1. In this question, you will be using the following trigonometric identities: cos? a + sin? a = 1 cos(a + B) sin(a + B) = cos a cos B – sin a sin B = sin a cos B + cos a sin B (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 sin a is exactly the set of unit vectors in R?. Now for any real number a, define cos a – sin cos a sin a Ra (a) Prove that for all a, B ER, RaR8 = Ra+B (b) Using part (a), or otherwise, prove that R. is invertible and that R 1 = R, for all a E R. %3D (c) Prove that for all a E R and all x, y E R², (Rax)· (Ray) = x y (d) Suppose A is a 2 × 2 matrix such that for all x, y E R?, (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). (e) Let B be any 2 × 2 matrix. %3D
1. In this question, you will be using the following trigonometric identities: cos? a + sin? a = 1 cos(a + B) sin(a + B) = cos a cos B – sin a sin B = sin a cos B + cos a sin B (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 sin a is exactly the set of unit vectors in R?. Now for any real number a, define cos a – sin cos a sin a Ra (a) Prove that for all a, B ER, RaR8 = Ra+B (b) Using part (a), or otherwise, prove that R. is invertible and that R 1 = R, for all a E R. %3D (c) Prove that for all a E R and all x, y E R², (Rax)· (Ray) = x y (d) Suppose A is a 2 × 2 matrix such that for all x, y E R?, (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). (e) Let B be any 2 × 2 matrix. %3D
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:
cos² a + sin² a = 1
(1)
(2)
cos(a + B)
sin(a + B)
cos a cos B – sin a sin B
sin a cos 3 + cOS a sin ß
(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
:a E
sin a
is exactly the set of unit vectors in R?.
Now for any real number a, define
cos a – sin a
sin a
R. =
COS a
(a) Prove that for all a, B ER,
RaR8 = Ra+B
(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 R?,
(Rax) · (Ray) =x•y
(d) Suppose A is a 2 x 2 matrix such that for all x, y E R?,
(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).
a
(e) Let B =
be any 2 x 2 matrix.
d
DI
DD
F9
F10
F11
F6
F7
F8
&
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