Consider the linear transformation T: RR whose matrix A relative to the standard basis is given. A = [ ²² ] (a) Find the eigenvalues of A. (Enter your answers from smallest to largest.) - (3₁ 3.4 (21, 2₂) = (b) Find a basis for each of the corresponding eigenspaces. (-2,1) B₁ = B₂ = A' = X (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).
Consider the linear transformation T: RR whose matrix A relative to the standard basis is given. A = [ ²² ] (a) Find the eigenvalues of A. (Enter your answers from smallest to largest.) - (3₁ 3.4 (21, 2₂) = (b) Find a basis for each of the corresponding eigenspaces. (-2,1) B₁ = B₂ = A' = X (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).
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter5: Orthogonality
Section5.4: Orthogonal Diagonalization Of Symmetric Matrices
Problem 27EQ
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![**Linear Transformation and Eigenvalues**
Consider the linear transformation \( T: \mathbb{R}^n \to \mathbb{R}^n \) whose matrix \( A \) relative to the standard basis is given by:
\[
A = \begin{bmatrix} 2 & 2 \\ -1 & 5 \end{bmatrix}
\]
**(a) Finding Eigenvalues**
Find the eigenvalues of \( A \). (Enter your answers from smallest to largest.)
\[
(\lambda_1, \lambda_2) = (3, 4) \quad \checkmark
\]
**(b) Basis for Eigenspaces**
Find a basis for each of the corresponding eigenspaces.
\[
B_1 = \left\{ \begin{pmatrix} -2 \\ 1 \end{pmatrix} \right\} \quad \textcolor{red}{\text{✗}}
\]
\[
B_2 = \left\{ \begin{pmatrix} \, \, \end{pmatrix} \right\}
\]
**(c) Finding the Matrix \( A' \)**
Find the matrix \( A' \) for \( T \) relative to the basis \( B' \), where \( B' \) is made up of the basis vectors found in part (b).
\[
A' = \begin{bmatrix} \, & \, \\ \, & \, \end{bmatrix} \quad \textcolor{green}{\Downarrow \, \Uparrow}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8eeaacd3-f9c7-4223-a8e6-a717e551a98a%2F7e9a2ca0-2990-496a-acb5-60d753486bad%2F8rn5mqc_processed.png&w=3840&q=75)
Transcribed Image Text:**Linear Transformation and Eigenvalues**
Consider the linear transformation \( T: \mathbb{R}^n \to \mathbb{R}^n \) whose matrix \( A \) relative to the standard basis is given by:
\[
A = \begin{bmatrix} 2 & 2 \\ -1 & 5 \end{bmatrix}
\]
**(a) Finding Eigenvalues**
Find the eigenvalues of \( A \). (Enter your answers from smallest to largest.)
\[
(\lambda_1, \lambda_2) = (3, 4) \quad \checkmark
\]
**(b) Basis for Eigenspaces**
Find a basis for each of the corresponding eigenspaces.
\[
B_1 = \left\{ \begin{pmatrix} -2 \\ 1 \end{pmatrix} \right\} \quad \textcolor{red}{\text{✗}}
\]
\[
B_2 = \left\{ \begin{pmatrix} \, \, \end{pmatrix} \right\}
\]
**(c) Finding the Matrix \( A' \)**
Find the matrix \( A' \) for \( T \) relative to the basis \( B' \), where \( B' \) is made up of the basis vectors found in part (b).
\[
A' = \begin{bmatrix} \, & \, \\ \, & \, \end{bmatrix} \quad \textcolor{green}{\Downarrow \, \Uparrow}
\]
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