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Math Help - Prove there is a matrix X

  1. #1
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    Prove there is a matrix X

    Let A be an mxn matrix and suppose that B is an mxp matrix such that every column of B is in the column space of A. Prove that there is a matrix X such that
    AX=B.
    (HITN: we know the equation Ax=v has a solution if and only if v \in col(A))
    Last edited by wopashui; February 26th 2010 at 03:58 PM.
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  2. #2
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    I got the same question, but there was a typo in it. I think it should read like this:

    Let A be an m\times n matrix and suppose that X is an n\times p matrix such that every column of X is in the column space of A.
    Prove that there is a matrix B such that AX=B ( B would be an m\times p size matrix).
    [HINT: we know that the equation Ax=v has a solution if and only if v\in col(A).]

    An addition to the hint was provided that for any column (i.e. v) of B, there is always a solution, x, to Ax=v, so we are to apply this successively to each column and determine how many x's we obtain.
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