- 1 In This Question You Will Be Using The Following Trigonometric Identities Cosa Sina 1 Cos A 8 Cos A Cos 1 (59.69 KiB) Viewed 30 times
1. In this question, you will be using the following trigonometric identities: cosa + sina = 1 cos(a + 8) = cos a cos - sin a sin B sin(a+B) sin a cos 3 + cos a sin (1) (2) (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 a sina is exactly the set of unit vectors in R2, Now for any real number a, define aer} R. = cosa - sina sina COS a (e) Let B = be any 2 x 2 matrix. (i) Show that there are real numbers up and a such that Til = U11 [cos a sin a ( Hint: erpress as a scalar multiple of a unit vector, and hence find an erpression for u11 in terms of a and c. (ii) Let a € R. Use the invertibility of R, to prove that there are unique U12, U22 ER such that cos a + u22 sin a COS (iii) 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 = RV, where a, B E R and U and V are upper- triangular. Prove that if B is invertible, then U = UV.