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 3 –- sin a sin 3 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 { CO a : αER sin a is exactly the set of unit vectors in R?. Now for any real number a, define CO a – sin a Ra sin a COS a (a) Prove that for all a, ß E R, R.R3 : Ra+ß (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 ER and all x,y € 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).
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 3 –- sin a sin 3 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 { CO a : αER sin a is exactly the set of unit vectors in R?. Now for any real number a, define CO a – sin a Ra sin a COS a (a) Prove that for all a, ß E R, R.R3 : Ra+ß (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 ER and all x,y € 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).
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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only need parts (c) and (d)
Transcribed Image Text:1. In this question, you will be using the following trigonometric identities:
cos? a + sin a
(1)
(2)
1
cos(a + B)
sin(a + B)
cos a cos B – sin a sin 3
sin a cos 3 + 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
{ a eR}
CO A
sin a
is exactly the set of unit vectors in R?.
Now for any real number ,
define
CO A
– sin a
Ra
sin a
COS a
(a) Prove that for all a, B E R,
R.R3 = Ra+8
(b) Using part (a), or otherwise, prove that Ra is invertible and that R = R-a, for
all a E R.
(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 R²,
(Ax) · (Ay) = x•y
Must it be true that A
Ra, for some a E R? Either prove this, or give a
||
counterexample (including justification).
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