Definition:Relative Matrix of Linear Transformation

Definition
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:
 * $\sqbrk {u; \sequence {b_m}, \sequence {a_n} }$

Also denoted as
If $u$ is an automorphism on an $n$-dimensional module $G$, we can write $\sqbrk {u; \sequence {b_m}, \sequence {a_n} }$ as $\sqbrk {u; \sequence {a_n} }$

Other otations include:
 * $M_{u, B, A}$
 * $\map { {M_B}^A} u$
 * $\map {M_{B, A} } u$

Warning
Note the order in which the bases are presented in this expression $\sqbrk {u; \sequence {b_m}, \sequence {a_n} }$.

The indication of the ordered basis for the domain, that is $\sequence {a_n}$, is given after that of the codomain, that is $\sequence {b_m}$.

Thus, the entries in the $j$th column of $\sqbrk {u; \sequence {b_m}, \sequence {a_n} }$ are the scalars occurring in the expression of $\map u {a_j}$ as a linear combination of the sequence $\tuple {b_1, \ldots, b_m}$.

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

Also see

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