Assume that T is a linear transformation. Find the standard matrix of T. T: R² R² first reflects points through the line x₂ = − -X₁ and then reflects points through the origin.
Assume that T is a linear transformation. Find the standard matrix of T. T: R² R² first reflects points through the line x₂ = − -X₁ and then reflects points through the origin.
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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![### Linear Transformation and Standard Matrix Calculation
**Problem Statement:**
Assume that \(T\) is a linear transformation. Find the standard matrix of \(T\).
**Transformation \(T\):**
\[ T: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \]
First, it reflects points through the line \( x_2 = -x_1 \) and then reflects points through the origin.
---
**Standard Matrix \(A\):**
\[ A = \begin{pmatrix}
0 & -1 \\
-1 & 0
\end{pmatrix} \]
*(Type an integer or simplified fraction for each matrix element.)*
---
**Explanation:**
The transformation involves two reflections:
1. **Reflection through the line \( x_2 = -x_1 \)**:
- The reflection matrix for this line is:
\[ \begin{pmatrix}
0 & -1 \\
-1 & 0
\end{pmatrix} \]
2. **Reflection through the origin**:
- Reflecting through the origin simply multiplies each coordinate by -1, which means each component of the matrix is multiplied by -1.
Combining these transformations, the final standard matrix representing this sequence of reflections is:
\[ A = \begin{pmatrix}
0 & -1 \\
-1 & 0
\end{pmatrix} \]
This matrix can be used to transform any vector in \(\mathbb{R}^2\) according to the described linear transformation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8b6c3689-2b60-4c25-801e-fccc9acdd031%2Fc060977b-0ec9-41a9-aea5-c7e1e1e29ec8%2F0pij6a_processed.png&w=3840&q=75)
Transcribed Image Text:### Linear Transformation and Standard Matrix Calculation
**Problem Statement:**
Assume that \(T\) is a linear transformation. Find the standard matrix of \(T\).
**Transformation \(T\):**
\[ T: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \]
First, it reflects points through the line \( x_2 = -x_1 \) and then reflects points through the origin.
---
**Standard Matrix \(A\):**
\[ A = \begin{pmatrix}
0 & -1 \\
-1 & 0
\end{pmatrix} \]
*(Type an integer or simplified fraction for each matrix element.)*
---
**Explanation:**
The transformation involves two reflections:
1. **Reflection through the line \( x_2 = -x_1 \)**:
- The reflection matrix for this line is:
\[ \begin{pmatrix}
0 & -1 \\
-1 & 0
\end{pmatrix} \]
2. **Reflection through the origin**:
- Reflecting through the origin simply multiplies each coordinate by -1, which means each component of the matrix is multiplied by -1.
Combining these transformations, the final standard matrix representing this sequence of reflections is:
\[ A = \begin{pmatrix}
0 & -1 \\
-1 & 0
\end{pmatrix} \]
This matrix can be used to transform any vector in \(\mathbb{R}^2\) according to the described linear transformation.
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