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.)

Fig: 1