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1. In this question, you will be using the following trigonometric identities: cos? cos(a + B) sin(a + B) a + sin? a = 1 (1) (2) cos a cos B – sin a sin ß sin a cos B + cos a sin B 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 : α R sin a is exactly the set of unit vectors in R². Now for any real number a, define cos a – sin a sin a R. = COS Q (a) Prove that for all a, ß E R, RaR3 = Ra+8 (b) Using part (a), or otherwise, prove that Ra is invertible and that R1 all a E R. = R_a, for (c) Prove that for all a E R and all x, y e R², (Rax) · (Ray) =x • y

1. In this question, you will be using the following trigonometric identities: cos? cos(a + B) sin(a + B) a + sin? a = 1 (1) (2) cos a cos B – sin a sin ß sin a cos B + cos a sin B 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 : α R sin a is exactly the set of unit vectors in R². Now for any real number a, define cos a – sin a sin a R. = COS Q (a) Prove that for all a, ß E R, RaR3 = Ra+8 (b) Using part (a), or otherwise, prove that Ra is invertible and that R1 all a E R. = R_a, for (c) Prove that for all a E R and all x, y e R², (Rax) · (Ray) =x • y

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1. In this question, you will be using the following trigonometric identities: cos? a + sin? a (1) (2) (3) 1 cos(a + B) sin(a + B) cos a cos B – sin a sin B sin a cos B + cos a sin 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 : αER sin a is exactly the set of unit vectors in R?. Now for any real number , define [cos a - sina Ra = sin a COS a (a) Prove that for all a, B E R, RaR3 = Ra+B (b) Using part (a), or otherwise, prove that Ra is invertible and that R1 all a E R. = R_a, for (c) Prove that for all a E R and all x, y E R², (Rax) · (Ray) =x · y (d) Suppose A is a 2 x 2 matrix such that for all x, y e R?, (Ах) (Ау) — х у 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. (i) Show that there are real numbers u11 and a such that [cos a = U11 sin a Hint: express as a scalar multiple of a unit vector, and hence find an expression for u11 in terms of a and c. (ii) Let a e R. Use the invertibility of R. to prove that there are unique U12, U22 E R such that [cos a = U12 sin a sin a + U22 Cos a (iii) Use parts (i) and (ii) to show that B can be expressed in the form B = RU for some a E R and some upper-triangular matrix U. (iv) Suppose that B = RaU = RgV, where , B E R and U and V are upper- triangular. Prove that if B is invertible, then U = ±V. 2
Transcribed Image Text:1. In this question, you will be using the following trigonometric identities: cos? a + sin? a (1) (2) (3) 1 cos(a + B) sin(a + B) cos a cos B – sin a sin B sin a cos B + cos a sin 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 : αER sin a is exactly the set of unit vectors in R?. Now for any real number , define [cos a - sina Ra = sin a COS a (a) Prove that for all a, B E R, RaR3 = Ra+B (b) Using part (a), or otherwise, prove that Ra is invertible and that R1 all a E R. = R_a, for (c) Prove that for all a E R and all x, y E R², (Rax) · (Ray) =x · y (d) Suppose A is a 2 x 2 matrix such that for all x, y e R?, (Ах) (Ау) — х у 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. (i) Show that there are real numbers u11 and a such that [cos a = U11 sin a Hint: express as a scalar multiple of a unit vector, and hence find an expression for u11 in terms of a and c. (ii) Let a e R. Use the invertibility of R. to prove that there are unique U12, U22 E R such that [cos a = U12 sin a sin a + U22 Cos a (iii) Use parts (i) and (ii) to show that B can be expressed in the form B = RU for some a E R and some upper-triangular matrix U. (iv) Suppose that B = RaU = RgV, where , B E R and U and V are upper- triangular. Prove that if B is invertible, then U = ±V. 2
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