1. In this question, you will be using the following trigonometric identities: cos? a + sin? a (1) (2) (3) 1 cos(a + B) cos a cos B – sin a sin B sin(a + B) 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 : αΕR sin a is exactly the set of unit vectors in R2. Now for any real number a, define cos a - sina 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 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?, (Rax) · (Ray) =x•y (d) Suppose A is a 2 x 2 matrix such that for all x, y € 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 x 2 matrix. cos a = u11 sin a (i) Show that there are real numbers u11 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 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 = R,U for some a e R and some upper-triangular matrix U. (iv) Suppose that B = RaU = R3V, where a, BER and U and V are upper- triangular. Prove that if B is invertible, then U = ±V.
1. In this question, you will be using the following trigonometric identities: cos? a + sin? a (1) (2) (3) 1 cos(a + B) cos a cos B – sin a sin B sin(a + B) 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 : αΕR sin a is exactly the set of unit vectors in R2. Now for any real number a, define cos a - sina 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 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?, (Rax) · (Ray) =x•y (d) Suppose A is a 2 x 2 matrix such that for all x, y € 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 x 2 matrix. cos a = u11 sin a (i) Show that there are real numbers u11 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 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 = R,U for some a e R and some upper-triangular matrix U. (iv) Suppose that B = RaU = R3V, where a, BER 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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Part (d) and (e) are the most important, please solve the whole question if possible ?
Transcribed Image Text:1. In this question, you will be using the following trigonometric identities:
cos?
(1)
(2)
a + sin² a
1
cos(a + B)
cos a cos B – sin a sin B
sin(a + B)
= sin a cos B+ cos a sin B
(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
:a E R
is exactly the set of unit vectors in R?.
Now for any real number a, define
cos a - sina
sin a
Ra =
Cos a
(a) Prove that for all a, B ER,
R,R8 = Ra+ß
(b) Using part (a), or otherwise, prove that Ra is invertible and that R = R-a, for
all α ε 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 × 2 matrix such that for all x, y € 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 =
|c
be any 2 x 2 matrix.
cos a
(i) Show that there are real numbers u11 and a such that
= 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 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 = RU
for some a E R and some upper-triangular matrix U.
(iv) Suppose that B = RQU = R;V, where a, B ER and U and V are upper-
triangular. Prove that if B is invertible, then U = ±V.
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