3 let t v v be linear and have distinct eigenvalues a ak for each eige

Question

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.)