**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} $.

Answered: Image /qna-images/answer/240f2c61-cde9-40fb-b229-14ed0c9a2dc8.jpg

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

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

Problem 1RQ

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Question

**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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