Find a basis B for the domain of T such that the matrix for T relative to B is diagonal. T: R² → R²: T(x, y) = (6x + 2y, 3x + y) (63) B =
Find a basis B for the domain of T such that the matrix for T relative to B is diagonal. T: R² → R²: T(x, y) = (6x + 2y, 3x + y) (63) B =
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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![Find a basis \( B \) for the domain of \( T \) such that the matrix for \( T \) relative to \( B \) is diagonal.
\[ T: \mathbb{R}^2 \rightarrow \mathbb{R}^2 : T(x, y) = (6x + 2y, 3x + y) \]
\[
B =
\left\{
\begin{bmatrix}
\ \ \ \ & \ \ \ \ \\
\ \ \ \ & \ \ \ \
\end{bmatrix},
\begin{bmatrix}
\ \ \ \ & \ \ \ \ \\
\ \ \ \ & \ \ \ \
\end{bmatrix}
\right\}
\]
The diagram features two 2x2 matrices represented by blank placeholders, indicating where the basis vectors are to be inserted. The task is to find vectors such that the transformation matrix \( T \) becomes diagonalized with respect to the basis \( B \). Green arrows show direction, suggesting the relationship or result of transformation using the basis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8f7d2653-b3fa-4350-8d3e-62930c027af3%2Ff1029cfe-efdb-4e4f-9540-5a7d9da83d91%2F3jgtqx_processed.png&w=3840&q=75)
Transcribed Image Text:Find a basis \( B \) for the domain of \( T \) such that the matrix for \( T \) relative to \( B \) is diagonal.
\[ T: \mathbb{R}^2 \rightarrow \mathbb{R}^2 : T(x, y) = (6x + 2y, 3x + y) \]
\[
B =
\left\{
\begin{bmatrix}
\ \ \ \ & \ \ \ \ \\
\ \ \ \ & \ \ \ \
\end{bmatrix},
\begin{bmatrix}
\ \ \ \ & \ \ \ \ \\
\ \ \ \ & \ \ \ \
\end{bmatrix}
\right\}
\]
The diagram features two 2x2 matrices represented by blank placeholders, indicating where the basis vectors are to be inserted. The task is to find vectors such that the transformation matrix \( T \) becomes diagonalized with respect to the basis \( B \). Green arrows show direction, suggesting the relationship or result of transformation using the basis.
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