Addition of Linear Transformations

Theorem
Let $\struct {G, +_G, \circ_G}$ and $\struct {H, +_H, \circ_H}$ 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 $\struct {H, +_H}$ is abelian.

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

Let $\lambda \in R, x \in G$.

Then: