Answer to Question #206116 in Linear Algebra for Sabelo

Question

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

Answer 1

By definition,$\forall v \in \text{Null} (S), S(v) = 0$

Since $ST = TS$, we have$S(T(v)) = T(S(v)) = T(0) = 0$

since $T$ is linear.

Hence $T(v) \in \text{Null} (S)$.

Thus $\text{Null} (S)$ is invariant under $T$.