Definition:Pointwise Addition of Linear Transformations

Definition

Let $\struct {G, +_G}$ and $\struct {H, +_H}$ be abelian groups.

Let $\struct {R, +_R, \times_R}$ be a ring.

Let $\struct {G, +_G, \circ_G}$ and $\struct {H, +_H, \circ_H}$ be $R$-modules.

Let $\map {\LL_R} {G, H}$ be the set of all linear transformations from $G$ to $H$.

Let $\oplus_H$ be the pointwise operation on $\map {\LL_R} {G, H}$ induced by $+_H$ defined as:

$\forall u, v \in \map {\LL_R} {G, H}: \forall x \in G: \map {\paren {u \oplus_H v} } x := \map u x +_H \map v x$

Then $\oplus_H$ is referred to as pointwise addition (of linear transformations) on $\map {\LL_R} {G, H}$.