Relative Matrix of Composition of Linear Transformations/Proof 1

Proof
Let $m\in M$, and $[m]_{\mathcal A}$ be its coordinate vector with respect to $\mathcal A$.

On the one hand:

On the other hand:

Thus $(\mathbf M_{(g\mathop\circ f), \mathcal C, \mathcal A} - \mathbf M_{g, \mathcal C, \mathcal B} \cdot \mathbf M_{f, \mathcal B, \mathcal A}) \cdot [m]_{\mathcal A} = 0$ for all $m\in M$.

The result follows.