10. A 5-3 -7
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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just number 10
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### Linear Algebra Exercises: Diagonalization
#### Transformations and Basis for \( \mathbb{R}^2 \)
In Exercises 9–12, define the linear transformation \( T : \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) by \( T(\mathbf{x}) = A\mathbf{x} \). The goal is to find a basis \( \mathcal{B} \) for \( \mathbb{R}^2 \) such that the matrix representation \([T]_{\mathcal{B}}\) is diagonal.
#### Exercise 9:
Given the matrix:
\[ A = \begin{pmatrix}
0 & 1 \\
-3 & 4
\end{pmatrix} \]
Determine the basis \( \mathcal{B} \) such that the matrix representation of \( T \) in terms of this basis is diagonal.
#### Exercise 10:
Given the matrix:
\[ A = \begin{pmatrix}
5 & -3 \\
-7 & 1
\end{pmatrix} \]
Identify the basis \( \mathcal{B} \) so that the matrix representation of \( T \) in this basis is diagonal.
---
The tasks involve finding eigenvalues and eigenvectors of the given matrices to construct the required basis \( \mathcal{B} \) that diagonalizes the transformation matrix \( A \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F324f16c4-c4e5-4b06-b074-f261b39e021a%2Fefa7d9e5-2e91-401a-a543-bfd9517615e3%2Frswdrnv_processed.png&w=3840&q=75)
Transcribed Image Text:---
### Linear Algebra Exercises: Diagonalization
#### Transformations and Basis for \( \mathbb{R}^2 \)
In Exercises 9–12, define the linear transformation \( T : \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) by \( T(\mathbf{x}) = A\mathbf{x} \). The goal is to find a basis \( \mathcal{B} \) for \( \mathbb{R}^2 \) such that the matrix representation \([T]_{\mathcal{B}}\) is diagonal.
#### Exercise 9:
Given the matrix:
\[ A = \begin{pmatrix}
0 & 1 \\
-3 & 4
\end{pmatrix} \]
Determine the basis \( \mathcal{B} \) such that the matrix representation of \( T \) in terms of this basis is diagonal.
#### Exercise 10:
Given the matrix:
\[ A = \begin{pmatrix}
5 & -3 \\
-7 & 1
\end{pmatrix} \]
Identify the basis \( \mathcal{B} \) so that the matrix representation of \( T \) in this basis is diagonal.
---
The tasks involve finding eigenvalues and eigenvectors of the given matrices to construct the required basis \( \mathcal{B} \) that diagonalizes the transformation matrix \( A \).
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