(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 = Roa, for some a E R? Either prove this, or give a counterexample (including justification).
(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 = Roa, 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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Can you please help me solve part d for this question, please?
Transcribed Image Text:1. In this question, you will be using the following trigonometric identities:
cos? a + sin? a = 1
cos(@ + B)
sin(a + B)
(1)
(2)
(3)
cos a cos B – sin a sin 3
sin a cos 3 + 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
: α R
sin a
is exactly the set of unit vectors in R?.
Now for any real number a, define
cos a - sin a
cos a
Ra
sin a
(a) Prove that for all a, 3 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 € R?,
(Rax) · (Ray) = x • y
(d) Suppose A is a 2 × 2 matrix such that for all x, y € R²,
(Ах) (Ау) —D х у
Must it be true that A = Ro, for some a e R? Either prove this, or give a
counterexample (including justification).
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