Definition:Relative Matrix of Linear Transformation

Definition
Let $\left({R, +, \circ}\right)$ be a commutative ring with unity.

Let $\left \langle {a_n} \right \rangle$ be an ordered basis of an $n$-dimensional $R$-module $G$.

Let $\left \langle {b_m} \right \rangle$ be an ordered basis of an $m$-dimensional $R$-module $H$.

Let $\mathcal L \left({G, H}\right)$ be the set of all linear transformations from $G$ to $H$.

Let $u \in \mathcal L \left({G, H}\right)$.

The matrix of $u$ relative to $\left \langle {a_n} \right \rangle$ and $\left \langle {b_m} \right \rangle$ is the $m \times n$ matrix $\left[{\alpha}\right]_{m n}$ where:


 * $\displaystyle \forall \left({i, j}\right) \in \left[{1 \,.\,.\, m}\right] \times \left[{1 \,.\,.\, n}\right]:u \left({a_j}\right) = \sum_{i \mathop = 1}^m \alpha_{i j} \circ b_i$

The matrix of such a linear transformation $u$ relative to the ordered bases $\left \langle {a_n} \right \rangle$ and $\left \langle {b_m} \right \rangle$ is denoted:
 * $\left[{u; \left \langle {b_m} \right \rangle, \left \langle {a_n} \right \rangle}\right]$

If $u$ is an automorphism on an $n$-dimensional module $G$, we can write $\left[{u; \left \langle {a_n} \right \rangle, \left \langle {a_n} \right \rangle}\right]$ as $\left[{u; \left \langle {a_n} \right \rangle}\right]$.

Comment
Note the order of the bases in this expression $\left[{u; \left \langle {b_m} \right \rangle, \left \langle {a_n} \right \rangle}\right]$. The indication of the ordered basis for the domain, i.e. $\left \langle {a_n} \right \rangle$, is given last, and that of the codomain, i.e. $\left \langle {b_m} \right \rangle$, given first.

Thus, the entries in the $j$th column of $\left[{u; \left \langle {b_m} \right \rangle, \left \langle {a_n} \right \rangle}\right]$ are the scalars occurring in the expression of $u \left({a_j}\right)$ as a linear combination of the sequence $\left({b_1, \ldots, b_m}\right)$.