1 In This Question You Will Be Using The Following Trigonometric Identities Cos A Sin A 1 1 Cos A 8 Cos A 1 (33.68 KiB) Viewed 26 times
1 In This Question You Will Be Using The Following Trigonometric Identities Cos A Sin A 1 1 Cos A 8 Cos A 2 (26.46 KiB) Viewed 26 times
1. In this question, you will be using the following trigonometric identities: cos? a + sin’a = 1 (1) cos(a + 8) = cos a cos 8-sin a sin 3 (2) sin(a + 8) sin a cos 3 + cos a sin 8 (3) where a, B e R. You do not need to prove these identities. You may also use without proof the fact that the set = {[sa] aer} is exactly the set of unit vectors in R. Now for any real number a, define cosa-sina R. sin a cosa (a) Prove that for all 0,8 € R, R.Rg = Ra+8 (b) Using part (a), or otherwise, prove that R, is invertible and that R' = R., for all a ER (c) Prove that for all a € R and all x,y ERP, (Rex). (R.y) = xy (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 = [ [a b] be any 2 x 2 matrix. (cosa = 11 • [aas a scalar multiple of a unit vector, and hence find an [cosa (i) Show that there are real numbers up and a such that sina) Hint: express expression for ui in terms of a and c. (ii) Let a € R. Use the invertibility of Ra to prove that there are unique 2012, U22 € R such that sinal sina cosa (iii) Use parts (1) and (ii) to show that B can be expressed in the form BR.U for some a ER and some upper-triangular matrix U. (iv) Suppose that B = R.U = RsV, where a, 8 € R and U and V are upper- triangular. Prove that if B is invertible, then U = UV. 1112 + ug2
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