Prove that the image of an invariant subspace under a morphism of representations is an invariant subspace.

Question

Prove that the image of an invariant subspace under a morphism of representations is an invariant subspace. Could anyone give me a hint on how to solve this Please?

Knowing the following:

And Remark 2 is given below:

MY THOUGHT:

I have proved this problem before:

And I think it will be helpful here but I do not know how .... could anyone help me please?

Answer 1

You have $\sigma \colon V_1 \to V_2$ such that $\sigma T_1(g) = T_2(g)\sigma$ for all $g\in G$. Let $U\subset V_1$ be an invariant subspace. We want see what happens to $\sigma(U)$.

Let $ v = \sigma(w) \in \sigma(U)$ for some $w\in V_1$ then $$ T_2(g)v = T_2(g)\sigma(w) = \sigma (T_1(g)w ) \in \sigma(U) $$ since $T_1(g)w\in U$.