\begin{definition}[Definition:Relative Matrix of Bilinear Form]
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:
:$\forall \tuple {i, j} \in \closedint 1 n \times \closedint 1 n: \sqbrk {\mathbf M_{f, \BB} }_{i j} = \map f {b_i, b_j}$
\end{definition}