### Linear Transformation and Standard MatrixGiven a linear transformation $ T : \mathbb{R}^2 \rightarrow \mathbb{R}^2 $ such that:\[ T \left( \begin{bmatrix} 1 \\ 0 \end{bmatrix} \right) = \begin{bmatrix} -4 \\ 4 \end{bmatrix}, \quad T \left( \begin{bmatrix} 0 \\ 1 \end{bmatrix} \right) = \begin{bmatrix} 1 \\ 1 \end{bmatrix}, \]we are required to find the standard matrix $ A $ of the transformation $ T $.### ExplanationTo determine the standard matrix $ A $, we use the given transformation properties. The columns of $ A $ can be found by applying the transformation $ T $ to the standard basis vectors of $ \mathbb{R}^2 $:1. The first column of $ A $ is $ T \left( \begin{bmatrix} 1 \\ 0 \end{bmatrix} \right) = \begin{bmatrix} -4 \\ 4 \end{bmatrix} $.2. The second column of $ A $ is $ T \left( \begin{bmatrix} 0 \\ 1 \end{bmatrix} \right) = \begin{bmatrix} 1 \\ 1 \end{bmatrix} $.Thus, the standard matrix $ A $ of the transformation $ T $ is:\[ A = \begin{bmatrix} -4 & 1 \\ 4 & 1 \end{bmatrix} \]### Matrix Representation\[ A = \begin{bmatrix} -4 & 1 \\ 4 & 1 \end{bmatrix} \]This matrix represents the linear transformation $ T $ in $\mathbb{R}^2 $.

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If T : R² → R² is a linear transformation such that T standard matrix of T is

Advanced Engineering Mathematics

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ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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Question

### Linear Transformation and Standard Matrix

Given a linear transformation $ T : \mathbb{R}^2 \rightarrow \mathbb{R}^2 $ such that:

\[ T \left( \begin{bmatrix} 1 \\ 0 \end{bmatrix} \right) = \begin{bmatrix} -4 \\ 4 \end{bmatrix}, \quad T \left( \begin{bmatrix} 0 \\ 1 \end{bmatrix} \right) = \begin{bmatrix} 1 \\ 1 \end{bmatrix}, \]

we are required to find the standard matrix $ A $ of the transformation $ T $.

### Explanation

To determine the standard matrix $ A $, we use the given transformation properties. The columns of $ A $ can be found by applying the transformation $ T $ to the standard basis vectors of $ \mathbb{R}^2 $:

1. The first column of $ A $ is $ T \left( \begin{bmatrix} 1 \\ 0 \end{bmatrix} \right) = \begin{bmatrix} -4 \\ 4 \end{bmatrix} $.
2. The second column of $ A $ is $ T \left( \begin{bmatrix} 0 \\ 1 \end{bmatrix} \right) = \begin{bmatrix} 1 \\ 1 \end{bmatrix} $.

Thus, the standard matrix $ A $ of the transformation $ T $ is:

\[ A = \begin{bmatrix} -4 & 1 \\ 4 & 1 \end{bmatrix} \]

### Matrix Representation

\[ A = \begin{bmatrix} -4 & 1 \\ 4 & 1 \end{bmatrix} \]

This matrix represents the linear transformation $ T $ in $\mathbb{R}^2 $.

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