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math294_hw4fa06 - Math 294 Homework 4 Section 3.3 28 Form a...

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Math 294 Homework 4 Section 3.3 28. Form a 4 X 4 matrix A with the given vectors as its columns. The matrix A reduces to 1 0 0 2 0 1 0 3 0 0 1 4 0 0 0 k - 29 . This matrix can be reduced further to I 4 if and only if k = 29. By Summary 3.3.9, the four given vectors form a basis of R 4 unless k = 29. 32. We need to find all vectors x in R 4 such that x 1 x 2 x 3 x 4 . 1 0 - 1 1 = 0 ; x 1 x 2 x 3 x 4 . 0 1 2 3 = 0 . Which amounts to finding the kernel of 1 0 - 1 1 0 1 2 3 . Using Kyle Numbers, we find the basis 1 - 2 1 0 , - 1 - 3 0 1 . 46. Use Exercise 45 with v 1 = 1 2 3 4 , v 2 = 1 4 6 8 , and w i = e i for i = 1 , 2 , 3 , 4. Now rref 1 1 1 0 0 0 2 4 0 1 0 0 3 6 0 0 1 0 4 8 0 0 0 1 = 1 0 2 0 0 - 1 / 4 0 1 - 1 0 0 1 / 4 0 0 0 1 0 - 1 / 2 0 0 0 0 1 - 3 / 4 .
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