Invariant subspace when $TS=ST$

Question

Assume $S,T: V\to V$ such that $TS=ST$.

Show that if $W\subset V$ is invariant under $T$ then it is also invariant under $S$

I am assuming the problem is trivial however I am a bit stuck.

$T(w_1) \in W$ for all $w_1$ in $W$, then exists a $w_2$ such that $T(w_2)=w_1$

$S(w_1)=ST(w_2)=TS(w_2)=\space? $

I am missing something completely trivial.. what have I missed?

Answer 1

This is false. Let $T$ be the identity map. Every $S$ commutes with the identity map and every subspace is invariant under the identity map, and so we find out that every subspace is invariant under every map.

Maybe you got the conditions wrong?