Relative Matrix of Composition of Linear Transformations/Proof 1

Proof
Let $m \in M$, and $\sqbrk m_\AA$ be its coordinate vector with respect to $\AA$.

On the one hand:

On the other hand:

Thus:


 * $\forall m \in M: \paren {\mathbf M_{g \mathop \circ f, \CC, \AA} - \mathbf M_{g, \CC, \BB} \cdot \mathbf M_{f, \BB, \AA} } \cdot \sqbrk m_\AA = 0$

The result follows.