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Solutions to Linear Algebra, Stephen H. Friedberg, Fourth Edition (Chapter 3) 본문

Math, Statistics/Linear Algebra

Solutions to Linear Algebra, Stephen H. Friedberg, Fourth Edition (Chapter 3)

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Solution maual to Linear Algebra, Fourth Edition, Stephen H. Friedberg. (Chapter 3)

Solutions to Linear Algebra, Fourth Edition, Stephen H. Friedberg. (Chapter 3)

Linear Algebra solution manual, Fourth Edition, Stephen H. Friedberg. (Chapter 3)

Linear Algebra solutions Friedberg. (Chapter 3)

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.
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