Let $S,T:V\to V$ be a linear transformation such that $ST=TS$. If $U$ is subspace of $V$ invariant under $T$, show that $S(U)$ is invariant under $T.$
If $U$ is subspace of $V$ invariant under $T$, then $T(U)\in U$. Want to show $T(S(U))\in S(U)$. Since $T$ and $S$ commute, then $T(S(U))\in S(U)$ iff $S(T(U))\in S(U)$. Since $U$ is invariant under assumption, the image of $S(T(U))$ is in $S(U)$. But I feel my explanation messy. Could someone clarify.
Here's how I would answer:
Note that since $T(U) \subset U$, we have $$ T(S(U)) = [TS](U) = [ST](U) = S(T(U)) \subset S(U) $$ the conclusion follows.