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February 25th, 2010, 12:32 AM   #1
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Prove there is a matrix X

Let A be an mxn matrix and suppose that B is an nxp 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 col(A))
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February 25th, 2010, 06:21 AM   #2
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Re: Prove there is a matrix X

It doesn't make sense. B should be MxP.
Anyway, assuming that B is MxP matrix :
{v1,v2,...,vm} in Rn are the columns of A. From the question, if b1 is a column of B then
b1 = a1v1+a2v2+...+amvm, where a1,a2,...,am in F. In other words, x1 = [a1,a2,...,am] is
a solution to this system : Ax1=b1. That brings us to Axi=bi, where 1<=i<=p. Therefore, if you'll
take all the x1,...xp and put them as columns in a matrix you'll get X such that AX=B.
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