Answer Happy

please answer question (iv), you can ignore the rest of the
question. Thank you.
a = 1. In this question, you will be using the following trigonometric identities: cos’ a + sin’a = 1 (1) cos(a +B) = cos a cosB - sin a sin 8 (2) sin(a+B) sin a cosB + cos a 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 sin is exactly the set of unit vectors in R2. Now for any real number a, define { [sua] 2 ER R = cosa - sina sina cosa se] = (a) Prove that for all a, B ER, R R3 = Ra+B (b) Using part (a), or otherwise, prove that Rg is invertible and that R. all a ER (c) Prove that for all a € R and all x, y € R2, = R-a, for (Rox). (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). [a bi (e) Let B= be any 2 x 2 matrix. cd (i) Show that there are real numbers u11 and a such that cosa U11 sina) (ii) Let a E R. Use the invertibility of R, to prove that there are unique U12, Un € R such that [a] (2 ) = = U12 cos a sin a [- sin al + 122 cosa (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 = R$V, where a, B E R and U and V are upper- triangular. Prove that if B is invertible, then U = IV. =