Change of Coordinate Vectors Under Linear Transformation

Theorem
Let $R$ be a ring with unity.

Let $M,N$ be free $R$-modules of finite dimension $m,n>0$ respectively.

Let $\mathcal A,\mathcal B$ be ordered bases of $M$ and $N$ respectively.

Let $f : M\to N$ be a linear mapping.

Let $\mathbf M_{f, \mathcal B, \mathcal A}$ be its matrix relative to $\mathcal A$ and $\mathcal B$.

Then for all $m\in M$:
 * $[f(m)]_{\mathcal B} = \mathbf M_{f, \mathcal B, \mathcal A} \cdot [m]_{\mathcal A}$

Where $[\cdot]_{-}$ denotes the coordinate vector with respect to a basis.

Proof
Both sides are linear in $m$ and they coincide on the elements of $\mathcal A$ by definition of $\mathbf M_{f, \mathcal B, \mathcal A}$.

So they are equal for all $m\in M$.

Also see

 * Change of Coordinate Vector Under Change of Basis, an analogous result for change of basis