Let b1 = and b2 = is a basis for R². Let T : R² → R² be a linear transformation such that The set B = T(51) = 851 + 202 and T(b) = 2b1 + 852.
Let b1 = and b2 = is a basis for R². Let T : R² → R² be a linear transformation such that The set B = T(51) = 851 + 202 and T(b) = 2b1 + 852.
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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![The text in the image is as follows:
"(b) The matrix of \( T \) relative to the standard basis \( E \) for \( \mathbb{R}^2 \) is
\[
[T]_E = \begin{bmatrix}
\boxed{} & \boxed{} \\
\boxed{} & \boxed{}
\end{bmatrix}
\]"
The diagram is a 2x2 matrix representation with empty cells, indicated by boxed areas for entries, where numerical values can be inputted. This matrix describes a linear transformation \( T \) in the context of a standard basis for two-dimensional space, \( \mathbb{R}^2 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2dbaa7d4-84b4-4619-a10e-5b2bde6f9c5e%2Fd3662dc5-d14c-4a4b-ab09-317d2c583074%2Fxy45vrl_processed.png&w=3840&q=75)
Transcribed Image Text:The text in the image is as follows:
"(b) The matrix of \( T \) relative to the standard basis \( E \) for \( \mathbb{R}^2 \) is
\[
[T]_E = \begin{bmatrix}
\boxed{} & \boxed{} \\
\boxed{} & \boxed{}
\end{bmatrix}
\]"
The diagram is a 2x2 matrix representation with empty cells, indicated by boxed areas for entries, where numerical values can be inputted. This matrix describes a linear transformation \( T \) in the context of a standard basis for two-dimensional space, \( \mathbb{R}^2 \).
![Let \(\vec{b_1} = \begin{bmatrix} -1 \\ -1 \end{bmatrix}\) and \(\vec{b_2} = \begin{bmatrix} -1 \\ -2 \end{bmatrix}\). The set \(B = \{\vec{b_1}, \vec{b_2}\}\) is a basis for \(\mathbb{R}^2\). Let \(T : \mathbb{R}^2 \to \mathbb{R}^2\) be a linear transformation such that
\[ T(\vec{b_1}) = 8\vec{b_1} + 2\vec{b_2} \]
and
\[ T(\vec{b_2}) = 2\vec{b_1} + 8\vec{b_2}. \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2dbaa7d4-84b4-4619-a10e-5b2bde6f9c5e%2Fd3662dc5-d14c-4a4b-ab09-317d2c583074%2Fk57q54r_processed.png&w=3840&q=75)
Transcribed Image Text:Let \(\vec{b_1} = \begin{bmatrix} -1 \\ -1 \end{bmatrix}\) and \(\vec{b_2} = \begin{bmatrix} -1 \\ -2 \end{bmatrix}\). The set \(B = \{\vec{b_1}, \vec{b_2}\}\) is a basis for \(\mathbb{R}^2\). Let \(T : \mathbb{R}^2 \to \mathbb{R}^2\) be a linear transformation such that
\[ T(\vec{b_1}) = 8\vec{b_1} + 2\vec{b_2} \]
and
\[ T(\vec{b_2}) = 2\vec{b_1} + 8\vec{b_2}. \]
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