∈ Mm×n (F ). Prove that A + At is symmetric for any square matrix A. Prove that tr(aA + bB) = a tr(A) + b tr(B) for any A, B ∈ Mn×n (F ). 7. Prove that diagonal matrices are symmetric matrices. 8.9.Determine whether the following sets are subspaces of R3 under the operations of addition and scalar multiplication defined on R3 . Justify your answers. (a) (b) (c) (d) (e) (f ) W1 = {(a1 , a2 , a3 ) ∈ R3 : a1 = 3a2 and a3 = −a2 } W2 = {(a1 , a2 , a3 ) ∈ R3 : a1 = a3 + 2} W3 = {(a1 , a2 , a3 ) ∈ R3 : 2a1 − 7a2 + a3 = 0} W4 = {(a1 , a2 , a3 ) ∈ R3 : a1 − 4a2 − a3 = 0} W5 = {(a1 , a2 , a3 ) ∈ R3 : a1 + 2a2 − 3a3 = 1} W6 = {(a1 , a2 , a3 ) ∈ R3 : 5a21 − 3a22 + 6a23 = 0} Let W1 , W3 , and W4 be as in Exercise 8. Describe W1 ∩ W3 , W1 ∩ W4 , and W3 ∩ W4 , and observe that each is a subspace of R3 . Sec. 1.3 Subspaces 21 10.11.Prove that W1 = {(a1 , a2 , . . . , an ) ∈ Fn : a1 + a2 + · · · + an = 0} is a subspace of Fn , but W2 = {(a1 , a2 , . . . , an ) ∈ Fn : a1 + a2 + · · · + an = 1} is not. Is the set W = {f (x) ∈ P(F ) : f (x) = 0 or f (x) has degree n} a subspace of P(F ) if n ≥ 1? Justify your answer. 12.13.14.An m×n matrix A is called upper triangular if all entries lying below the diagonal entries are zero, that is, if Aij = 0 whenever i > j. Prove that the upper triangular matrices form a subspace of Mm×n (F ). Let S be a nonempty set and F a field. Prove that for any s0 ∈ S, {f ∈ F(S, F ) : f (s0 ) = 0}, is a subspace of F(S, F ). Let S be a nonempty set and F a field. Let C(S, F ) denote the set of all functions f ∈ F(S, F ) such that f (s) = 0 for all but a finite number of elements of S. Prove that C(S, F ) is a subspace of F(S, F ). 15.Is the set of all differentiable real-valued functions defined on R a sub- space of C(R)? Justify your answer. 16.Let Cn (R) denote the set of all real-valued functions defined on the real line that have a continuous nth derivative. Prove that Cn (R) is a subspace of F(R, R). 17.Prove that a subset W of a vector space V is a subspace of V if and only if W = ∅, and, whenever a ∈ F and x, y ∈ W, then ax ∈ W and x + y ∈ W. 18.Prove that a subset W of a vector space V is a subspace of V if and only if 0 ∈ W and ax + y ∈ W whenever a ∈ F and x, y ∈ W . 19.Let W1 and W2 be subspaces of a vector space V. Prove that W1 ∪ W2 is a subspace of V if and only if W1 ⊆ W2 or W2 ⊆ W1 . 20. † Prove that if W is a subspace of a vector space V and w1 , w2 , . . . , wn are in W, then a1 w1 + a2 w2 + · · · + an wn ∈ W for any scalars a1 , a2 , . . . , an . 21.Show that the set of convergent sequences {an } (i.e., those for which limn→∞ an exists) is a subspace of the vector space V in Exercise 20 of Section 1.2. 22.Let F1 and F2 be fields. A function g ∈ F(F1 , F2 ) is called an even function if g(−t) = g(t) for each t ∈ F1 and is called an odd function if g(−t) = −g(t) for each t ∈ F1 . Prove that the set of all even functions in F(F1 , F2 ) and the set of all odd functions in F(F1 , F2 ) are subspaces of F(F1 , F2 ). † A dagger means that this exercise is essential for a later section. 22 Chap. 1 Vector Spaces The following definitions are used in Exercises 23–30. Definition. If S1 and S2 are nonempty subsets of a vector space V, then the sum of S1 and S2 , denoted S1 + S2 , is the set {x + y : x ∈ S1 and y ∈ S2 }. Definition. A vector space V is called the direct sum of W1 and W2 if W1 and W2 are subspaces of V such that W1 ∩ W2 = {0 } and W1 + W2 = V. We denote that V is the direct sum of W1 and W2 by writing V = W1 ⊕ W2 . 23. Let W1 and W2 be subspaces of a vector space V. (a) Prove that W1 + W2 is a subspace of V that contains both W1 and W2 . (b) Prove that any subspace of V that contains both W1 and W2 must also contain W1 + W2 . 24. Show that Fn is the direct sum of the subspaces W1 = {(a1 , a2 , . . . , an ) ∈ Fn : an = 0} and W2 = {(a1 , a2 , . . . , an ) ∈ Fn : a1 = a2 = · · · = an−1 = 0}. 