Suppose H is a n-dimensional subspace. Let A = Let T H H be a linear transformation. Then [T] = [[T(1)]® [T(ả₂)]® B B True False {ả₁, ···, ản }, B = {₁,...,n} be two bases of H. [T(an)]].
Suppose H is a n-dimensional subspace. Let A = Let T H H be a linear transformation. Then [T] = [[T(1)]® [T(ả₂)]® B B True False {ả₁, ···, ản }, B = {₁,...,n} be two bases of H. [T(an)]].
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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![### Linear Transformation and Basis Change
#### Problem Statement:
Suppose \( H \) is an \( n \)-dimensional subspace.
Let \( \mathcal{A} = \{ \vec{a}_1, \cdots, \vec{a}_n \} \) and \( \mathcal{B} = \{ \vec{b}_1, \cdots, \vec{b}_n \} \) be two bases of \( H \).
Let \( T: H \to H \) be a linear transformation.
Then
\[
[T]^{\mathcal{A}}_{\mathcal{B}} = [ [T(\vec{a}_1)]_{\mathcal{B}} \; [T(\vec{a}_2)]_{\mathcal{B}} \; \cdots \; [T(\vec{a}_n)]_{\mathcal{B}} ]
\]
#### Verification:
- **True**
- **False**
For students studying linear algebra, understanding the basis and transformations is crucial. The notation and the logic here represent how a linear transformation \( T \) can be described in terms of its effect on the basis vectors of \( H \), and how it translates between two different bases \( \mathcal{A} \) and \( \mathcal{B} \). This helps in constructing the matrix representation of \( T \) in the new basis \( \mathcal{B} \).
You can verify this statement by checking how the transformation matrix \( [T]^{\mathcal{A}}_{\mathcal{B}} \) is constructed using the images of the basis vectors of \( \mathcal{A} \) expressed in terms of the basis \( \mathcal{B} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F238584dd-2c47-447e-aaba-b9fce1ec1b5c%2F785bb71b-72eb-4617-b0da-5df7c958a6f3%2Fvuhsd3p_processed.png&w=3840&q=75)
Transcribed Image Text:### Linear Transformation and Basis Change
#### Problem Statement:
Suppose \( H \) is an \( n \)-dimensional subspace.
Let \( \mathcal{A} = \{ \vec{a}_1, \cdots, \vec{a}_n \} \) and \( \mathcal{B} = \{ \vec{b}_1, \cdots, \vec{b}_n \} \) be two bases of \( H \).
Let \( T: H \to H \) be a linear transformation.
Then
\[
[T]^{\mathcal{A}}_{\mathcal{B}} = [ [T(\vec{a}_1)]_{\mathcal{B}} \; [T(\vec{a}_2)]_{\mathcal{B}} \; \cdots \; [T(\vec{a}_n)]_{\mathcal{B}} ]
\]
#### Verification:
- **True**
- **False**
For students studying linear algebra, understanding the basis and transformations is crucial. The notation and the logic here represent how a linear transformation \( T \) can be described in terms of its effect on the basis vectors of \( H \), and how it translates between two different bases \( \mathcal{A} \) and \( \mathcal{B} \). This helps in constructing the matrix representation of \( T \) in the new basis \( \mathcal{B} \).
You can verify this statement by checking how the transformation matrix \( [T]^{\mathcal{A}}_{\mathcal{B}} \) is constructed using the images of the basis vectors of \( \mathcal{A} \) expressed in terms of the basis \( \mathcal{B} \).
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