3 Consider the basis B of R2 given by v1 2 3 (a) Find the B-coordinates []B of the vector x = -2 | (b) Let T : R² → R² be the linear transformation defined by th -5 8 standard matrix A = Find the B-matrix for T. -3 5

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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**Consider the basis \( \mathcal{B} \) of \( \mathbb{R}^2 \) given by**

\[
\vec{v}_1 = \begin{bmatrix} 1 \\ 1 \end{bmatrix}, \quad \vec{v}_2 = \begin{bmatrix} 3 \\ 2 \end{bmatrix}.
\]

**(a)** Find the \( \mathcal{B} \)-coordinates \([ \vec{x} ]_{\mathcal{B}}\) of the vector \( \vec{x} = \begin{bmatrix} 3 \\ -2 \end{bmatrix} \).

**(b)** Let \( \mathcal{T} : \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) be the linear transformation defined by the standard matrix

\[
A = \begin{bmatrix} -5 & 8 \\ -3 & 5 \end{bmatrix}.
\]

Find the \( \mathcal{B} \)-matrix for \( \mathcal{T} \).
Transcribed Image Text:**Consider the basis \( \mathcal{B} \) of \( \mathbb{R}^2 \) given by** \[ \vec{v}_1 = \begin{bmatrix} 1 \\ 1 \end{bmatrix}, \quad \vec{v}_2 = \begin{bmatrix} 3 \\ 2 \end{bmatrix}. \] **(a)** Find the \( \mathcal{B} \)-coordinates \([ \vec{x} ]_{\mathcal{B}}\) of the vector \( \vec{x} = \begin{bmatrix} 3 \\ -2 \end{bmatrix} \). **(b)** Let \( \mathcal{T} : \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) be the linear transformation defined by the standard matrix \[ A = \begin{bmatrix} -5 & 8 \\ -3 & 5 \end{bmatrix}. \] Find the \( \mathcal{B} \)-matrix for \( \mathcal{T} \).
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