25.Let W1 denote the set of all polynomials f (x) in P(F ) such that in the representation f (x) = an xn + an−1 xn−1 + · · · + a1 x + a0 , we have ai = 0 whenever i is even. Likewise let W2 denote the set of all polynomials g(x) in P(F ) such that in the representation g(x) = bm xm + bm−1xm−1 + · · · + b1 x + b0 , we have bi = 0 whenever i is odd. Prove that P(F ) = W1 ⊕ W2 . 26.In Mm×n (F ) define W1 = {A ∈ Mm×n(F ) : Aij = 0 whenever i > j} and W2 = {A ∈ Mm×n (F ) : Aij = 0 whenever i ≤ j}. (W1 is the set of all upper triangular matrices defined in Exercise 12.) Show that Mm×n (F ) = W1 ⊕ W2 . 27.Let V denote the vector space consisting of all upper triangular n × n matrices (as defined in Exercise 12), and let W1 denote the subspace of V consisting of all diagonal matrices. Show that V = W1 ⊕ W2 , where W2 = {A ∈ V : Aij = 0 whenever i ≥ j}. Sec. 1.3 Subspaces 23 28.29.A matrix M is called skew-symmetric if M t = −M . Clearly, a skew- symmetric matrix is square. Let F be a field. Prove that the set W1 of all skew-symmetric n × n matrices with entries from F is a subspace of Mn×n (F ). Now assume that F is not of characteristic 2 (see Ap- pendix C), and let W2 be the subspace of Mn×n (F ) consisting of all symmetric n × n matrices. Prove that Mn×n (F ) = W1 ⊕ W 2 . Let F be a field that is not of characteristic 2. Define W1 = {A ∈ Mn×n (F ) : Aij = 0 whenever i ≤ j} and W2 to be the set of all symmetric n × n matrices with entries from F . Both W1 and W2 are subspaces of Mn×n (F ). Prove that Mn×n(F ) = W1 ⊕ W2 . Compare this exercise with Exercise 28. 30.31.Let W1 and W2 be subspaces of a vector space V. Prove that V is the direct sum of W1 and W2 if and only if each vector in V can be uniquely written as x 1 + x 2 , where x1 ∈ W1 and x2 ∈ W2 . Let W be a subspace of a vector space V over a field F . For any v ∈ V the set {v} + W = {v + w : w ∈ W} is called the coset of W containing v. It is customary to denote this coset by v + W rather than {v} + W. (a) Prove that v + W is a subspace of V if and only if v ∈ W. (b) Prove that v1 + W = v2 + W if and only if v1 − v2 ∈ W. Addition and scalar multiplication by scalars of F can be defined in the collection S = {v + W : v ∈ V} of all cosets of W as follows: (v1 + W) + (v2 + W) = (v1 + v2 ) + W for all v1 , v2 ∈ V and a(v + W) = av + W for all v ∈ V and a ∈ F . (c) Prove that the preceding operations are well defined; that is, show that if v1 + W = v1 + W and v2 + W = v2 + W, then (v1 + W) + (v2 + W) = (v1 + W) + (v2 + W) and a(v1 + W) = a(v1 + W) for all a ∈ F . (d) Prove that the set S is a vector space with the operations defined in (c). This vector space is called the quotient space of V modulo W and is denoted by V/W. 1. Label the following statements as true or false. (a)(b)(c)(d)(e)(f )The zero vector is a linear combination of any nonempty set of vectors. The span of ∅ is ∅. If S is a subset of a vector space V, then span(S) equals the inter- section of all subspaces of V that contain S. In solving a system of linear equations, it is permissible to multiply an equation by any constant. In solving a system of linear equations, it is permissible to add any multiple of one equation to another. Every system of linear equations has a solution. Sec. 1.4 Linear Combinations and Systems of Linear Equations 33 2. Solve the following systems of linear equations by the method intro- duced in this section. 