Consider the linear transformation T: RR whose matrix A relative to the standard basis is given. A = (a) Find the eigenvalues of A. (Enter your answers from smallest to largest.) -C (21, 2₂) = (b) Find a basis for each of the corresponding eigenspaces. -{\ B1 = 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: RR whose matrix A relative to the standard basis is given. A = (a) Find the eigenvalues of A. (Enter your answers from smallest to largest.) -C (21, 2₂) = (b) Find a basis for each of the corresponding eigenspaces. -{\ B1 = 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
Problem 1RQ
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![### Linear Transformation and Eigenvalues
Consider the linear transformation \( T: \mathbb{R}^n \rightarrow \mathbb{R}^n \) whose matrix \( A \) relative to the standard basis is given:
\[
A = \begin{pmatrix}
3 & -2 \\
1 & 6
\end{pmatrix}
\]
#### (a) Find the Eigenvalues of \( A \)
To determine the eigenvalues of \( A \), solve the characteristic equation of the matrix. Enter your answers from smallest to largest.
\[
(\lambda_1, \lambda_2) = \left( \boxed{\,} , \boxed{\,} \right)
\]
#### (b) Find a Basis for Each of the Corresponding Eigenspaces
Next, find a basis for each of the eigenspaces associated with the eigenvalues found in part (a).
\[
B_1 = \left\{ \boxed{\,} \right\}
\]
\[
B_2 = \left\{ \boxed{\,} \right\}
\]
#### (c) Find the Matrix \( A' \) for \( T \) Relative to the Basis \( B' \)
Finally, find the matrix \( A' \) for \( T \) relative to the basis \( B' \), where \( B' \) is made up of the basis vectors found in part (b).
\[
A' = \begin{pmatrix}
\boxed{\,} & \boxed{\,} \\
\boxed{\,} & \boxed{\,}
\end{pmatrix}
\]
Understanding and solving these steps will allow you to find the new matrix representation of the linear transformation \( T \) in the context of eigenvalues and eigenspaces.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F03210149-7bd1-493b-9cd1-96815f185546%2Fa85b6d1a-b00f-48f5-8c2c-7b7ade850247%2Fsc3eh57_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Linear Transformation and Eigenvalues
Consider the linear transformation \( T: \mathbb{R}^n \rightarrow \mathbb{R}^n \) whose matrix \( A \) relative to the standard basis is given:
\[
A = \begin{pmatrix}
3 & -2 \\
1 & 6
\end{pmatrix}
\]
#### (a) Find the Eigenvalues of \( A \)
To determine the eigenvalues of \( A \), solve the characteristic equation of the matrix. Enter your answers from smallest to largest.
\[
(\lambda_1, \lambda_2) = \left( \boxed{\,} , \boxed{\,} \right)
\]
#### (b) Find a Basis for Each of the Corresponding Eigenspaces
Next, find a basis for each of the eigenspaces associated with the eigenvalues found in part (a).
\[
B_1 = \left\{ \boxed{\,} \right\}
\]
\[
B_2 = \left\{ \boxed{\,} \right\}
\]
#### (c) Find the Matrix \( A' \) for \( T \) Relative to the Basis \( B' \)
Finally, find the matrix \( A' \) for \( T \) relative to the basis \( B' \), where \( B' \) is made up of the basis vectors found in part (b).
\[
A' = \begin{pmatrix}
\boxed{\,} & \boxed{\,} \\
\boxed{\,} & \boxed{\,}
\end{pmatrix}
\]
Understanding and solving these steps will allow you to find the new matrix representation of the linear transformation \( T \) in the context of eigenvalues and eigenspaces.
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