## 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:

 $\displaystyle \left({\phi +_H \psi}\right) \left({\lambda \circ x}\right)$ $=$ $\displaystyle \phi \left({\lambda \circ x}\right) +_H \psi \left({\lambda \circ x}\right)$ $\displaystyle$ $=$ $\displaystyle \lambda \circ \phi \left({x}\right) +_H \lambda \circ \psi \left({x}\right)$ $\displaystyle$ $=$ $\displaystyle \lambda \circ \left({\phi \left({x}\right) +_H \psi \left({x}\right)}\right)$ $\displaystyle$ $=$ $\displaystyle \lambda \circ \left({\phi +_H \psi}\right) \left({x}\right)$

$\blacksquare$