2x1 − 2x2 − 3x3 = −2 (a) 3x1 − 3x2 − 2x3 + 5x4 = 7 x1 − x2 − 2x3 − x4 = −3 3x1 − 7x2 + 4x3 = 10 (b) x1 − 2x2 + x3 = 3 2x1 − x2 − 2x3 = 6 x1 + 2x2 − x3 + x4 = 5 (c) x1 + 4x2 − 3x3 − 3x4 = 6 2x1 + 3x2 − x3 + 4x4 = 8 x1 + 2x 2 + 2x3 = 2 (d) x1 + 8x3 + 5x4 = −6 x1 + x 2 + 5x3 + 5x4 = 3 x1 + 2x2 − 4x3 − x4 + x5 = 7 −x1 + 10x3 − 3x4 − 4x5 = −16 (e) 2x1 + 5x2 − 5x3 − 4x4 − x5 = 2 4x1 + 11x2 − 7x3 − 10x4 − 2x5 = 7 x1 + 2x2 + 6x3 = −1 2x1 + x2 + x3 = 8 (f ) 3x1 + x2 − x3 = 15 x1 + 3x2 + 10x3 = −5 3.For each of the following lists of vectors in R3 , determine whether the first vector can be expressed as a linear combination of the other two. (a) (b) (c) (d) (e) (f ) (−2, 0, 3), (1, 3, 0), (2, 4, −1) (1, 2, −3), (−3, 2, 1), (2, −1, −1) (3, 4, 1), (1, −2, 1), (−2, −1, 1) (2, −1, 0), (1, 2, −3), (1, −3, 2) (5, 1, −5), (1, −2, −3), (−2, 3, −4) (−2, 2, 2), (1, 2, −1), (−3, −3, 3) 4.For each list of polynomials in P3 (R), determine whether the first poly- nomial can be expressed as a linear combination of the other two. (a) (b) (c) (d) (e) (f ) x3 − 3x + 5, x3 + 2x2 − x + 1, x3 + 3x2 − 1 4x3 + 2x2 − 6, x3 − 2x2 + 4x + 1, 3x3 − 6x2 + x + 4 −2x3 − 11x2 + 3x + 2, x3 − 2x2 + 3x − 1, 2x3 + x2 + 3x − 2 x3 + x2 + 2x + 13, 2x3 − 3x2 + 4x + 1, x3 − x2 + 2x + 3 x3 − 8x2 + 4x, x3 − 2x2 + 3x − 1, x3 − 2x + 3 6x3 − 3x2 + x + 2, x3 − x2 + 2x + 3, 2x3 − 3x + 1 34 Chap. 1 Vector Spaces 5. In each part, determine whether the given vector is in the span of S. (a) (2, −1, 1), S = {(1, 0, 2), (−1, 1, 1)} (b) (−1, 2, 1), S = {(1, 0, 2), (−1, 1, 1)} (c) (−1, 1, 1, 2), S = {(1, 0, 1, −1), (0, 1, 1, 1)} (d) (2, −1, 1, −3), S = {(1, 0, 1, −1), (0, 1, 1, 1)} (e) −x3 + 2x2 + 3x + 3, S = {x3 + x2 + x + 1, x2 + x + 1, x + 1} (f ) 2x3 − x2 + x + 3, S = {x3 + x2 + x + 1, x2 + x + 1, x + 1} 1 2 1 0 0 1 1 1 (g) , S = , , −3 4 −1 0 0 1 0 0 1 0 1 0 0 1 1 1 (h) , S = , , 0 1 −1 0 0 1 0 0 6.7.8.Show that the vectors (1, 1, 0), (1, 0, 1), and (0, 1, 1) generate F3 . In Fn , let ej denote the vector whose jth coordinate is 1 and whose other coordinates are 0. Prove that {e1 , e2, . . . , en} generates Fn . Show that Pn (F ) is generated by {1, x, . . . , xn }. 9. Show that the matrices 1 0 0 1 0 0 0 0 , , , and 0 0 0 0 1 0 0 1 generate M2×2 (F ). 10. Show that if 1 0 0 0 0 1 M1 = , M2 = , and M3 = , 0 0 0 1 1 0 then the span of {M1 , M2 , M3 } is the set of all symmetric 2×2 matrices. 11. † Prove that span({x}) = {ax : a ∈ F } for any vector x in a vector space. Interpret this result geometrically in R3 . 12.Show that a subset W of a vector space V is a subspace of V if and only if span(W) = W. 13. † Show that if S1 and S2 are subsets of a vector space V such that S1 ⊆ S2 , then span(S1 ) ⊆ span(S2 ). In particular, if S1 ⊆ S2 and span(S1 ) = V, deduce that span(S2 ) = V. 14.Show that if S1 and S2 are arbitrary subsets of a vector space V, then span(S1 ∪S2 ) = span(S1 )+span(S2 ). (The sum of two subsets is defined in the exercises of Section 1.3.) Sec. 1.5 Linear Dependence and Linear Independence 35 15.Let S1 and S2 be subsets of a vector space V. Prove that span(S1 ∩S2 ) ⊆ span(S1 ) ∩ span(S2 ). Give an example in which span(S1 ∩ S2 ) and span(S1 ) ∩ span(S2 ) are equal and one in which they are unequal. 16.Let V be a vector space and S a subset of V with the property that whenever v1 , v2 , . . . , vn ∈ S and a1 v1 + a2 v2 + · · · + an vn = 0 , then a1 = a2 = · · · = an = 0. Prove that every vector in the span of S can be uniquely written as a linear combination of vectors of S. 17.Let W be a subspace of a vector space V. Under what conditions are there only a finite number of distinct subsets S of W such that S gen- erates W? EXERCISES 1. Label the following statements as true or false. (a) If S is a linearly dependent set, then each vector in S is a linear combination of other vectors in S. (b) Any set containing the zero vector is linearly dependent. (c) The empty set is linearly dependent. (d) Subsets of linearly dependent sets are linearly dependent. (e) Subsets of linearly independent sets are linearly independent. (f ) If a1 x1 + a2 x2 + · · · + an xn = 0 and x1 , x2 , . . . , xn are linearly independent, then all the scalars ai are zero. 2. 