C3093305 - ·· c k-1 x k-1 = Hence c 1 x 1 ·· c k-1 x...

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5. (Statement of the problem.) Solution. (a) Not necessary. If x k +1 is a linear combination of x 1 , · · · , x k , then by Observation (II) in page 135, x 1 , · · · , x k , x k +1 are linearly dependent. ( Note: It is important to indicate which theorem, definition, etc. you use. ) (b) Yes. We prove it by contradiction. Suppose that x 1 , · · · , x k - 1 are linearly dependent. By definition, there exist scalars c 1 , · · · , c k - 1 , not all zero, such that c 1 x
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Unformatted text preview: ··· + c k-1 x k-1 = . Hence, c 1 x 1 + ··· + c k-1 x k-1 +0 · x n = . Since c 1 , ··· ,c k-1 , 0 are not all zero, we conclude by definition that x 1 , ··· , x k-1 , x k are linearly dependent. It contradicts to the fact that x 1 , ··· , x k-1 , x k are linearly independent. Hence x 1 , ··· , x k-1 must be linearly independent. / 1...
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