Definition:Relative Matrix of Linear Transformation

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:

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

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 range, 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 $$\left({b_1, \ldots, b_m}\right)$$.