midterm1 Practice Solutions

midterm1 Practice Solutions - Math 136 Additional Practice...

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Unformatted text preview: Math 136 Additional Practice for Term Test 1 Solutions 1: Let B = 1 1 − 2 , 1 − 1 3 , 5 1 . (a) Determine if vectorv = 5 5 is in span B . Solution: We need to determine if there exists c 1 , c 2 , c 3 such that 5 5 = c 1 1 1 − 2 + c 2 1 − 1 3 + c 3 5 1 = c 1 + c 2 + 5 c 3 c 1 − c 2 + c 3 − 2 c 1 + 3 c 2 Comparing entries, we see that we need to determine whether the system c 1 + c 2 + 5 c 3 = 0 c 1 − c 2 + c 3 = 5 − 2 c 1 + 3 c 2 = 5 is consistent or not. Row reducing the corresponding augmented matrix to RREF gives 1 1 5 1 − 1 1 5 − 2 3 5 ∼ 1 0 3 0 1 2 0 0 0 1 Therefore, the rank of the coefficient matrix is less than the rank of the augmented matrix, thus the system is inconsistent. Therefore, vectorv is not in the span of B . (b) Determine if B is linearly independent or linearly dependent. Solution: Consider 5 = c 1 1 1 − 2 + c 2 1 − 1 3 + c 3 5 1 = c 1 + c 2 + 5 c 3 c 1 − c 2 + c 3 − 2 c 1 + 3 c 2 We see that we get a homogeneous system with the same coefficient matrix as in (a)....
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This note was uploaded on 03/03/2011 for the course MATH 136 taught by Professor All during the Spring '08 term at Waterloo.

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midterm1 Practice Solutions - Math 136 Additional Practice...

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