3 Determine whether the following sets are linearly dependent or linearly independent. 1 −3 −2 6 (a) , in M2×2 (R) −2 4 4 −8 1 −2 −1 1 (b) , in M2×2 (R) −1 4 2 −4 (c) {x3 + 2x2 , −x2 + 3x + 1, x3 − x2 + 2x − 1} in P3 (R) 3The computations in Exercise 2(g), (h), (i), and (j) are tedious unless technology is used. Sec. 1.5 Linear Dependence and Linear Independence 41 (d) {x3 − x, 2x2 + 4, −2x3 + 3x2 + 2x + 6} in P3 (R) (e) {(1, −1, 2), (1, −2, 1), (1, 1, 4)} in R3 (f ) {(1, −1, 2), (2, 0, 1), (−1, 2, −1)} in R3 1 0 0 −1 −1 2 2 1 (g) , , , in M2×2 (R) −2 1 1 1 1 0 −4 4 1 0 0 −1 −1 2 2 1 (h) , , , in M2×2 (R) −2 1 1 1 1 0 2 −2 (i)(j){x4 − x3 + 5x2 − 8x + 6, −x4 + x3 − 5x2 + 5x − 3, x4 + 3x2 − 3x + 5, 2x4 + 3x3 + 4x2 − x + 1, x 3 − x + 2} in P4 (R) {x4 − x3 + 5x2 − 8x + 6, −x4 + x3 − 5x2 + 5x − 3, x4 + 3x2 − 3x + 5, 2x4 + x3 + 4x2 + 8x} in P4 (R) 3. In M2×3 (F ), prove that the set ⎧⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎫ ⎨ 1 1 0 0 0 0 1 0 0 1 ⎬ ⎝0 0⎠ , ⎝1 1⎠ , ⎝0 0⎠ , ⎝1 0⎠ , ⎝0 1⎠ ⎩ ⎭ 0 0 0 0 1 1 1 0 0 1 4.5.6.is linearly dependent. In Fn , let ej denote the vector whose jth coordinate is 1 and whose other coordinates are 0. Prove that {e1 , e2 , · · · , en } is linearly independent. Show that the set {1, x, x2 , . . . , xn } is linearly independent in Pn (F ). In Mm×n (F ), let E ij denote the matrix whose only nonzero entry is 1 in the ith row and jth column. Prove that {E ij : 1 ≤ i ≤ m, 1 ≤ j ≤ n} is linearly independent. 7.Recall from Example 3 in Section 1.3 that the set of diagonal matrices in M2×2 (F ) is a subspace. Find a linearly independent set that generates this subspace. 8. Let S = {(1, 1, 0), (1, 0, 1), (0, 1, 1)} be a subset of the vector space F3 . (a) Prove that if F = R, then S is linearly independent. (b) Prove that if F has characteristic 2, then S is linearly dependent. 9. † Let u and v be distinct vectors in a vector space V. Show that {u, v} is linearly dependent if and only if u or v is a multiple of the other. 10. Give an example of three linearly dependent vectors in R3 such that none of the three is a multiple of another. 42 Chap. 1 Vector Spaces 11.Let S = {u1 , u2 , . . . , un } be a linearly independent subset of a vector space V over the field Z2 . How many vectors are there in span(S)? Justify your answer. 12. Prove Theorem 1.6 and its corollary. 13.Let V be a vector space over a field of characteristic not equal to two. (a) Let u and v be distinct vectors in V. Prove that {u, v} is linearly independent if and only if {u + v, u − v} is linearly independent. (b) Let u, v, and w be distinct vectors in V. Prove that {u, v, w} is linearly independent if and only if {u + v, u + w, v + w} is linearly independent. 14.15.Prove that a set S is linearly dependent if and only if S = {0 } or there exist distinct vectors v, u1 , u2 , . . . , un in S such that v is a linear combination of u1 , u2 , . . . , un . Let S = {u1 , u2 , . . . , un } be a finite set of vectors. Prove that S is linearly dependent if and only if u1 = 0 or uk+1 ∈ span({u1 , u2 , . . . , uk }) for some k (1 ≤ k < n). 16.Prove that a set S of vectors is linearly independent if and only if each finite subset of S is linearly independent. 17.Let M be a square upper triangular matrix (as defined in Exercise 12 of Section 1.3) with nonzero diagonal entries. Prove that the columns of M are linearly independent. 18.19.20.Let S be a set of nonzero polynomials in P(F ) such that no two have the same degree. Prove that S is linearly independent. Prove that if {A1 , A2 , . . . , Ak } is a linearly independent subset of Mn×n (F ), then {At 1 , At 2 , . . . , Atk } is also linearly independent. Let f, g, ∈ F(R, R) be the functions defined by f (t) = ert and g(t) = est , where r = s. Prove that f and g are linearly independent in F(R, R). 