Define T: R² →R² by T(x) = Ax, where A is the matrix defined below. Find a basis B for R2 with the property that [T] is diagonal. -1 - 3 418 A = 6 -{₁ B= -1 [T]B =
Define T: R² →R² by T(x) = Ax, where A is the matrix defined below. Find a basis B for R2 with the property that [T] is diagonal. -1 - 3 418 A = 6 -{₁ B= -1 [T]B =
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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![# Matrix Transformation and Basis Selection
## Problem Statement
Define the transformation \( T: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) by \( T(\mathbf{x}) = A\mathbf{x} \), where \( A \) is the matrix defined below. Our task is to find a basis \( B \) for \( \mathbb{R}^2 \) such that the matrix representation \([T]_B\) is diagonal.
## Given Matrix
The matrix \( A \) is:
\[
A = \begin{bmatrix} -1 & -3 \\ 4 & 6 \end{bmatrix}
\]
## Objective
- Find a basis \( B \) for \( \mathbb{R}^2 \) such that \([T]_B\) is a diagonal matrix.
## Basis Representation
The basis \( B \) is provided as follows:
\[
B = \left\{ \begin{bmatrix} 1 \\ -1 \end{bmatrix}, \begin{bmatrix} \, \, \, \, \, \, \end{bmatrix} \right\}, \quad [T]_B = \begin{bmatrix} \, \, \, \, \, \, \end{bmatrix}
\]
### Note
- The second vector of the basis \( B \) is not fully specified, as well as the diagonal elements of \([T]_B\).
In a comprehensive solution, we need to:
- Determine the eigenvectors of matrix \( A \) to complete the basis \( B \).
- Calculate the eigenvalues of \( A \) to fill in the diagonal of \([T]_B\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F46029127-96a5-4dc2-997a-3c090d1aab2d%2Fc4064883-672f-4935-8c95-a34b5b8d4abe%2Fgxsv0ij_processed.png&w=3840&q=75)
Transcribed Image Text:# Matrix Transformation and Basis Selection
## Problem Statement
Define the transformation \( T: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) by \( T(\mathbf{x}) = A\mathbf{x} \), where \( A \) is the matrix defined below. Our task is to find a basis \( B \) for \( \mathbb{R}^2 \) such that the matrix representation \([T]_B\) is diagonal.
## Given Matrix
The matrix \( A \) is:
\[
A = \begin{bmatrix} -1 & -3 \\ 4 & 6 \end{bmatrix}
\]
## Objective
- Find a basis \( B \) for \( \mathbb{R}^2 \) such that \([T]_B\) is a diagonal matrix.
## Basis Representation
The basis \( B \) is provided as follows:
\[
B = \left\{ \begin{bmatrix} 1 \\ -1 \end{bmatrix}, \begin{bmatrix} \, \, \, \, \, \, \end{bmatrix} \right\}, \quad [T]_B = \begin{bmatrix} \, \, \, \, \, \, \end{bmatrix}
\]
### Note
- The second vector of the basis \( B \) is not fully specified, as well as the diagonal elements of \([T]_B\).
In a comprehensive solution, we need to:
- Determine the eigenvectors of matrix \( A \) to complete the basis \( B \).
- Calculate the eigenvalues of \( A \) to fill in the diagonal of \([T]_B\).
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