\begin{definition}[Definition:Relative Matrix of Linear Transformation]
Let $\struct {R, +, \circ}$ be a ring with unity.
Let $G$ be a free $R$-module of finite dimension $n>0$
Let $H$ be a free $R$-module of finite dimension $m>0$
Let $\sequence {a_n}$ be an ordered basis of $G$.
Let $\sequence {b_m}$ be an ordered basis of $H$.
Let $u : G \to H$ be a linear transformation.
The '''matrix of $u$ relative to $\sequence {a_n}$ and $\sequence {b_m}$''' is the $m \times n$ matrix $\sqbrk \alpha_{m n}$ where:
:$\ds \forall \tuple {i, j} \in \closedint 1 m \times \closedint 1 n: \map u {a_j} = \sum_{i \mathop = 1}^m \alpha_{i j} \circ b_i$
That is, the matrix whose columns are the coordinate vectors of the image of the basis elements of $\AA$ relative to the basis $\BB$.
The matrix of such a linear transformation $u$ relative to the ordered bases $\sequence {a_n}$ and $\sequence {b_m}$ is denoted:
:$\sqbrk {u; \sequence {b_m}, \sequence {a_n} }$
\end{definition}