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If R^n=span{X1, X2,..., Xk} and A does not equal 0 is an mxn matrix, show that AXi does not equal 0 for some i?

Please give as much detail as you can. I think this may be on a test so I really need to understand how you solve the proof.

1 Answer

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  • Anonymous
    9 years ago
    Favorite Answer

    Suppose A is in RREF (otherwise, reduce it via a series of elementary operations). Since A is not the zero matrix, there exists a row i in which the j^th column is a leading one.

    Then AX_j has at least one non-zero entry, namely its i^th entry is 1.

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