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 $\AA, \BB$ be ordered bases of $M$ and $N$ respectively.

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

Let $\mathbf M_{f, \BB, \AA}$ be its matrix relative to $\AA$ and $\BB$.

Then for all $m \in M$:
 * $\sqbrk {\map f m}_\BB = \mathbf M_{f, \BB, \AA} \cdot \sqbrk m_\AA$

where $\sqbrk {\, \cdot \,}_-$ denotes the coordinate vector with respect to a basis.

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

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