# Prove there is a matrix X

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

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