Answer Happy

1. In this question, you will be using the following trigonometric identities: cos² a + sin² a = 1 = cos a cos 3-sin a sin 3 cos(a + 3) sin(a + 3) = (2) (3) sin a cos 3 + cos a sin 3 where a, ß ER. You do not need to prove these identities. You may also use without proof the fact that the set cos a :a ER sin a is exactly the set of unit vectors in R2. Now for any real number a, define Ra= cos a-sin a sin a cos a (a) Prove that for all a, 6 € R, RaRs = Ra+B (b) Using part (a), or otherwise, prove that Ro is invertible and that R¹ = R-a, for all a € R. (c) Prove that for all a ER and all x, y € R², (Rax) (Ray) = x.y (d) Suppose A is a 2 x 2 matrix such that for all x, y € R², (Ax). (Ay) = x.y Must it be true that A = Ra, for some a ER? 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 u₁1 and a such that = 211 [cos a] sin a (ii) Let a € R. Use the invertibility of Ro to prove that there are unique U12, U22 ER such that - sin a a = 212 cos a sin a +422 [ COS a (iii) Use parts (i) and (ii) to show that B can be expressed in the form B = RaU for some a ER and some upper-triangular matrix U. (iv) Suppose that B = RU = RV, where a, ß ER and U and V are upper- triangular. Prove that if B is invertible, then U = +V.