Negative Linear Transformation

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

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

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

Then: