Let b1 = and b = E: {61, 62 is a basis for R?. Let T : R? → R² be a linear transformation such that The set B = T(51) = 851 + 2b, and T(52) = 2b1 + 862. (a) The matrix of T relative to the basis B is (T]B =
Let b1 = and b = E: {61, 62 is a basis for R?. Let T : R? → R² be a linear transformation such that The set B = T(51) = 851 + 2b, and T(52) = 2b1 + 862. (a) The matrix of T relative to the basis B is (T]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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![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
\[T(\vec{b}_1) = 8\vec{b}_1 + 2\vec{b}_2 \quad \text{and} \quad T(\vec{b}_2) = 2\vec{b}_1 + 8\vec{b}_2.\]
(a) The matrix of \(T\) relative to the basis \(B\) is
\[
[T]_B = \begin{bmatrix}
\editable{} & \editable{} \\
\editable{} & \editable{}
\end{bmatrix}.
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2dbaa7d4-84b4-4619-a10e-5b2bde6f9c5e%2F83408155-2499-4ad5-8c9c-d3338ad56ff4%2Fxoujl77_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
\[T(\vec{b}_1) = 8\vec{b}_1 + 2\vec{b}_2 \quad \text{and} \quad T(\vec{b}_2) = 2\vec{b}_1 + 8\vec{b}_2.\]
(a) The matrix of \(T\) relative to the basis \(B\) is
\[
[T]_B = \begin{bmatrix}
\editable{} & \editable{} \\
\editable{} & \editable{}
\end{bmatrix}.
\]
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