Answer Happy

= 1. In this question, you will be using the following trigonometric identities: cosa + sina 1 (1) cos(a + b) cos a cos8 - sin a sin 8 sin(a+B) sin a cosB + cos a sin (3) where a, ß E R. You do not need to prove these identities. You may also use without proof the fact that the set cos a QER sin a is exactly the set of unit vectors in R2. Now for any real number a, define Ra= cosa - sin a sin a COS a (a) Prove that for all a, B ER, R R3 = Ra+3 (b) Using part (a), or otherwise, prove that Rg is invertible and that R1 = R-a, for all a ER (c) Prove that for all a € R and all x, y € R2, (Rex) · (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 € R? Either prove this, or give a counterexample (including justification). (e) Let B= be any 2 x 2 matrix. cosa [el Hint: express (i) Show that there are real numbers uji and a such that = U11 sin a as a scalar multiple of a unit vector, and hence find an expression for uji in terms of a and c. (ii) Let a € R. Use the invertibility of Rg to prove that there are unique U12, U22 € R such that [al cos a = U12 sin a + U22 sin a cos a (iii) Use parts (i) and (ii) to show that B can be expressed in the form B = RaU for some a € R and some upper-triangular matrix U. (iv) Suppose that B = R U = R3V, where a, B E R and U and V are upper- triangular. Prove that if B is invertible, then U = EV.