You just need to answer e), but please well-written, Thank you! 8. Let V be a finite dimensional inner product space and let P :V + V be a linear map satisfying p2 = P. (a) Prove that the only eigenvalues of P are 0 and 1. Notice that in this problem you are not proving that the eigenvalues exist. You only show that if they do exist, they have to be 0 or 1. (b) Prove that P is diagonalizable. Here you need to show that you do have eigenvectors and that you have enough eigenvectors to diagonalize P. When F = R, linear maps don't necessarily have eigenvectors. (c) Prove that if P is selfadjoint, then to EV, ||P(v)|| = ||0||. (d) Suppose that Vv E V, ||P(0)|| = ||0|| and that vi, v2 E V satisfy: P(vi) = 0 and P(v2) = V2. Prove that (v1, v2) = 0. (e) Use the previous result to prove that if VU E V, ||P(v)|| = ||0||, then P is selfadjoint. You just need to answer e), but please well-written, Thank you! 8. Let V be a finite dimensional inner product space and let P :V + V be a linear map satisfying p2 = P. (a) Prove that the only eigenvalues of P are 0 and 1. Notice that in this problem you are not proving that the eigenvalues exist. You only show that if they do exist, they have to be 0 or 1. (b) Prove that P is diagonalizable. Here you need to show that you do have eigenvectors and that you have enough eigenvectors to diagonalize P. When F = R, linear maps don't necessarily have eigenvectors. (c) Prove that if P is selfadjoint, then to EV, ||P(v)|| = ||0||. (d) Suppose that Vv E V, ||P(0)|| = ||0|| and that vi, v2 E V satisfy: P(vi) = 0 and P(v2) = V2. Prove that (v1, v2) = 0. (e) Use the previous result to prove that if VU E V, ||P(v)|| = ||0||, then P is selfadjoint