**Linear Transformation Example**Let $ \mathbf{e}_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix} $ and $ \mathbf{e}_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix} $, $ \mathbf{y}_1 = \begin{bmatrix} 3 \\ 7 \end{bmatrix} $, and $ \mathbf{y}_2 = \begin{bmatrix} -1 \\ 8 \end{bmatrix} $, and let $ T: \mathbb{R}^2 \rightarrow \mathbb{R}^2 $ be a linear transformation that maps $ \mathbf{e}_1 $ into $ \mathbf{y}_1 $ and maps $ \mathbf{e}_2 $ into $ \mathbf{y}_2 $. Find the images of $ \begin{bmatrix} 5 \\ -3 \end{bmatrix} $ and $ \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} $.---The image of $ \begin{bmatrix} 5 \\ -3 \end{bmatrix} $ is $[ \enspace ]$.---**Explanation:**The task is to find the result of a specific linear transformation on given vectors. Here, the basis vectors $ \mathbf{e}_1 $ and $ \mathbf{e}_2 $ in $ \mathbb{R}^2 $ are mapped to new vectors $ \mathbf{y}_1 $ and $ \mathbf{y}_2 $ respectively by the transformation $ T $. Once the transformation matrix is determined, it will be applied to the vector $ \begin{bmatrix} 5 \\ -3 \end{bmatrix} $ to determine its image.

Given that and , , and .The linear transformation such that and .The aim is to find the image of…

Answered: and e₂= Y₁ = 0 1 [8] [8] 3-8 5 and -3…

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

**Linear Transformation Example**

Let $ \mathbf{e}_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix} $ and $ \mathbf{e}_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix} $, $ \mathbf{y}_1 = \begin{bmatrix} 3 \\ 7 \end{bmatrix} $, and $ \mathbf{y}_2 = \begin{bmatrix} -1 \\ 8 \end{bmatrix} $, and let $ T: \mathbb{R}^2 \rightarrow \mathbb{R}^2 $ be a linear transformation that maps $ \mathbf{e}_1 $ into $ \mathbf{y}_1 $ and maps $ \mathbf{e}_2 $ into $ \mathbf{y}_2 $. Find the images of $ \begin{bmatrix} 5 \\ -3 \end{bmatrix} $ and $ \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} $.

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The image of $ \begin{bmatrix} 5 \\ -3 \end{bmatrix} $ is $[ \enspace ]$.

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**Explanation:**

The task is to find the result of a specific linear transformation on given vectors. Here, the basis vectors $ \mathbf{e}_1 $ and $ \mathbf{e}_2 $ in $ \mathbb{R}^2 $ are mapped to new vectors $ \mathbf{y}_1 $ and $ \mathbf{y}_2 $ respectively by the transformation $ T $. Once the transformation matrix is determined, it will be applied to the vector $ \begin{bmatrix} 5 \\ -3 \end{bmatrix} $ to determine its image.

Expert Solution

Step 1: Given the information

Given that $e subscript 1 equals open square brackets table row 1 row 0 end table close square brackets$ and $e subscript 2 equals open square brackets table row 0 row 1 end table close square brackets$ , $y subscript 1 equals open square brackets table row 3 row 7 end table close square brackets$ , and $y subscript 2 equals open square brackets table row cell negative 1 end cell row 8 end table close square brackets$ .

The linear transformation $T colon straight real numbers squared rightwards arrow straight real numbers squared$ such that $T open parentheses e subscript 1 close parentheses equals y subscript 1$ and $T open parentheses e subscript 2 close parentheses equals y subscript 2$ .

The aim is to find the image of $open square brackets table row 5 row cell negative 3 end cell end table close square brackets$ and $open square brackets table row cell x subscript 1 end cell row cell x subscript 2 end cell end table close square brackets$ .