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

From Addition of Linear Transformations:

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

From Negative Linear Transformation:

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

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 |

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 28$. Linear Transformations