3. Let T: V → V be linear and have distinct eigenvalues A₁,...,Ak. For each eigenvalue Aj, we

will denote by

E(T, Aj) = {v € V: Tv=Xjv}

the set of all eigenvectors corresponding to the eigenvalue xj.

1

Suppose that S: V → V linear is such that ST = TS. Then, show that for all eigenvalues X;

S: E(T, Aj) → E(T, λj).

(The above formula is a compressed way of writing that if v € E(T, X;) then Sv € E(T, Aj). This is

what you need to show.)