Question

Suppose S; T 2 L(V ) are such that ST = T S. Prove that null S is invariant under T.

Answer 1

We need to prove $T(\operatorname{null}S)\subset\operatorname{null}S$

Take arbitrary $x\in T(\operatorname{null}S)$

Since $x\in T(\operatorname{null}S)$, there exists $y\in\operatorname{null}S$ such that $x=Ty$

Then $Sx=STy$. Since $ST=TS$, we have $Sx=TSy$

$Sy=0$, because $y\in\operatorname{null}S$, then $Sx=TSy=T0=0$, that is $x\in\operatorname{null}S$.

Since we take arbitrary $x\in T(\operatorname{null}S)$, we have $T(\operatorname{null}S)\subset\operatorname{null}S$