Product of Change of Basis Matrices

Theorem
Let $R$ be a ring with unity.

Let $M$ be a free $R$-module of finite dimension $n>0$.

Let $\AA$, $\BB$ and $\CC$ be ordered bases of $M$.

Let $\mathbf M_{\AA, \BB}$, $\mathbf M_{\BB, \CC}$ and $\mathbf M_{\AA, \CC}$ be the change of basis matrices from $\AA$ to $\BB$, $\BB$ to $\CC$ and $\AA$ to $\CC$ respectively.

Then:
 * $\mathbf M_{\AA, \CC} = \mathbf M_{\AA, \BB} \cdot \mathbf M_{\BB, \CC}$

Proof
Let $m \in M$.

Let $\sqbrk m_\AA$ be its coordinate vector relative to $\AA$, and similarly for $\BB$ and $\CC$.

On the one hand:

On the other hand:

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

Because $m$ is arbitrary, the result follows.

Also see

 * Relative Matrix of Composition of Linear Transformations, an analogous result for linear transformations, of which this is a special case