Suppose that B = {V₁, V2,..., Vk) is an orthonormal basis for a subspace W of R". Let C be the nx k matrix whose columns are the vectors in B. Prove the following statements: (a) CTC = Ik. (b) Pw = CCT.

6. Let B be the following ordered basis for R?: в %3 {(1,—1), (2, -3)} . Find the coordinate vector of v = (-5,8) with respect to B. 1 2 3 (a) [v]B (b) [v]B (c) [v]B = -3 -3 -2 3 -3 (d) [v]B = (e) [v]B 2

V has bases B 2v1 + 5v2 + bv3, W3 = v1 + v2 + 5v3. (v1, v2, V3) and B' = (w1, w2, w3). If wi vi + 2v2 + 4v3, W2 = Determine the change of basis matrices, PgB and Pg¬B'-

Let {v1, v2, . .. , vn} be an orthogonal set in R" with respect to the Euclidean inner product. Let A be the n x n matrix with row vectors v1, v2, ... , Vn. b) What extra condition(s) do we need from the vectors v1, v2, ..., Vn so that the matrix AAT is an identity matrix? Justify your answer!