# Prove there is a matrix B

• Feb 26th 2010, 09:24 AM
Runty
Prove there is a matrix B
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 there is a matrix $\displaystyle B$ such that $\displaystyle AX=B$
[Hint: we know that the equation $\displaystyle Ax=v$ has a solution if and only if $\displaystyle v\in col(A)$]

I don't know if this is exactly how the question is meant to be said. Our Prof. made a severe typo in the original, so I'm asking him to simply rewrite the question in its intended form rather than just provide us with a quick correction (which could be easily misinterpreted).
• Feb 28th 2010, 07:52 AM
Runty
I've consulted a few other sources for a possible answer to this question, but I am still in need of a formulaic answer. One such answer read as this:

"For every column b of B, you get a column solution to Ax=b.

Assemble all of those columns x into X (in the same order that you got them from b in B). For example, the third column of X will be a solution to Ax=b, where b is the third column of B."

How would I show this in terms of mathematical formulas? Or is that thinking the wrong way?