1.Label the following statements as true or false. (a) The zero vector space has no basis. (b) Every vector space that is generated by a finite set has a basis. (c) Every vector space has a finite basis. (d) A vector space cannot have more than one basis. 54 Chap. 1 Vector Spaces (e) If a vector space has a finite basis, then the number of vectors in every basis is the same. (f ) The dimension of Pn (F ) is n. (g) The dimension of Mm×n (F ) is m + n. (h) Suppose that V is a finite-dimensional vector space, that S1 is a linearly independent subset of V, and that S2 is a subset of V that generates V. Then S1 cannot contain more vectors than S2 . (i) If S generates the vector space V, then every vector in V can be written as a linear combination of vectors in S in only one way. (j) Every subspace of a finite-dimensional space is finite-dimensional. (k) If V is a vector space having dimension n, then V has exactly one subspace with dimension 0 and exactly one subspace with dimen- sion n. (l) If V is a vector space having dimension n, and if S is a subset of V with n vectors, then S is linearly independent if and only if S spans V. 2. Determine which of the following sets are bases for R3 . (a) (b) (c) (d) (e) {(1, 0, −1), (2, 5, 1), (0, −4, 3)} {(2, −4, 1), (0, 3, −1), (6, 0, −1)} {(1, 2, −1), (1, 0, 2), (2, 1, 1)} {(−1, 3, 1), (2, −4, −3), (−3, 8, 2)} {(1, −3, −2), (−3, 1, 3), (−2, −10, −2)} 3.Determine which of the following sets are bases for P2 (R). (a) (b) (c) (d) (e) {−1 − x + 2x2 , 2 + x − 2x2 , 1 − 2x + 4x2 } {1 + 2x + x2, 3 + x2 , x + x2 } {1 − 2x − 2x2 , −2 + 3x − x 2 , 1 − x + 6x2 } {−1 + 2x + 4x2 , 3 − 4x − 10x2 , −2 − 5x − 6x2 } {1 + 2x − x2, 4 − 2x + x2 , −1 + 18x − 9x2 } 4.Do the polynomials x3 −2x2 +1, 4x2 −x+3, and 3x−2 generate P3 (R)? Justify your answer. 5.Is {(1, 4, −6), (1, 5, 8), (2, 1, 1), (0, 1, 0)} a linearly independent subset of R3 ? Justify your answer. 6. Give three different bases for F2 and for M2×2 (F ). 7.The vectors u1 = (2, −3, 1), u2 = (1, 4, −2), u3 = (−8, 12, −4), u4 = (1, 37, −17), and u5 = (−3, −5, 8) generate R3. Find a subset of the set {u1 , u2 , u3, u4 , u5 } that is a basis for R3 . Sec.8.1.6 9.10.11.12.13.14.Bases and Dimension 55 Let W denote the subspace of R5 consisting of all the vectors having coordinates that sum to zero. The vectors u1 = (2, −3, 4, −5, 2), u2 = (−6, 9, −12, 15, −6), u3 = (3, −2, 7, −9, 1), u4 = (2, −8, 2, −2, 6), u5 = (−1, 1, 2, 1, −3), u6 = (0, −3, −18, 9, 12), u7 = (1, 0, −2, 3, −2), u8 = (2, −1, 1, −9, 7) generate W. Find a subset of the set {u1 , u2 , . . . , u8 } that is a basis for W. The vectors u1 = (1, 1, 1, 1), u2 = (0, 1, 1, 1), u3 = (0, 0, 1, 1), and u4 = (0, 0, 0, 1) form a basis for F4 . Find the unique representation of an arbitrary vector (a1 , a2 , a3 , a4 ) in F4 as a linear combination of u1 , u2 , u3 , and u4 . In each part, use the Lagrange interpolation formula to construct the polynomial of smallest degree whose graph contains the following points. (a) (−2, −6), (−1, 5), (1, 3) (b) (−4, 24), (1, 9), (3, 3) (c) (−2, 3), (−1, −6), (1, 0), (3, −2) (d) (−3, −30), (−2, 7), (0, 15), (1, 10) Let u and v be distinct vectors of a vector space V. Show that if {u, v} is a basis for V and a and b are nonzero scalars, then both {u + v, au} and {au, bv} are also bases for V. Let u, v, and w be distinct vectors of a vector space V. Show that if {u, v, w} is a basis for V, then {u + v + w, v + w, w} is also a basis for V. The set of solutions to the system of linear equations x1 − 2x2 + x3 = 0 2x1 − 3x2 + x3 = 0 is a subspace of R3 . Find a basis for this subspace. Find bases for the following subspaces of F5 : W1 = {(a1 , a2 , a3 , a4 , a5 ) ∈ F5 : a1 − a3 − a4 = 0} and W2 = {(a1 , a2 , a 3 , a4 , a5 ) ∈ F5 : a2 = a3 = a4 and a1 + a5 = 0}. What are the dimensions of W1 and W2 ? 