(-2, 0, 6, – 5)} 19. S = {(-3, 2, 5, 28), (–6, 1, – 8, – 1), (14, – 10, 12, – 10), (0, 5, 12, 50)} 20. S = {(2, 5, –3, – 2), (–2, – 3, 2, – 5), (1, 3, – 2, 2), 4 - 1 40. A= 4 1 (-1, – 5, 3, 5)} Copyrighe 2017 Cenpage Leaming All Rigles Reserved. May not be copied scanned, or dupliculed in whole r in part. Due to electonic rights, sme dthird purty coteut may be supprosed from the clook andiar eClupteri Editorial seview has deomed tht any suppresed contout does ot materially allect the overall leaing experience. Cengage Leaming reserves the right to semne additional content al any time if suhioquet rights sestrictions equire it 206 Chapter 4 Vector Spaces Rank, Nullity, Bases, and Linear Independence In Exercises 41 and 42, use the fact that matrices A and B Nonhomogeneous System In Exercises 49-56, determine whether the nonhomogeneous system Ax = b is consistent. If it is, write the solution in the form are row-equivalent. x = x, + X, where x, is a particular solution of Ax = b and x, is a solution of Ax = 0. (a) Find the rank and nullity of A. (b) Find a basis for the nullspace of A. (c) Find a basis for the row space of A. 49. 4y = 17 50. x + 2y - 4z = -1 3x – 12y = 51 - 3x – 6y + 12z = 3 (d) Find a basis for the column space of A. - 2x + 8y = - 34 (e) Determine whether the rows of A are linearly independent. (f) Let the columns of A be denoted by a,, a,, a, a, and ag. Determine whether each set is linearly independent. x + 3y + 10z = 18 52. – 2x + 7y + 32z = 29 51. 2x - 4y + 5z = - 7x + 14y + 4z = - 28 -x + 3y + 14z = 12 3x - 6y + z = 12 (i) {a,, a,, a,} (ii) {a,, a,, a,} (iii) {a,, a, a<} 2z = 53. 3x – 8y + 4z = 19 [1 2 1 1 - 6y + 2z + 4w = 41. A = 3 + 22z + -2 5x w = 29 -1 4 x - 2y + 27 8. Г1 -4 54. 3w – 2x + 16y – 2z = -7 - 1 -w + 5x - 14y + 18z = 29 -2 3w - x+ 14y + 2z = O 55. x, + 2x, + x, + x, + 5x, = -2 -5 -17 - 5x, 10x, + 3x, + 3x4 + 55x, = -8 3 42. A = 3 X1 + 2x, + 2x, – 3x, - 5x, = 2.x, + x, + x, + 15x, = -2 14 11 -19 7 7 - 13 -x, O 56. 5x, – 4x2 + 12x, – 33x4 + 14xs = -4 - 2x, + x, - 6x, + 12x, - 8x, = | 1 -2 3 2x, - x + 6x, - 12x + 8x, = -1 Consistency of Ax = b In Exercises 57-62, determine Finding a Basis and Dimension In Exercises 43–48, whether b is in the column space of A. If it is, write b as find (a) a basis for and (b) the dimension of the solution a linear combination of the column vectors of A. space of the homogeneous system of linear equations. 43. -x + y + z = 0 44. x - 2y + 3z = 0 57. A = 4 3x - v - 3x + 6y – 9z = 0 2x – 4y – 5z = 0 58. A = 45. Зх, + 3х, + 15х, + 11х, - 0 3. 2 х, — Зх, + X = 0 59. A = - 1 b =|1 2x, + 3x, + 11x, + 8x, = 0 1 46. 2x, + 2x, + 4x3 – 2x4 = 0 1 x, + 2x, + x3 + 2x, = 0 60. A = -1 1 –x, + x, + 4x, – 2x, = 0 47. 9х, — 4х, 1 -3 2r, - 20x, = 0 -1 - 1 1 12x, - 6x, - 4x, – 29x, = 0 61. A = Зx, — 2х, - 7x, = 0 -3 -2 -3 3x, - 2x, - x, - 8x, = 0 В 48. х, + Зх, + 2х, + 22х, + 13х, — 0 -9 62. A = -3 -2 11 X - 2x, + X = 0 - 25 3x, + 6x, + 5x, + 42x, + 27xg = 0 Cipyright 2017 Cenpage Leaming All Rigles Reserved. May not be copiel scanned, oe dupliouted in whole e in part. Eue to electronic rights, some third party cotent may be sappressed from the cBook andiar eChapteri Editorial review has deemed tht any suppresed content does ot mterialy affect the overal leaing esperience. Cengage Leaming reserves the right to remve dditional content at any time if suhsoquet rights restrictions require it 4.6 Exercises 20 True or False? In Exercises 73–76, determine whethe 63. Proof Prove that if A is not square, then either the row vectors of A or the column vectors of A form a linearly dependent set. each statement is true or false. If a statement is true, giv a reason or cite an appropriate statement from the text If a statement is false, provide an example that shows th statement is not true in all cases or cite an appropriat 64. Give an example showing that the rank of the product of two matrices can be less than the rank of either matrix. statement from the text. 65. Give examples of matrices A and R of the same size such that 73. (a) The nullspace of a matrix A is the solution space cC (a) rank(A + B) < rank(A) and rank(A + B) < rank(B) the homogeneous system Ax = 0. (b) The dimension of the nullspace of a matrix A is th (b) rank(A + B) = rank(A) and rank(A + B) = rank(B) nullity of A. (c) rank(A + B)> rank(A) and rank(A + B) > rank(B). 74. (a) If an m x n matrix A is row-equivalent to an m x matrix R. then the rou enace of A is eguivalent t 66. Proof Prove that the nonzero row vectors of a matrix Copy Select All Look Up Share... Highlight sio (b) How many vectors are in a basis for the row space be obtained from elementar s on an i x n matrix A, then the colum space of Bs equal to the column space of A. 75. (a) If an m x nmatrix of A? row opera (c) How many vectors are in a basis for the column (b) The system of linear equations Ax = bis inconsister if ond onlu if h ic in tha column cnace of space of A?