[sin a 1. In this question, you will be using the following trigonometric identities: cos? a + sin? a 1 (1) cos(a + 3) cos a cos 3 - sin a sin 3 (2) sin(a + B) = sin a cos 3 + cos a sin 3 (3) where a, 3 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 :a € R is eractly the set of unit vectors in R2. Now for any real number a, define cos a - sin a Ra = sin a COS 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, Ra, for all a E R. (c) Prove that for all a ER and all x, y e R?, (R,x) · (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). (e) Let B = be any 2 x 2 matrix. [cos a = u11 sin a (i) Show that there are real numbers uu and a such that 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 Roa to prove that there are unique U12, U22 E R such that [cos a = u12 - sin a U22 COS a (iii) Use parts (i) and (ii) to show that B can be expressed in the form B = R,U for some a E R and some upper-triangular matrix U. (iv) Suppose that B RaU = R3V, where a, 3 ER and U and V are upper- triangular. Prove that if B is invertible, then U = +V.
[sin a 1. In this question, you will be using the following trigonometric identities: cos? a + sin? a 1 (1) cos(a + 3) cos a cos 3 - sin a sin 3 (2) sin(a + B) = sin a cos 3 + cos a sin 3 (3) where a, 3 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 :a € R is eractly the set of unit vectors in R2. Now for any real number a, define cos a - sin a Ra = sin a COS 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, Ra, for all a E R. (c) Prove that for all a ER and all x, y e R?, (R,x) · (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). (e) Let B = be any 2 x 2 matrix. [cos a = u11 sin a (i) Show that there are real numbers uu and a such that 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 Roa to prove that there are unique U12, U22 E R such that [cos a = u12 - sin a U22 COS a (iii) Use parts (i) and (ii) to show that B can be expressed in the form B = R,U for some a E R and some upper-triangular matrix U. (iv) Suppose that B RaU = R3V, where a, 3 ER and U and V are upper- triangular. Prove that if B is invertible, then U = +V.
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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Question
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Need part (e) only
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)
cos a cos 3 – sin a sin B
%3D
sin(a + B) =
sin a cos 3 + cos a sin 3
(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
: αER
is exactly the set of unit vectors in R2.
Now for any real number a, define
cos a - sin a
sin a
R. =
COs 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 ER and all x, y E R?,
(R.x) · (Ray) =x y
(d) Suppose A is a 2 x 2 matrix such that for all x, y € R?,
(Ах) (Ау) 3D 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.
[cos a
= U11
sin a
(i) Show that there are real numbers u1u and a such that
Нint: егpress
as a scalar multiple of a unit vector, and hence find an
expression for u1 in terms of a and c.
(ii) Let a e R. Use the invertibility of Ra 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 = R,U
for some a E R and some upper-triangular matrix U.
(iv) Suppose that B = RU = R3V, where a, 3 ER and U and V are upper-
triangular. Prove that if B is invertible, then U = +V.
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