56 Chap. 1 Vector Spaces 15.The set of all n × n matrices having trace equal to zero is a subspace W of Mn×n (F ) (see Example 4 of Section 1.3). Find a basis for W. What is the dimension of W? 16. The set of all upper triangular n × n matrices is a subspace W of Mn×n (F ) (see Exercise 12 of Section 1.3). Find a basis for W. What is the dimension of W? 17. The set of all skew-symmetric n × n matrices is a subspace W of Mn×n (F ) (see Exercise 28 of Section 1.3). Find a basis for W. What is the dimension of W? 18.Find a basis for the vector space in Example 5 of Section 1.2. Justify your answer. 19. Complete the proof of Theorem 1.8. 20. † Let V be a vector space having dimension n, and let S be a subset of V that generates V. (a) Prove that there is a subset of S that is a basis for V. (Be careful not to assume that S is finite.) (b) Prove that S contains at least n vectors. 21.Prove that a vector space is infinite-dimensional if and only if it contains an infinite linearly independent subset. 22.Let W1 and W2 be subspaces of a finite-dimensional vector space V. Determine necessary and sufficient conditions on W1 and W2 so that dim(W1 ∩ W2 ) = dim(W1 ). 23.Let v1 , v2 , . . . , vk , v be vectors in a vector space V, and define W1 = span({v1 , v2 , . . . , vk }), and W2 = span({v1 , v2, . . . , vk , v}). (a)(b)Find necessary and sufficient conditions on v such that dim(W1 ) = dim(W2 ). State and prove a relationship involving dim(W1 ) and dim(W2 ) in the case that dim(W1 ) = dim(W2 ). 24.Let f (x) be a polynomial of degree n in Pn(R). Prove that for any g(x) ∈ Pn (R) there exist scalars c0 , c1 , . . . , cn such that g(x) = c0 f (x) + c1 f (x) + c2 f (x) + · · · + cn f (n) (x), where f (n) (x) denotes the nth derivative of f (x). 25. Let V, W, and Z be as in Exercise 21 of Section 1.2. If V and W are vector spaces over F of dimensions m and n, determine the dimension of Z. Sec. 1.6 Bases and Dimension 57 26.For a fixed a ∈ R, determine the dimension of the subspace of Pn (R) defined by {f ∈ Pn (R) : f (a) = 0}. 27.Let W1 and W2 be the subspaces of P(F ) defined in Exercise 25 in Section 1.3. Determine the dimensions of the subspaces W1 ∩ Pn (F ) and W2 ∩ Pn (F ). 28.Let V be a finite-dimensional vector space over C with dimension n. Prove that if V is now regarded as a vector space over R, then dim V = 2n. (See Examples 11 and 12.) Exercises 29–34 require knowledge of the sum and direct sum of subspaces, as defined in the exercises of Section 1.3. 29.(a)(b)Prove that if W1 and W2 are finite-dimensional subspaces of a vector space V, then the subspace W1 + W2 is finite-dimensional, and dim(W1 + W2 ) = dim(W1) + dim(W2 ) − dim(W1 ∩ W2 ). Hint: Start with a basis {u1 , u2 , . . . , uk } for W1 ∩ W2 and extend this set to a basis {u1 , u2 , . . . , uk , v1 , v2 , . . . vm } for W 1 and to a basis {u1 , u2 , . . . , uk , w1 , w2 , . . . wp } for W2 . Let W1 and W2 be finite-dimensional subspaces of a vector space V, and let V = W1 + W2 . Deduce that V is the direct sum of W1 and W2 if and only if dim(V) = dim(W1 ) + dim(W2 ). 30. Let V = M2×2(F ), W1= a c b a ∈ V : a, b, c ∈ F , and W2= 0−aa b ∈ V : a, b ∈ F . Prove that W1 and W2 are subspaces of V, and find the dimensions of W1 , W2 , W1 + W2 , and W1 ∩ W2 . 31.Let W1 and W2 be subspaces of a vector space V having dimensions m and n, respectively, where m ≥ n. (a) Prove that dim(W1 ∩ W2 ) ≤ n. (b) Prove that dim(W1 + W2 ) ≤ m + n. 32.(a)(b)Find an example of subspaces W1 and W2 of R3 with dimensions m and n, where m > n > 0, such that dim(W1 ∩ W2 ) = n. Find an example of subspaces W1 and W2 of R3 with dimensions m and n, where m > n > 0, such that dim(W1 + W2 ) = m + n. 58 Chap. 1 Vector Spaces (c)Find an example of subspaces W1 and W2 of R3 with dimensions m and n, where m ≥ n, such that both dim(W1 ∩ W2 ) < n and dim(W1 + W2 ) < m + n. 33.