Addition of Linear Transformations

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

Let $\phi: G \to H$ and $\psi: G \to H$ be linear transformations.

Let $\phi +_H \psi$ be the operation on $H^G$ induced by $+_H$ as defined in Induced Structure.

Then $\phi +_H \psi: G \to H$ is a linear transformation.

Poof
From the definition of a module, the group $\left({H, +_H}\right)$ is abelian.

Therefore we can apply Homomorphism on Induced Structure to show that $\phi +_H \psi: G \to H$ is a homomorphism.

Then: