Prove the following variant of Theorem EMMVP that has a weaker hypothesis: Suppose that $C=\set{\vectorlist{u}{p}}$ is a linearly independent spanning set for $\complex{n}$. Suppose also that $A$ and $B$ are $m\times n$ matrices such that $A\vect{u}_i=B\vect{u}_i$ for every $1\leq i\leq n$. Then $A=B$.

Can you weaken the hypothesis even further while still preserving the conclusion?