I'm looking for a solution verification.
Question: Suppose $S,T \in L(V)$ are such that $ST = TS$. Prove that null S is invariant under T.
My solution: Let $s \in \text{null }S$. Then $Ss=0$, and so $TSs=0$, and so $TSs=STs=0$.
$STs=0$ implies that $Ts \in \text{null }S$, hence we have $s \in \text{null }S \implies Ts \in \text{null }S$. Hence $\text{null }S$ is invariant under $T$