Let T be a linear operator on a finite-dimensional vector space V, and let B and B' be ordered bases for V. Suppose that Q is the change of coordinate matrix that changes B'- coordinates into B-coordinates. Then [T] B¹ = Q=¹[T] BQ. Proof. Let I be the identity transformation on V. Then T = IT = TI; hence, by Theorem 2.11 (p. 89), Q[T], [1] [T] = [IT]} = [TI] = [T] [I]} = [T],Q. = Therefore [T] =Q-¹[T] BQ. B¹
Let T be a linear operator on a finite-dimensional vector space V, and let B and B' be ordered bases for V. Suppose that Q is the change of coordinate matrix that changes B'- coordinates into B-coordinates. Then [T] B¹ = Q=¹[T] BQ. Proof. Let I be the identity transformation on V. Then T = IT = TI; hence, by Theorem 2.11 (p. 89), Q[T], [1] [T] = [IT]} = [TI] = [T] [I]} = [T],Q. = Therefore [T] =Q-¹[T] BQ. B¹
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![Theorem 2.23.
Let T be a linear operator on a finite-dimensional vector space V, and let B and B' be
ordered bases for V. Suppose that Q is the change of coordinate matrix that changes B'-
coordinates into B-coordinates. Then
[T] B = Q-¹[T] BQ.
Proof.
Let I be the identity transformation on V. Then T = IT = TI; hence, by Theorem 2.11
(p. 89),
Q[T], = [1] [T] = [IT]} = [TI] = [T]/I]} = [T],Q.
Therefore [T] =Q-¹[T] µQ.
B¹](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F45e9ceaf-0062-410b-addf-404f0a3b8197%2Fc9b1698a-fbf6-454f-8200-b165d0563065%2F95rhwhj_processed.png&w=3840&q=75)
Transcribed Image Text:Theorem 2.23.
Let T be a linear operator on a finite-dimensional vector space V, and let B and B' be
ordered bases for V. Suppose that Q is the change of coordinate matrix that changes B'-
coordinates into B-coordinates. Then
[T] B = Q-¹[T] BQ.
Proof.
Let I be the identity transformation on V. Then T = IT = TI; hence, by Theorem 2.11
(p. 89),
Q[T], = [1] [T] = [IT]} = [TI] = [T]/I]} = [T],Q.
Therefore [T] =Q-¹[T] µQ.
B¹

Transcribed Image Text:5. Let T be the linear operator on P₁ (R) defined by T(p(x)) = p'(x), the derivative of
p(x). Let ß = {1, x} and ß′ = {1 + x, 1 − x}. Use Theorem 2.23 and the fact
that
1
1
2 2
( D ' ( ))
(1
1 1
2 2
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