Definition:Relative Matrix of Bilinear Form

Definition
Let $R$ be a ring with unity.

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

Let $\mathcal B = \left \langle {b_m} \right \rangle$ be an ordered basis of $M$.

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

The matrix of $f$ relative to $\mathcal B$ is the $n\times n$ matrix $\mathbf M_{f, \mathcal B}$ where:
 * $\displaystyle \forall \left({i, j}\right) \in \left[{1 \,.\,.\, n}\right] \times \left[{1 \,.\,.\, n}\right] : (\mathbf M_{f, \mathcal B})_{ij} = f \left({b_i, b_j}\right)$

Also see

 * Matrix of Bilinear Form Under Change of Basis
 * Definition:Change of Basis Matrix