Matrix of Bilinear Form Under Change of Basis

Theorem
Let $R$ be a ring with unity.

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

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

Let $\mathbf M_{\mathcal A, \mathcal B}$ be the change of basis matrix from $\mathcal A$ to $\mathcal B$.

Let $f : M\times M \to R$ be a bilinear form.

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

Then its matrix relative to $\mathcal B$ equals:
 * $\mathbf M_{f, \mathcal B} = \mathbf M_{\mathcal A, \mathcal B}^\intercal \mathbf M_{f, \mathcal A} \mathbf M_{\mathcal A, \mathcal B}$

Proof
Let $m\in M$, and let $[m]_{\mathcal A}$ and $[m]_{\mathcal B}$ denote its coordinate vectors relative to $\mathcal A$ and $\mathcal B$.

We have:

Thus $\mathbf M_{f, \mathcal B} = \mathbf M_{\mathcal A, \mathcal B}^\intercal \mathbf M_{f, \mathcal A} \mathbf M_{\mathcal A, \mathcal B}$.