Let V be a vector space over

Question 7. Let V be a vector space over R and let P : V - V be a linear map such that P? = P. a) Prove that if ve Im(P), then P(v) = v. (2 points) b) Let U1, U2, ..., Un be linearly independent vectors in Im(P) and let w1, w2, ..., Wm be linearly in- dependent vectors in ker(P). Prove that {71, U2, ..., Un, W1, W2, ..., Wm} is a linearly independent set of vectors in V. (6 points) c) Prove that if (U, w) = 0 for all v E Im(P) and w E ker(), then P is the orthogonal projection onto Im(P). (4 points)