**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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Find a basis for the eigenspace corresponding to the eigenvalue. 4 3 -2 A = 9 - 4 A= 3 -1-3 A basis for the eigenspace corresponding to à = 3 is {).

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

**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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