Find a basis for the eigenspace of B below. 20 8-[2₁9]-X-3.2 = 3, B = a. B={(1,0),(1,1)) b. B=((0,1),(1,1)} c. B={(1,0),(1,0)} d. B={(1, 1), (1,1)}
Find a basis for the eigenspace of B below. 20 8-[2₁9]-X-3.2 = 3, B = a. B={(1,0),(1,1)) b. B=((0,1),(1,1)} c. B={(1,0),(1,0)} d. B={(1, 1), (1,1)}
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter7: Distance And Approximation
Section7.1: Inner Product Spaces
Problem 10AEXP
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![### Problem Statement
**Objective:** Find a basis for the eigenspace of matrix \( B \).
**Matrix \( B \):**
\[ B = \begin{bmatrix} 2 & 0 \\ -1 & 3 \end{bmatrix} \]
**Eigenvalues \( \lambda \):**
\[ \lambda = 3, 2 \]
**Options for the basis:**
a. \( B = \{ (1,0), (1,1) \} \)
b. \( B = \{ (0,1), (1,1) \} \)
c. \( B = \{ (1,0), (1,0) \} \)
d. \( B = \{ (1, -1), (1,1) \} \)
### Explanation
To find the basis for the eigenspace of matrix \( B \), we solve the characteristic equation for \( B \) and find the respective eigenvectors associated with each eigenvalue \( \lambda \). The correct option will provide the valid basis for the eigenspace, where each basis vector is an eigenvector corresponding to a specific eigenvalue.
### Understanding the Options
- **Option a** presents the vectors \( (1,0) \) and \( (1,1) \).
- **Option b** presents the vectors \( (0,1) \) and \( (1,1) \).
- **Option c** presents the vectors \( (1,0) \) and \( (1,0) \), which appear not to be linearly independent.
- **Option d** presents the vectors \( (1, -1) \) and \( (1,1) \).
We need to verify each option to see which set of vectors forms a basis for the eigenspace of \( B \).
When evaluating candidate bases, linear independence and the correct span relative to the eigenspaces are crucial criteria.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5483afab-7850-40f6-959f-e698dea1419a%2F47aaccdf-5062-4c99-880c-c1eb887f72af%2Fr78gkx_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
**Objective:** Find a basis for the eigenspace of matrix \( B \).
**Matrix \( B \):**
\[ B = \begin{bmatrix} 2 & 0 \\ -1 & 3 \end{bmatrix} \]
**Eigenvalues \( \lambda \):**
\[ \lambda = 3, 2 \]
**Options for the basis:**
a. \( B = \{ (1,0), (1,1) \} \)
b. \( B = \{ (0,1), (1,1) \} \)
c. \( B = \{ (1,0), (1,0) \} \)
d. \( B = \{ (1, -1), (1,1) \} \)
### Explanation
To find the basis for the eigenspace of matrix \( B \), we solve the characteristic equation for \( B \) and find the respective eigenvectors associated with each eigenvalue \( \lambda \). The correct option will provide the valid basis for the eigenspace, where each basis vector is an eigenvector corresponding to a specific eigenvalue.
### Understanding the Options
- **Option a** presents the vectors \( (1,0) \) and \( (1,1) \).
- **Option b** presents the vectors \( (0,1) \) and \( (1,1) \).
- **Option c** presents the vectors \( (1,0) \) and \( (1,0) \), which appear not to be linearly independent.
- **Option d** presents the vectors \( (1, -1) \) and \( (1,1) \).
We need to verify each option to see which set of vectors forms a basis for the eigenspace of \( B \).
When evaluating candidate bases, linear independence and the correct span relative to the eigenspaces are crucial criteria.
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