Set of Linear Transformations under Pointwise Addition forms Abelian Group

Theorem
Let $\left({G, +_G, \circ}\right)_R$ and $\left({H, +_H, \circ}\right)_R$ be $R$-modules.

Let $\mathcal L_R \left({G, H}\right)$ be the set of all linear transformations from $G$ to $H$.

Let $+_H$ be the operation on $H^G$ as defined in Addition of Linear Transformations.

Then $\left({\mathcal L_R \left({G, H}\right), +_H}\right)$ is an abelian group.

Proof
Let $\phi, \psi \in \mathcal L_R \left({G, H}\right)$.


 * From Addition of Linear Transformations, $\phi +_H \psi \in \mathcal L_R \left({G, H}\right)$.


 * From Negative Linear Transformation, $- \phi \in \mathcal L_R \left({G, H}\right)$.


 * Thus, from the Two-Step Subgroup Test, $\left({\mathcal L_R \left({G, H}\right), +_H}\right)$ is a subgroup of the Induced Group $\left({H^G, +_H}\right)$.