Answer Happy

1. In this question, you will be using the following trigonometric identities: cosa + sind a 1 (1) cos(a +B) cos a cos B - sin a sin 8 (2) sin(a+B) sin a cosB + cos a sin B (3) where a, B ER. You do not need to prove these identities. You may also use without proof the fact that the set :QER sina is exactly the set of unit vectors in R2. Now for any real number a, define cosa - sin a R sin a COS a
= Q (a) Prove that for all a, BER, R Rg = Ra+ (b) Using part (a), or otherwise, prove that R, is invertible and that R1 = R-a, for all a ER (c) Prove that for all a E R and all x, y € R2, (Rex). (Ray) = xy (d) Suppose A is a 2 x 2 matrix such that for all x, y € R2, (Ax). (Ay)=xy Must it be true that A = Ra, for some a e R? Either prove this, or give a counterexample (including justification). [a b] (e) Let B be any 2 x 2 matrix. cal =U Hint: erpress (i) Show that there are real numbers un and a such that Tcos al sina as a scalar multiple of a unit vector, and hence find an expression for un terms of a and c. (ii) Let a € R. Use the invertibility of Ra to prove that there are unique U12, U22 ER such that a Icos al + 1122 = 12 sina sin a 2 - [o COS (iii) Use parts (i) and (ii) to show that B can be expressed in the form B = RU for some a e R and some upper-triangular matrix U. (iv) Suppose that B = R U = RV, where a, B e R and U and V are upper- triangular. Prove that if B is invertible, then U = IV.