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 $\BB = \sequence {b_m}$ be an ordered basis of $M$.

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

The matrix of $f$ relative to $\BB$ is the $n \times n$ matrix $\mathbf M_{f, \BB}$ where:
 * $\forall \tuple {i, j} \in \closedint 1 n \times \closedint 1 n: \sqbrk {\mathbf M_{f, \BB} }_{i j} = \map f {b_i, b_j}$

Also see

 * Matrix of Bilinear Form Under Change of Basis


 * Definition:Change of Basis Matrix