I am studying representation theory. Consider that $V$ and $W$ are two finite dimension vector spaces such that $V$ is isomorphic to $W$. Consider also that $A$ is an algebra. I would like to know If exists an homomorphism of representations $$\phi: V \to W$$
such that $\phi(av) = a\phi(v)$ to $v \in V $ and $\phi \neq 0$
Thank you in advance
The answer is no. Let $A=\mathbb{C}[x]/(x^2-1)$. Let $V$ be the one dimensional $A$-module with $x.v=v$ for all $v\in V$, and let $W$ be the one dimensional $A$-module with $x.w=-w$ for all $w\in W$. Then $V$ and $W$ are linearly isomorphic, but they are not isomorphic as $A$-modules.