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 $\AA$ and $\BB$ be ordered bases of $M$.

Let $\mathbf M_{\AA, \BB}$ be the change of basis matrix from $\AA$ to $\BB$.

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

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

Then its matrix relative to $\BB$ equals:
 * $\mathbf M_{f, \BB} = \mathbf M_{\AA, \BB}^\intercal \mathbf M_{f, \AA} \mathbf M_{\AA, \BB}$

Proof
Let $m \in M$, and let $\sqbrk m_\AA$ and $\sqbrk m_\BB$ denote its coordinate vectors relative to $\AA$ and $\BB$.

We have:

Thus $\mathbf M_{f, \BB} = \mathbf M_{\AA, \BB}^\intercal \mathbf M_{f, \AA} \mathbf M_{\AA, \BB}$.