20. S = {(5, 6, 5), (2, 50. A = 21. S = {(-2, 5, 0), (4, 6, 3)} 22. S = {(1, 0, 1), (1, 1, 0), (0, 1, 1)} 23. S = {(1, –2, 0), (0, 0, 1), (–1, 2, 0)} 24. S = {(1, 0, 3), (2, 0, – 1), (4, 0, 5), (2, 0, 6)} 51. A = 52. A = Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronie Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserve. Showing Linear Dependence In Exercises 53–56, show that the set is linearly dependent by finding a Proc vect nontrivial linear combination of vectors in the set whose 65. sum is the zero vector. Then express one of the vectors 66. in the set as a linear combination of the other vectors in the set. 67. 53. S = {(3, 4), (-1, 1), (2, 0)} {(2, 4), (– 1, – 2), (0, 6)} {(1, 1, 1), (1, 1, 0), (0, 1, 1), (0, 0, 1)} {(1, 2, 3, 4), (1, 0, 1, 2), (1, 4, 5, 6)} 54. S = 55. S = 56. S = 57. For which values of t is each set linearly independent? (a) S = {(t, 1, 1), (1, t, 1), (1, 1, t)} (b) S = {(t, 1, 1), (1, 0, 1), (1, 1, 3t)} 58. For which values of t is each set linearly independent? (a) S = {(t, 0, 0), (0, 1, 0), (0, 0, 1)} (b) S = {(t, t, t), (t, 1, 0), (t, 0, 1)} 59. Proof Complete the proof of Theorem 4.7. 68. 60. CAPSTONE By inspection, determine why each of the sets is linearly dependent. 69. (a) S = {(1, –2), (2, 3), (– 2, 4)} (b) S = {(1, –6, 2), (2, – 12, 4)} 70. (c) S = {(0,0), (1, 0)} 71. Spanning the Same Subspace In Exercises 61 and 62, show that the sets S, and S, span the same subspace of R³. 61. S, = {(1, 2, –1), (0, 1, 1), (2, 5, –1)} S, = {(-2, –6, 0), (1, 1, – 2)} 62. S, = {(0, 0, 1), (0, 1, 1), (2, 1, 1)} S, = {(1, 1, 1), (1, 1, 2), (2, 1, 1)} 72. True or False? In Exercises 63 and 64, determine 73. whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example 74. I that shows the statement is not true in all cases or cite an appropriate statement from the text.