1. In this question, you will be using the following trigonometric identities: cos" a + sin" a = 1 (1) cos(a + 8) = cos a cos 3 – sin a sin 3 sin(a + B) = sin a cos 3 + cos a sin 3 where a, 3 € R. You do not need to prove these identities. You may also use without proof the fact that the set [cos a :a €R is exactly the set of unit vectors in R°. Now for any real number a, define cos a - sin a R. = sin a Cos a (a) Prove that for all a, 3 €R, Ra Ra = Ra+a (b) Using part (a), or otherwise, prove that R, is invertible and that R,' = Ra, for all a € R. (c) Prove that for all a € R and all x, y e R², (R,x) · (Ray) = x y (d) Suppose A is a 2 x 2 matrix such that for all x, y e IRª, (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 = b] be any 2 x 2 matrix. [cos a (i) Show that there are real numbers u1 and a such that sin a Hint: erpress as a scalar multiple of a unit vector, and hence find an erpression for ui in terms of a and c. (ii) Let a e R. Use the invertibility of R, to prove that there are unique U12, 22 € R such that [cos a] 12 sin a [- sin a' + ua 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 = R,U = R,V, where a, 3 € R 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 (1) cos(a + 8) = cos a cos 3 – sin a sin 3 sin(a + B) = sin a cos 3 + cos a sin 3 where a, 3 € R. You do not need to prove these identities. You may also use without proof the fact that the set [cos a :a €R is exactly the set of unit vectors in R°. Now for any real number a, define cos a - sin a R. = sin a Cos a (a) Prove that for all a, 3 €R, Ra Ra = Ra+a (b) Using part (a), or otherwise, prove that R, is invertible and that R,' = Ra, for all a € R. (c) Prove that for all a € R and all x, y e R², (R,x) · (Ray) = x y (d) Suppose A is a 2 x 2 matrix such that for all x, y e IRª, (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 = b] be any 2 x 2 matrix. [cos a (i) Show that there are real numbers u1 and a such that sin a Hint: erpress as a scalar multiple of a unit vector, and hence find an erpression for ui in terms of a and c. (ii) Let a e R. Use the invertibility of R, to prove that there are unique U12, 22 € R such that [cos a] 12 sin a [- sin a' + ua 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 = R,U = R,V, where a, 3 € R and U and V are upper- triangular. Prove that if B is invertible, then U = ±V.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.2: Determinants
Problem 10AEXP
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can i get help with 1)e)iv please
![1. In this question, you will be using the following trigonometric identities:
cos* a + sin a = 1
(1)
(2)
(3)
cos(a + B) = cos a cos 3 – sin a sin 3
sin(a + 8) = sina cos 3+ cos a sin 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
:a €R
sin a
is eractly the set of unit vectors in R².
Now for any real number a, define
cos a
- sin o
R. =
sin a
Cos a
(a) Prove that for all a, 3 €R,
Ra Ra = Ra+s
(b) Using part (a), or otherwise, prove that R, is invertible and that R,' = Ra, for
all a €R.
(e) Prove that for all a €R 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 € R2,
(Ax) · (Ay) = x y
Must it be true that A = R, for some a € R? Either prove this, or give a
counterexample (including justification).
- : -
[a b]
(e) Let B =
E a be any 2 x 2 matrix.
Le d]
COs a
(i) Show that there are real numbers u1 and a such that
= U11
sin a
Hint: erpress
as a scalar multiple of a unit vector, and hence find an
erpression for un in terms of a and c.
(ii) Let a € R. Use the invertibility of R. to prove that there are unique
U12, Uz2 € R such that
cos a
sin a
= U12
sin a
Cos a
(iii) Use parts (i) and (ii) to show that B can be expressed in the form
B = R„U
for some a € R and some upper-triangular matrix U.
(iv) Suppose that B = R,U = RV, where a, 3 € R and U and V are upper-
triangular. Prove that if B is invertible, then U = ±V.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4a444b5d-e347-43ad-b303-ece389659713%2F595cf9ae-56ed-4a34-b82b-5ab1009ef043%2Fp9s1onc_processed.png&w=3840&q=75)
Transcribed Image Text:1. In this question, you will be using the following trigonometric identities:
cos* a + sin a = 1
(1)
(2)
(3)
cos(a + B) = cos a cos 3 – sin a sin 3
sin(a + 8) = sina cos 3+ cos a sin 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
:a €R
sin a
is eractly the set of unit vectors in R².
Now for any real number a, define
cos a
- sin o
R. =
sin a
Cos a
(a) Prove that for all a, 3 €R,
Ra Ra = Ra+s
(b) Using part (a), or otherwise, prove that R, is invertible and that R,' = Ra, for
all a €R.
(e) Prove that for all a €R 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 € R2,
(Ax) · (Ay) = x y
Must it be true that A = R, for some a € R? Either prove this, or give a
counterexample (including justification).
- : -
[a b]
(e) Let B =
E a be any 2 x 2 matrix.
Le d]
COs a
(i) Show that there are real numbers u1 and a such that
= U11
sin a
Hint: erpress
as a scalar multiple of a unit vector, and hence find an
erpression for un in terms of a and c.
(ii) Let a € R. Use the invertibility of R. to prove that there are unique
U12, Uz2 € R such that
cos a
sin a
= U12
sin a
Cos a
(iii) Use parts (i) and (ii) to show that B can be expressed in the form
B = R„U
for some a € R and some upper-triangular matrix U.
(iv) Suppose that B = R,U = RV, where a, 3 € R and U and V are upper-
triangular. Prove that if B is invertible, then U = ±V.
Transcribed Image Text:la
(e) Let B =
be any 2 x 2 matrix.
d
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
sin a
= U12
+ U22
sin a
CO a
(iii) Use parts (i) and (ii) to show that B can be expressed in the form
B = RaU
for some a E R and some upper-triangular matrix U.
(iv) Suppose that B = RaU = R3V, where a, ß E R and U and V are upper-
triangular. Prove that if B is invertible, then U = ±V.
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