Definition:Relative Matrix of Linear Transformation

Definition
Let $\left({R, +, \circ}\right)$ 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 $\left \langle {a_n} \right \rangle$ be an ordered basis of $G$.

Let $\left \langle {b_m} \right \rangle$ be an ordered basis of $H$.

Let $u : G \to H$ be a linear transformation.

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:


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

That is, the matrix whose columns are the coordinate vectors of the image of the basis elements of $\mathcal A$ relative to the basis $\mathcal B$.

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

A motivation for this choice is the intuitive cancellation law in Change of Coordinate Vectors Under Linear Mapping.

Also denoted as
Alternative notations include $M_{f, B, A}$, $M_{B}^A(f)$ and $M_{B, A}(f)$.

Also see

 * Definition:Change of Basis Matrix
 * Linear Transformation as Matrix Product
 * Matrix Product as Linear Transformation