Prove that $S(W)$ is Invariant subspace

Question

Let $S, T: V\to V$ such that $ST=TS$. Let $W\subseteq V$. Prove that if $W$ is invariant subspace of $T$ then also $S(W)$ is invariant subspace of $T$.

Let $w\in W$. $$T(S(w)) = S(T(w)) = S(w')$$

where $w'\in W$ because $W$ is invariant of $T$.

I am not sure how to conclude that $S(W)$ is invariant of $T$.

Can you help me complete this?

Answer 1

Let $y\in S(W)$ so there's $x\in W$ such that $y=S(x)$. So $T(y)=T(S(x))=S(T(x))\in S(W)$ because $W$ is invariant by $T$ and then $T(x)\in W$. Conclude.