Problem 1. Let V be a subspace of R" of dimension r. Let projy : R" - R" denote orthogonal projection onto V. Consider the map py : R" - R" defined by pv (x) = 2projv () - for all a E R". This problem should help you understand why py is called the reflection over V. (a) Prove that for T E VI, PV(J) = -T. (b) Prove that for T E V, pv (a) = . (c) Prove that py o py = Id as transformations of R". (d) Prove that there is a basis B for R" such that [projv]s = I, 0 0 and [puls = 6 2. ] [HINT: Let B be the union of a basis for V and a basis for VA.] A GENERAL HINT FOR ALL PARTS OF PROBLEM 1: Remember that by Thm 5.1.4, every & E R" can be written uniquely as all + al where...n Problem 2. Let V be the one-dimensional subspace of R" spanned by the vector w = ) e; = E R. - ... i=1 (a) Find the standard matrix of the orthogonal projection projy : R" -> R" on V. (b) Find the standard matrix of projyl : R" - R" where VI. [Hint: Use Theorem 5.1.4 to show that = = projva + projvid for all a ER".]