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1. In this question, you will be using the following trigonometric identities: wa+sina = 1 (1) cos(+3) = cos Co-sino sin 3 (2) sina + 8) = sin cos + cosasin 3 (3) where a. 3 ER. 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 mumber o, define COS-a R sino (*) Prove that for all 0.3 ER RRER (b) Using part (a), or otherwise, prone that is invertible and that R' = R.,, for alla ER (c) Prove that for all a E Randall x y ER. (Rx)-(R.y) = xy (d) Suppose A is a 2 x 2 matrix stach that for all xyR? (Ax) - (Ay) = x y Must it be true that A = R for some a € R? Either prove this, or give a counterexample (including justification). (e) Let B =[ : be any 2x 2 matrix cosa (i) Show that there are real numbers and a such that Hint: capress as a scator multiple of a unit rcctor, and hence find an expression for in terms of aande (1) Let E R. Use the invertibility of R. to prove that there are unique ER such that Teos sino sina (11) Use parts (i) and (m) to show that can be expressed in the form BRU for some a € R and some upper-triangular matrix U. (iv) Suppose that B-RU - RV, where 3 € R and U and V are upper- triangular. Prove that if B is invertible, then U = V. - 12 (1- COS 2

1. In this question, you will be using the following trigonometric identities: cosa + sina = 1 (1) cos(+3) = cos a cos 3 - sina sin 3 (2) sin(a+ 8) = sin a cos 3 + cosa sin 3 (3) where 0.8 ER. You do not need to prove these identities. You may also use without proof the fact that the set COS Ha sin - sina COS is exactly the set of unit vectors in R? Now for any real number a, define [eos a R = sina (a) Prove that for all o, 3 ER RR = R. + (b) Using part (a), or otherwise, prove that R. is invertible and that R' = R., for all o ER (c) Prove that for all a € R and all x y € R, (Rex) (Ray) = x y (d) 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 = Ra, for some a € R? Either prove this, or give a counterexample (including justification) (e) Let B = Toasa (1) Show that there are real numbers and a such that = 1111 sina) Hint: erpress as a scalar multiple of a unit vector, and hence find an expression for us in terms of a and c. (ii) Let a E R. Use the invertibility of R. to prove that there are unique 112, 42 R such that [oos a - sina La be any 2 x 2 matrix • Caº [u 1412 COS (ii) Use parts (i) and (ii) to show that B can be expressed in the form B=RU for some a € R and some upper-triangular matrix U (iv) Suppose that B = R.U = R.V, where a, 3 € R and U and V are upper- triangular. Prove that if B is invertible, then U = UV. =