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 transformation.

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

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

where $\left[{\, \cdot \,}\right]_{-}$ 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