Define T:R2 R2 by T(x) = Ax. Find a basis B for R2 with the property that [T] is diagonal. A = 51 15 A basis for R2 with the property that [T]g is diagonal is (Use a comma to separate answers as needed.)
Define T:R2 R2 by T(x) = Ax. Find a basis B for R2 with the property that [T] is diagonal. A = 51 15 A basis for R2 with the property that [T]g is diagonal is (Use a comma to separate answers as needed.)
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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Question
![**Problem Statement:**
Define \( T:\mathbb{R}^2 \to \mathbb{R}^2 \) by \( T(x) = Ax \). Find a basis \( B \) for \( \mathbb{R}^2 \) with the property that \([T]_B\) is diagonal.
**Matrix:**
\[ A = \begin{bmatrix} 5 & 1 \\ 1 & 5 \end{bmatrix} \]
**Solution Prompt:**
A basis for \( \mathbb{R}^2 \) with the property that \([T]_B\) is diagonal is \(\{ \}\).
(Use a comma to separate answers as needed.)
---
**Explanation:**
The task is to determine a suitable basis for \( \mathbb{R}^2 \) such that the matrix representation of the linear transformation \( T \) with respect to that basis is a diagonal matrix.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F46029127-96a5-4dc2-997a-3c090d1aab2d%2Fa905afc7-b685-4f50-af69-fa257a631e84%2Fjl09m5a_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Define \( T:\mathbb{R}^2 \to \mathbb{R}^2 \) by \( T(x) = Ax \). Find a basis \( B \) for \( \mathbb{R}^2 \) with the property that \([T]_B\) is diagonal.
**Matrix:**
\[ A = \begin{bmatrix} 5 & 1 \\ 1 & 5 \end{bmatrix} \]
**Solution Prompt:**
A basis for \( \mathbb{R}^2 \) with the property that \([T]_B\) is diagonal is \(\{ \}\).
(Use a comma to separate answers as needed.)
---
**Explanation:**
The task is to determine a suitable basis for \( \mathbb{R}^2 \) such that the matrix representation of the linear transformation \( T \) with respect to that basis is a diagonal matrix.
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