Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![**Title: Finding a Basis for the Eigenspace Corresponding to a Given Eigenvalue**
**Problem Statement:**
Find a basis for the eigenspace corresponding to the eigenvalue.
\[ A = \begin{bmatrix} 4 & 3 & -2 \\ 2 & 9 & -4 \\ -1 & -3 & 5 \end{bmatrix} , \quad \lambda = 3 \]
**Question:**
A basis for the eigenspace corresponding to \( \lambda = 3 \) is _____________ (Type a vector or list of vectors. Type an integer or simplified fraction for each matrix element).
**Explanation:**
In this exercise, you are asked to find a basis for the eigenspace corresponding to the eigenvalue \( \lambda = 3 \) for the given matrix \( A \).
To approach this problem:
1. **Compute \( \mathbf{A} - \lambda \mathbf{I} \) :**
- Subtract \( \lambda \mathbf{I} \) from \( \mathbf{A} \), where \( \mathbf{I} \) is the identity matrix of the same dimension as \( \mathbf{A} \), and \( \lambda \) is the eigenvalue.
2. **Solve the Homogeneous System \( (\mathbf{A} - \lambda \mathbf{I})\mathbf{x} = \mathbf{0} \) :**
- Find the solutions to the equation \( (\mathbf{A} - \lambda \mathbf{I})\mathbf{x} = \mathbf{0} \).
3. **Find the Basis for the Eigenspace:**
- The solution vector(s) \( \mathbf{x} \) corresponding to this equation form a basis for the eigenspace.
By finding these steps, you will identify a set of vector(s) that span the eigenspace for \( \lambda = 3 \).
(Note: Ensure that your final answer is a list of vectors or a vector with integer or simplified fraction entries.)](https://content.bartleby.com/qna-images/question/780f9839-f737-4aba-91a2-6210989911b1/95d8ea2c-12de-4da2-8d34-0ebdb1c3c345/z9xm8jr_processed.jpeg)
Transcribed Image Text:**Title: Finding a Basis for the Eigenspace Corresponding to a Given Eigenvalue**
**Problem Statement:**
Find a basis for the eigenspace corresponding to the eigenvalue.
\[ A = \begin{bmatrix} 4 & 3 & -2 \\ 2 & 9 & -4 \\ -1 & -3 & 5 \end{bmatrix} , \quad \lambda = 3 \]
**Question:**
A basis for the eigenspace corresponding to \( \lambda = 3 \) is _____________ (Type a vector or list of vectors. Type an integer or simplified fraction for each matrix element).
**Explanation:**
In this exercise, you are asked to find a basis for the eigenspace corresponding to the eigenvalue \( \lambda = 3 \) for the given matrix \( A \).
To approach this problem:
1. **Compute \( \mathbf{A} - \lambda \mathbf{I} \) :**
- Subtract \( \lambda \mathbf{I} \) from \( \mathbf{A} \), where \( \mathbf{I} \) is the identity matrix of the same dimension as \( \mathbf{A} \), and \( \lambda \) is the eigenvalue.
2. **Solve the Homogeneous System \( (\mathbf{A} - \lambda \mathbf{I})\mathbf{x} = \mathbf{0} \) :**
- Find the solutions to the equation \( (\mathbf{A} - \lambda \mathbf{I})\mathbf{x} = \mathbf{0} \).
3. **Find the Basis for the Eigenspace:**
- The solution vector(s) \( \mathbf{x} \) corresponding to this equation form a basis for the eigenspace.
By finding these steps, you will identify a set of vector(s) that span the eigenspace for \( \lambda = 3 \).
(Note: Ensure that your final answer is a list of vectors or a vector with integer or simplified fraction entries.)
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