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 $$\left({H^G, +_H}\right)$$.