7. Let K be the plane 3x - y + 2z = 0. Let v = -1 . Find the orthogonal projection of v onto K by 1 using the formula Pv = W(WTW)-1WTv, where W is a basis matrix for K. 8. The purpose of this problem is to generalize the orthogonal projection formula in Problem 7. Let K be a subspace of F", and let W be a basis matrix for K. Prove that Pv = W(WTW)-1WTv is the orthogonal projection of F" onto K. (EXTRA CREDIT) 9. Prove that the orthogonal projection formula in Problem 8 is independent of the basis matrix of K. That is, if W and U are basis matrices for K, then W(WW)-1WT = U(UTU)-BUT. (EXTRA CREDIT)11. The purpose of this problem is to generalize the equation in 10b. Let K be a subspace of R" and let W1, W2, ..., Wm be a basis of K. Extend W1, W2, ..., Wm into a basis W1, W2, ..., Wm, S1, S2, ..., Sn-m for R", where S1, $2, ..., Sn-m are orthogonal to K. Show that the orthogonal projection P of R" onto K is given by Pv = [W1 ... Wm S1 ... Sn-m][e1 ... em0 ... O][w1 ... Wm S1 ... Sn-m]-1v. (EXTRA CREDIT) 12. Let I be a line with the equation y = mx and let s be the line that goes through the origin and is perpendicular to l. Let r be the reflection transformation about l. Show that a. r(v) = 1 1 - m2 2m 1+m2 2m m2 - 1. v for each VER2. (EXTRA CREDIT) b. r(v) = cos2a sin2a sin2a -cos2al v for each vER, where a is the angle between I and the positive x-axis. (EXTRA CREDIT) Use the following definition for r: If v is in l, then r(v) = v. If v is not in l, then r(v) = u, where l is the perpendicular bisector of the segment connecting the tips of v and u