Consider the linear transformation T: R" → R^ whose matrix A relative to the standard basis is given. з -1 (a) Find the eigenvalues of A. (Enter your answers from smallest to largest.) (^1, 2) =| (b) Find a basis for each of the corresponding eigenspaces. {[ B = B2 (c) Find the matrix A' for T relative to the basis B', where B' is made up of the basis vectors found in part (b). A' =
Consider the linear transformation T: R" → R^ whose matrix A relative to the standard basis is given. з -1 (a) Find the eigenvalues of A. (Enter your answers from smallest to largest.) (^1, 2) =| (b) Find a basis for each of the corresponding eigenspaces. {[ B = B2 (c) Find the matrix A' for T relative to the basis B', where B' is made up of the basis vectors found in part (b). A' =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![**Linear Transformations and Eigenvalues: A Detailed Example**
Consider the linear transformation \( T: \mathbb{R}^n \to \mathbb{R}^n \) whose matrix \( A \) relative to the standard basis is given:
\[ A = \begin{bmatrix} 3 & -1 \\ 2 & 6 \end{bmatrix} \]
(a) **Finding Eigenvalues:**
To find the eigenvalues of matrix \( A \), solve for:
\[
(\lambda_1, \lambda_2) = \boxed{\phantom{\text{Eigenvalues}}}
\]
(Enter your answers from smallest to largest.)
(b) **Finding Basis for Each Corresponding Eigenspace:**
For each eigenvalue found, determine the basis for the corresponding eigenspace.
For the eigenvalue \(\lambda_1\):
\[
B_1 = \left\{ \boxed{\phantom{\text{Basis vector for } \lambda_1}} \right\}
\]
For the eigenvalue \(\lambda_2\):
\[
B_2 = \left\{ \boxed{\phantom{\text{Basis vector for } \lambda_2}} \right\}
\]
(c) **Finding the Matrix \( A' \) Relative to Basis \( B' \):**
Where \( B' \) is made up of the basis vectors found in part (b), transform matrix \( A \) to its representation \( A' \) relative to the new basis \( B' \):
\[
A' = \begin{bmatrix}
\boxed{\phantom{\text{Value}}} & \boxed{\phantom{\text{Value}}} \\
\boxed{\phantom{\text{Value}}} & \boxed{\phantom{\text{Value}}}
\end{bmatrix}
\]
---
This example guides you through the process of finding eigenvalues and eigenspaces for a given matrix, and how to transform a matrix relative to a new basis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F737222ef-5c70-4907-93aa-2c2d7afd58f4%2F93cac5d3-a3c7-4493-8a1f-392ce2efc732%2Fc6zx2y_processed.png&w=3840&q=75)
Transcribed Image Text:**Linear Transformations and Eigenvalues: A Detailed Example**
Consider the linear transformation \( T: \mathbb{R}^n \to \mathbb{R}^n \) whose matrix \( A \) relative to the standard basis is given:
\[ A = \begin{bmatrix} 3 & -1 \\ 2 & 6 \end{bmatrix} \]
(a) **Finding Eigenvalues:**
To find the eigenvalues of matrix \( A \), solve for:
\[
(\lambda_1, \lambda_2) = \boxed{\phantom{\text{Eigenvalues}}}
\]
(Enter your answers from smallest to largest.)
(b) **Finding Basis for Each Corresponding Eigenspace:**
For each eigenvalue found, determine the basis for the corresponding eigenspace.
For the eigenvalue \(\lambda_1\):
\[
B_1 = \left\{ \boxed{\phantom{\text{Basis vector for } \lambda_1}} \right\}
\]
For the eigenvalue \(\lambda_2\):
\[
B_2 = \left\{ \boxed{\phantom{\text{Basis vector for } \lambda_2}} \right\}
\]
(c) **Finding the Matrix \( A' \) Relative to Basis \( B' \):**
Where \( B' \) is made up of the basis vectors found in part (b), transform matrix \( A \) to its representation \( A' \) relative to the new basis \( B' \):
\[
A' = \begin{bmatrix}
\boxed{\phantom{\text{Value}}} & \boxed{\phantom{\text{Value}}} \\
\boxed{\phantom{\text{Value}}} & \boxed{\phantom{\text{Value}}}
\end{bmatrix}
\]
---
This example guides you through the process of finding eigenvalues and eigenspaces for a given matrix, and how to transform a matrix relative to a new basis.
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