Answer Happy

1. In this question, you will be using the following trigonometric identities: cos² a + sin² a = 1 (1) (2) cos(a + B) = cos a cos sin a sin sin(a + 3) = sin a cos + 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 {[cosa]: a ER} sin is exactly the set of unit vectors in R². Now for any real number α, define COS a - sin a Ra sin a COS a (a) Prove that for all a, ß ER, Ra RB - Ra+ - (b) Using part (a), or otherwise, prove that Ra is invertible and that R=¹ = R-a, for all a € R. (c) Prove that for all a E R and all x, y € R², (Rax) · (Ray): = x.y (d) Suppose A is a 2 × 2 matrix such that for all x, y ≤ R², (Ax). (Ay) = = x.y Must it be true that A = Ray for some a ER? Either prove this, or give a counterexample (including justification).