Definition:Relative Matrix of Linear Transformation/Warning

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.

Consider the matrix of $u$ relative to $\sequence {a_n}$ and $\sequence {b_m}$:
 * $\sqbrk {u; \sequence {b_m}, \sequence {a_n} }$

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.