1. In this question, you will be using the following trigonometric identities: cos a + sin a = 1 (1) cos(a + B) = cos a cos B - sin a sin B (2) sin(a + B) = sin a cos B + cos a sin B (3) where a, B E R. You do not need to prove these identities. You may also use without proof the fact that the set is exactly the set of unit vectors in R2. Now for any real number a, define Ra = cos a - sin a sin a COS O (a) Prove that for all a, B ER, RaRe = Rats (b) Using part (a), or otherwise, prove that Ro is invertible and that Ral = R_a, for all a E R. (c) Prove that for all o E R and all x, y E R2, (Rox) . (Ray) = x . y (d) Suppose A is a 2 x 2 matrix such that for all x, y E R2, (Ax) . ( Ay) = x . y Must it be true that A = Ra, for some a E R? Either prove this, or give a counterexample (including justification). (e) Let B = be any 2 x 2 matrix . (i) Show that there are real numbers un and a such that [o] - was [ cows a]. Hint: express as a scalar multiple of a unit vector, and hence find an expression for uni in terms of a and c. (ii) Let a E R. Use the invertibility of Ro to prove that there are unique U12, U22 E R such that = 212 [cos a + 2 22 [- sin a] Isin a COS OF (iii) Use parts (i) and (ii) to show that B can be expressed in the form B = RQU for some o E R and some upper-triangular matrix U. (iv) Suppose that B = ROU = RV, where a, B E R and U and V are upper- triangular. Prove that if B is invertible, then U = V