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1. In this question, you will be using the following trigonometric identities: cosa + sina 1 (1) cos(a + 8) cos a cos 8-sin osin 8 (2) sin(a + 8) = sin cos B+cos a sin 8 (3) where a, 8 e R. You do not need to prove these identities. You may also use without proof the fact that the set : sina] is exactly the set of unit vectors in R? Now for any real number a, define cosa - sina R- sina cosa (a) Prove that for all 0,8 € R, R, R=R+# (b) Using part (a), or otherwise, prove that is invertible and that R' = R., for all a ER (c) Prove that for all o e R and all x,y € R? (R.). (R.y)-xy (a) Suppose A is a 2 x 2 matrix such that for all X,Y ER, (Ax). (Ay) X y Must it be true that A = R., for some a € R? Either prove this, or give a counterexample (including justification). 67 (e) Let B- be any 2 x 2 matrix. cos a () Show that there are real numbers 11 and a such that sin Hint: erpress as a scalar multiple of a unit vector, and hence find an expression for url in terms of a and c. (ii) Let o E R. Use the invertibility of R. to prove that there are unique U12, U22 e R such that cosa sina = 12 sina 005 (in) Une parts (1) and (ii) to show that B can be expressed in the form BERU for some a e R and some upper-triangular matrix U. (iv) Suppose that B = R.U = RAV, where a, 8 e R and U and V are upper- triangular. Prove that if B is invertible, then U = UV. с 12 2