(e) Let B = a b d be any 2 x 2 matrix. C (i) Show that there are real numbers un and a such that cos a = ull sin a (ii) Let a E R. Use the invertibility of Ra to prove that there are unique U12, U22 E R such that cos ou - sin a = U12 sin a + U22 COS a (iii) Use parts (i) and (ii) to show that B can be expressed in the form B = RQU for some a E R and some upper-triangular matrix U. (iv) Suppose that B = RQU = RV, where a, B E R and U and V are upper- triangular. Prove that if B is invertible, then U = #V.In this question, you will be using the following trigonometric identities: cos2 oz + sin2 a = 1 (1) cos(a + {3) = cos (1 cos {3 sin a: sin ,8 (2) sin(oz + ) = sin 0: cos ,8 + cos oz sin ,8 (3) where 01,5 6 R. You do not need to prove these identities. You may also use without proof the fact that the set cos a { |: . :| : 0: E R} s1n 0: is exactly the set of unit vectors in R2. New for any real number a, dene cos oz sin oz sin 0: cos a