What is Product?

For $A \in \mathcal{M}_{n,p}$ and $B \in \mathcal{M}_{p,q}$:

This is precisely the matrix of the composition: $\operatorname{Mat}(v \circ u) = \operatorname{Mat}(v)\, \operatorname{Mat}(u)$ (bases matching in the middle). Similarly, if $X$ is the column of coordinates of $x$, the column of $u(x)$ is $AX$.