Negative Linear Transformation

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

Let $$\phi: G \to H$$ be a linear transformation.

Let $$- \phi$$ be the negative of $$\phi$$ as defined in Induced Structure Inverse.

Then $$- \phi: G \to H$$ is also a linear transformation.

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

Therefore we can apply Inverse Mapping in Induced Structure to show that $$- \phi: G \to H$$ is a homomorphism.

Then:

$$ $$ $$ $$