(a)(b)Let W1 and W2 be subspaces of a vector space V such that V = W1 ⊕W2 . If β1 and β2 are bases for W1 and W2 , respectively, show that β1 ∩ β2 = ∅ and β1 ∪ β2 is a basis for V. Conversely, let β1 and β2 be disjoint bases for subspaces W1 and W2 , respectively, of a vector space V. Prove that if β1 ∪ β2 is a basis for V, then V = W1 ⊕ W2 . 34. (a) Prove that if W1 is any subspace of a finite-dimensional vector space V, then there exists a subspace W2 of V such that V = W1 ⊕ W2 . (b) Let V = R2 and W1 = {(a1 , 0) : a1 ∈ R}. Give examples of two different subspaces W2 and W2 such that V = W1 ⊕ W2 and V = W1 ⊕ W2 . The following exercise requires familiarity with Exercise 31 of Section 1.3. 35.Let W be a subspace of a finite-dimensional vector space V, and consider the basis {u1 , u2 , . . . , uk } for W. Let {u1 , u2 , . . . , uk , uk+1 , . . . , un } be an extension of this basis to a basis for V. (a)(b)Prove that {uk+1 + W, uk+2 + W, . . . , un + W} is a basis for V/W. Derive a formula relating dim(V), dim(W), and dim(V/W). 1. Label the following statements as true or false. (a) Every family of sets contains a maximal element. (b) Every chain contains a maximal element. (c) If a family of sets has a maximal element, then that maximal element is unique. (d) If a chain of sets has a maximal element, then that maximal ele- ment is unique. (e) A basis for a vector space is a maximal linearly independent subset of that vector space. (f ) A maximal linearly independent subset of a vector space is a basis for that vector space. 2. Show that the set of convergent sequences is an infinite-dimensional subspace of the vector space of all sequences of real numbers. (See Exercise 21 in Section 1.3.) 3. Let V be the set of real numbers regarded as a vector space over the field of rational numbers. Prove that V is infinite-dimensional. Hint: 62 Chap. 1 Vector Spaces Use the fact that π is transcendental, that is, π is not a zero of any polynomial with rational coefficients. 4.Let W be a subspace of a (not necessarily finite-dimensional) vector space V. Prove that any basis for W is a subset of a basis for V. 5.Prove the following infinite-dimensional version of Theorem 1.8 (p. 43): Let β be a subset of an infinite-dimensional vector space V. Then β is a basis for V if and only if for each nonzero vector v in V, there exist unique vectors u1 , u2 , . . . , un in β and unique nonzero scalars c1 , c2 , . . . , cn such that v = c1 u1 + c2 u2 + · · · + cn un . 6.Prove the following generalization of Theorem 1.9 (p. 44): Let S1 and S2 be subsets of a vector space V such that S1 ⊆ S2 . If S1 is linearly independent and S2 generates V, then there exists a basis β for V such that S1 ⊆ β ⊆ S2 . Hint: Apply the maximal principle to the family of all linearly independent subsets of S2 that contain S1 , and proceed as in the proof of Theorem 1.13. 7. Prove the following generalization of the replacement theorem. Let β be a basis for a vector space V, and let S be a linearly independent subset of V. There exists a subset S1 of β such that S ∪ S1 is a basis for V. </span></div>
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<p><figure class='imageblock alignCenter'><span data-url='https://k.kakaocdn.net/dn/D6dCe/btqv6CPu0y6/KR3RHaOHOjmoczvJlk6iK0/img.png' data-lightbox='lightbox' data-alt=''><img src='https://k.kakaocdn.net/dn/D6dCe/btqv6CPu0y6/KR3RHaOHOjmoczvJlk6iK0/img.png' srcset='https://img1.daumcdn.net/thumb/R1280x0/?scode=mtistory2&fname=https%3A%2F%2Fk.kakaocdn.net%2Fdn%2FD6dCe%2Fbtqv6CPu0y6%2FKR3RHaOHOjmoczvJlk6iK0%2Fimg.png' data-filename="Piano.png"></span></figure></p>
<p> </p>
<p><b>1</b><span><b>월</b></span></p>
<p>페이스북 해커컵(Facebook HackerCup)<br /><br /></p>
