**Finding a Basis for the Eigenspace**In this lesson, we will learn how to find a basis for the eigenspace corresponding to a given eigenvalue of a matrix. Consider the problem below, where we are given a matrix $A$ and an eigenvalue $\lambda$:Given matrix $A$ and eigenvalue $\lambda$:\[A = \begin{bmatrix}3 & 0 & 2 \\3 & 0 & 4 \\2 & -1 & 5 \end{bmatrix}, \quad \lambda = 2\]We are required to find a basis for the eigenspace corresponding to the eigenvalue $\lambda = 2$.To proceed, follow these steps:1. Subtract $\lambda I$ (where $I$ is the identity matrix of the same dimension as $A$) from the matrix $A$.2. Solve the resulting system of linear equations to find the null space of the matrix $A - 2I$.3. The basis for the null space found in step 2 will form the basis for the eigenspace corresponding to the given eigenvalue.We then get the augmented matrix equation and solve it step-by-step to find the eigenvectors. Note that the specific computations are subject to detailed mathematical operations involving linear algebra techniques.### Example Solution Format\[ \text{A basis for the eigenspace corresponding to} \ \lambda = 2 \ \text{is} \quad [\text{Vectors here}].\]Remember to use a comma to separate multiple vectors in the basis if needed.

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Find a basis for the eigenspace corresponding to the eigenvalue of A given below. 0 2 A=3 0 4 1 = 2 2 -1 5 A basis for the eigenspace corresponding to A = 2 is | (Use a comma to separate answers as needed.)

Find a basis for the eigenspace corresponding to the eigenvalue of A given below. 0 2 A=3 0 4 1 = 2 2 -1 5 A basis for the eigenspace corresponding to A = 2 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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**Finding a Basis for the Eigenspace**

In this lesson, we will learn how to find a basis for the eigenspace corresponding to a given eigenvalue of a matrix. Consider the problem below, where we are given a matrix $A$ and an eigenvalue $\lambda$:

Given matrix $A$ and eigenvalue $\lambda$:

\[A = \begin{bmatrix}
3 & 0 & 2 \\
3 & 0 & 4 \\
2 & -1 & 5
\end{bmatrix}, \quad \lambda = 2\]

We are required to find a basis for the eigenspace corresponding to the eigenvalue $\lambda = 2$.

To proceed, follow these steps:
1. Subtract $\lambda I$ (where $I$ is the identity matrix of the same dimension as $A$) from the matrix $A$.
2. Solve the resulting system of linear equations to find the null space of the matrix $A - 2I$.
3. The basis for the null space found in step 2 will form the basis for the eigenspace corresponding to the given eigenvalue.

We then get the augmented matrix equation and solve it step-by-step to find the eigenvectors. Note that the specific computations are subject to detailed mathematical operations involving linear algebra techniques.

### Example Solution Format
\[ \text{A basis for the eigenspace corresponding to} \ \lambda = 2 \ \text{is} \quad [\text{Vectors here}].\]

Remember to use a comma to separate multiple vectors in the basis if needed.

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