Prove there is a matrix X

• February 24th 2010, 11:31 PM
wopashui
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))
• February 26th 2010, 02:21 PM
Runty
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.