# Set of Linear Transformations under Pointwise Addition forms Abelian Group

## Theorem

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

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

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

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

Let $\oplus_H$ be defined as pointwise addition on $\map {\LL_R} {G, H}$:

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

Then $\struct {\map {\LL_R} {G, H}, \oplus_H}$ is an abelian group.

## Proof

From Structure Induced by Group Operation is Group, $\struct {H^G, \oplus_H}$ is a group

Let $\phi, \psi \in \map {\LL_R} {G, H}$.

$\phi \oplus_H \psi \in \map {\LL_R} {G, H}$
$-\phi \in \map {\LL_R} {G, H}$

Thus, from the Two-Step Subgroup Test:

$\struct {\map {\LL_R} {G, H}, \oplus_H}$ is a subgroup of $\struct {H^G, \oplus_H}$.

It remains to be shown that $\struct {\map {\LL_R} {G, H}, \oplus_H}$ is abelian.

Let $u$ and $v$ be arbitrary elements of $\map {\LL_R} {G, H}$.

Indeed, we have that:

 $\ds \map {\paren {u \oplus_H v} } x$ $=$ $\ds \map u x +_H \map v x$ Definition of Pointwise Addition of Linear Transformations $\ds$ $=$ $\ds \map v x +_H \map u x$ as $H$ is abelian $\ds$ $=$ $\ds \map {\paren {v \oplus_H u} } x$ Definition of Pointwise Addition of Linear Transformations

$\blacksquare$