Addition of Linear Transformations

From ProofWiki
Jump to navigation Jump to search

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:

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

$\blacksquare$


Sources