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:

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