<p><b>2</b><span><b>월</b></span></p>
<p> </p>
<p><br /><b>3</b><span><b>월</b></span></p>
<p><span>접수 – 구글 코드잼</span>(Google Codejam)</p>
<p><br /><br /></p>
<p><b>4</b><span><b>월</b></span></p>
<p><span>접수 – 글로벌 </span>SW <span>공모대전</span></p>
<p><span>접수 – 데프콘</span>(Defcon) CTF</p>
<p> </p>
<p><br /><b>5</b><span><b>월</b></span></p>
<p><span>접수 – 차세대 보안리더 양성 교육 </span>(Best of Best – BoB)</p>
<p> </p>
<p><b>6</b><span><b>월</b></span></p>
<p><span>접수 – 대학생 디지털 솔루션 챌린지</span></p>
<p><span>접수 </span>- <span>삼성 </span>Capture The Flag 2017 (<u><a href="https://codeground.org">https://codeground.org</a></u>)</p>
<p><span>접수 – 삼성 대학생 프로그래밍 대회</span></p>
<p><span>접수 – 국방 사이버 안보 콘테스트 </span><span>(</span><u><span></span><a href="http://www.whitehatcontest.kr">http://www.whitehatcontest.kr</a></u><span>) </span></p>
<p><span><i><b>예선 – 국방 사이버 안보 콘테스트 </b></i></span><span><i><b>(</b></i></span><u><span><i><b></b></i></span><i><b><a href="http://www.whitehatcontest.kr">http://www.whitehatcontest.kr</a></b></i></u><span><i><b>) </b></i></span></p>
<p><br /><br /></p>
<p><b>7</b><span><b>월</b></span></p>
<p><span>접수 – 한국공학한림원 소프트웨어 챌린지 </span>(<u><a href="https://naek.or.kr">https://naek.or.kr</a></u>)</p>
<p><span><i><b>예선 </b></i></span><i><b>- </b></i><span><i><b>삼성 </b></i></span><i><b>Capture The Flag 2017 (</b></i><u><i><b></b></i><i><b><a href="https://codeground.org">https://codeground.org</a></b></i></u><i><b>)</b></i></p>
<p><span><i><b>본선 – 국방 사이버 안보 콘테스트 </b></i></span><i><b>(</b></i><u><i><b></b></i><i><b><a href="http://www.whitehatcontest.kr">http://www.whitehatcontest.kr</a></b></i></u><i><b>) </b></i></p>
<p><br /><br /></p>
<p><b>8</b><span><b>월</b></span></p>
<p><span><i><b>본선 </b></i></span><i><b>- </b></i><span><i><b>삼성 </b></i></span><i><b>Capture The Flag 2017 (</b></i><u><i><b></b></i><i><b><a href="https://codeground.org">https://codeground.org</a></b></i></u><i><b>)</b></i></p>
<p><span>접수 – 카카오 </span>Code Festival</p>
<p><span>접수 – </span>ACPC-ICPC <span>알고리즘 대회</span></p>
<p><br /><br /></p>
<p><b>9</b><span><b>월</b></span></p>
<p>접수 - LG CNS 코드 몬스터</p>
<p>접수 - ACM ICPC</p>
<p>접수 - 디지털 과거 Code+</p>
<p><span style="color: #333333;">접수<span> </span></span><span style="color: #333333;">-<span> </span></span><span style="color: #333333;">사이버 공격 방어 대회<span> </span></span><span style="color: #333333;">(</span><u><a href="http://cce.cstec.kr">http://cce.cstec.kr</a></u><span style="color: #333333;">)</span></p>
<p><span>접수 – </span>HDCON (<span>해킹 방어 대회</span>)</p>
<p><span><i><b>예선 </b></i></span><i><b>- </b></i><span><i><b>사이버 공격 방어 대회 </b></i></span><i><b>(</b></i><u><i><b></b></i><i><b><a href="https://cce.cstec.kr">https://cce.cstec.kr</a></b></i></u><i><b>)</b></i></p>
<p><span><i><b>예선 – 한국공학한림원 소프트웨어 챌린지 </b></i></span><i><b>(</b></i><u><i><b></b></i><i><b><a href="https://naek.or.kr">https://naek.or.kr</a></b></i></u><i><b>)</b></i></p>
<p> </p>
<p><b>10</b><span><b>월</b></span></p>
<p> </p>
<p><b>11</b><span><b>월</b></span></p>
<p><span><i><b>본선 </b></i></span><i><b>- </b></i><span><i><b>사이버 공격 방어 대회 </b></i></span><i><b>(</b></i><u><i><b></b></i><i><b><a href="https://cce.cstec.kr">https://cce.cstec.kr</a></b></i></u><i><b>)</b></i></p>
<p><span><i><b>본선 – 한국공학한림원 소프트웨어 챌린지 </b></i></span><i><b>(</b></i><u><i><b></b></i><i><b><a href="https://naek.or.kr">https://naek.or.kr</a></b></i></u><i><b>)</b></i></p>
<p> </p>
<p> </p>
<p><b>12</b><span><b>월</b></span></p><div style="text-align:center;margin:10px 0 10px 0;clear:both"><div style="display:inline;text-align:center;"><script async src="https://pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script>
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Tutorialshashnuthttps://koreanfoodie.me/88https://koreanfoodie.me/88#entry88commentSat, 15 Jun 2019 09:57:12 +0900