Is null(T-\lambdaI) invariant under S for every \lambda \in F?

[pte419]

Suppose S, T \in L(V) are such that ST=TS. Prove that null(T-\lambdaI) is invariant under S for every \lambda \in F.


I can't find anything helpful in my book, I'm not sure where to start...

 

[Dick]

null(T-lambda*I) is just the set of all vectors x such that (T-lambda*I)x=0. So Tx-lambda*x=0. Or Tx=lambda*x. To show this is invariant under S you just have to show that if x satisfies that property, then so does Sx.