Let T R2 R2 be a linear transfor [3] into the vector a (i) T maps vector u
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 Problem
#### Problem Statement
Let \( T: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) be a linear transformation such that:
(i) \( T \) maps vector \( \mathbf{u} = \begin{bmatrix} 1 \\ 5 \end{bmatrix} \) into the vector \( \mathbf{a} = \begin{bmatrix} -2 \\ 1 \end{bmatrix} \),
(ii) \( T \) maps vector \( \mathbf{v} = \begin{bmatrix} 3 \\ 1 \end{bmatrix} \) into the vector \( \mathbf{b} = \begin{bmatrix} -4 \\ 5 \end{bmatrix} \).
Use the fact that \( T \) is a linear transformation to find \( T(3\mathbf{u} + 2\mathbf{v}) \).
#### Explanation
Given the properties of the linear transformation \( T \), the vectors \( \mathbf{u} \) and \( \mathbf{v} \) are mapped to specific vectors \( \mathbf{a} \) and \( \mathbf{b} \) respectively. The goal is to apply the linearity of \( T \) to find the image of the linear combination \( 3\mathbf{u} + 2\mathbf{v} \).
##### Step-by-Step Solution
1. **Express \( T \left( 3\mathbf{u} + 2\mathbf{v} \right) \) using linearity:**
\[
T \left( 3\mathbf{u} + 2\mathbf{v} \right) = 3T(\mathbf{u}) + 2T(\mathbf{v})
\]
2. **Substitute the known mappings:**
\[
T(\mathbf{u}) = \mathbf{a} = \begin{bmatrix} -2 \\ 1 \end{bmatrix}
\]
\[
T(\mathbf{v}) = \mathbf{b} = \begin{bmatrix} -4 \\ 5 \end{bmatrix}
\]
3. **Compute \( 3T(\mathbf{u}) \) and \( 2T(\mathbf{v}) \):**
\[
3T(\mathbf{u}) = 3 \begin{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad281aeb-79af-4eda-b9b4-346ba477171b%2Fe7b64e4b-014a-41a6-b0fb-136567759201%2F11ep616w_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Linear Transformation Problem
#### Problem Statement
Let \( T: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) be a linear transformation such that:
(i) \( T \) maps vector \( \mathbf{u} = \begin{bmatrix} 1 \\ 5 \end{bmatrix} \) into the vector \( \mathbf{a} = \begin{bmatrix} -2 \\ 1 \end{bmatrix} \),
(ii) \( T \) maps vector \( \mathbf{v} = \begin{bmatrix} 3 \\ 1 \end{bmatrix} \) into the vector \( \mathbf{b} = \begin{bmatrix} -4 \\ 5 \end{bmatrix} \).
Use the fact that \( T \) is a linear transformation to find \( T(3\mathbf{u} + 2\mathbf{v}) \).
#### Explanation
Given the properties of the linear transformation \( T \), the vectors \( \mathbf{u} \) and \( \mathbf{v} \) are mapped to specific vectors \( \mathbf{a} \) and \( \mathbf{b} \) respectively. The goal is to apply the linearity of \( T \) to find the image of the linear combination \( 3\mathbf{u} + 2\mathbf{v} \).
##### Step-by-Step Solution
1. **Express \( T \left( 3\mathbf{u} + 2\mathbf{v} \right) \) using linearity:**
\[
T \left( 3\mathbf{u} + 2\mathbf{v} \right) = 3T(\mathbf{u}) + 2T(\mathbf{v})
\]
2. **Substitute the known mappings:**
\[
T(\mathbf{u}) = \mathbf{a} = \begin{bmatrix} -2 \\ 1 \end{bmatrix}
\]
\[
T(\mathbf{v}) = \mathbf{b} = \begin{bmatrix} -4 \\ 5 \end{bmatrix}
\]
3. **Compute \( 3T(\mathbf{u}) \) and \( 2T(\mathbf{v}) \):**
\[
3T(\mathbf{u}) = 3 \begin{
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