- 1 In This Question You Will Be Using The Following Trigonometric Identities Cosa Sind A 1 Cos A B Cos A Cos Sin 1 (93.04 KiB) Viewed 47 times
- 1 In This Question You Will Be Using The Following Trigonometric Identities Cosa Sind A 1 Cos A B Cos A Cos Sin 2 (59.69 KiB) Viewed 47 times
cooperate.
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= 1. In this question, you will be using the following trigonometric identities: cosa + sind a 1 cos(a +B) cos a cos sin a sin sin(a + 3) = sin a cos + cos a sin 8 (1) (2) (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 cos a :QER ER} sin a is exactly the set of unit vectors in R2. Now for any real number a, define cosa - sin a Ra= sin a COS a (a) Prove that for all a, B e R, RR = Ra+B (b) Using part (a), or otherwise, prove that R, is invertible and that Rol = R-a, for all a ER (c) Prove that for all a E R and all x,y e R2, (R,x). (Ray) = x.y (d) Suppose A is a 2 x 2 matrix such that for all x, y € R2, (Ax). (Ay) = xy Must it be true that A = Ra, for some a E R? Either prove this, or give a counterexample (including justification). (e) Let B= be any 2 x 2 matrix. [a o [ ] cos a (i) Show that there are real numbers u11 and a such that [a] = U11 sina Hint: express as a scalar multiple of a unit vector, and hence find an expression for uni in terms of a and c. (ii) Let a € R. Use the invertibility of Ra to prove that there are unique U12, U22 E R such that cos a sin a = U12 + 122 sina COS (iii) Use parts (i) and (ii) to show that B can be expressed in the form B = R.U for some a ER and some upper-triangular matrix U. (iv) Suppose that B = R U = R3V, where a, B E R and U and V are upper- triangular. Prove that if B is invertible, then U = V. 2
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.