(Solved): 1. In this question, you will be using the following trigonometric identities: cosa + sina = 1 (1) ...

1. In this question, you will be using the following trigonometric identities: cosa + sina = 1 (1) cos(a + 8) = cos a cos 3 - sin a sin 8 (2) sin(a + 8) = sin a cos 3 + cosa sin 8 (3) where a,B ER. You do not need to prove these identities. You may also use without proof the fact that the set cos al sina :0€R} - sina COS is eractly the set of unit vectors in Rº. Now for any real number o, define cosa R = sina (a) Prove that for all a,BER, R.R3 = R.+B (b) Using part (a), or otherwise, prove that Ra is invertible and that R' = R-a, for all a ER (c) Prove that for all a € R and all x, y € R2, (R.x). (Ray) = x y (d) Suppose A is a 2 x 2 matrix such that for all x, y ER?, (Ax). (Ay) = xy 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. - [ U11 cos a sin a (i) Show that there are real numbers uu and a such that Hint: erpress expression for us in terms of a and e. (ii) Let a ER. Use the invertibility of R, to prove that there are unique 112, tly2 R such that » [a ) as a scalar multiple of a unit vector, and hence find an [cos a S2 +12 sino COS X sina (iii) Use parts (i) and (ii) to show that B can be expressed in the form B = R.U for some a E R and some upper-triangular matrix U. (iv) Suppose that B = R U = RV, where a, 8 € R and U and V are upper- triangular. Prove that if B is invertible, then U = UV. 2