Find the standard matrix of the given linear transformation from R² to R². Reflection in the line y = x
Find the standard matrix of the given linear transformation from R² to R². Reflection in the line y = x
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
![**Topic: Linear Transformations and Matrices**
**Objective: Understanding the Standard Matrix for Reflection**
In this lesson, we will find the standard matrix of the linear transformation which involves the reflection in the line \( y = x \) in the plane \( \mathbb{R}^2 \).
### Problem Statement
We are tasked with determining the matrix representation of the reflection transformation across the line \( y = x \).
**Linear Transformation Details:**
- **Reflection Line:** \( y = x \)
### Matrix for Reflection
The standard matrix for reflecting a vector across the line \( y = x \) is expressed as:
\[
\begin{bmatrix}
0 & 1 \\
1 & 0
\end{bmatrix}
\]
This matrix results from exchanging the coordinates (i.e., switching \( x \) and \( y \) components) of any point it is applied to.
**Calculation Steps:**
1. **Initial Point:** Consider a point \( (x, y) \).
2. **Reflection Process:** Reflecting over the line \( y = x \) swaps its coordinates to \( (y, x) \).
3. **Matrix Representation:** As a matrix operation, this is expressed by multiplying the point vector with the transformation matrix.
### Application:
The matrix is then used to perform the reflection operation on any vector in the 2D plane \( \mathbb{R}^2 \).
### Conclusion:
Understanding the application of this matrix can be an essential tool for various real-world graphics and engineering problems involving geometric transformations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F67ad69fc-d7b2-4082-8b18-7ae4b62544b8%2F9d53d7ee-96f2-43b7-81dc-2c688dd77d2d%2F0l6bwbj_processed.png&w=3840&q=75)
Transcribed Image Text:**Topic: Linear Transformations and Matrices**
**Objective: Understanding the Standard Matrix for Reflection**
In this lesson, we will find the standard matrix of the linear transformation which involves the reflection in the line \( y = x \) in the plane \( \mathbb{R}^2 \).
### Problem Statement
We are tasked with determining the matrix representation of the reflection transformation across the line \( y = x \).
**Linear Transformation Details:**
- **Reflection Line:** \( y = x \)
### Matrix for Reflection
The standard matrix for reflecting a vector across the line \( y = x \) is expressed as:
\[
\begin{bmatrix}
0 & 1 \\
1 & 0
\end{bmatrix}
\]
This matrix results from exchanging the coordinates (i.e., switching \( x \) and \( y \) components) of any point it is applied to.
**Calculation Steps:**
1. **Initial Point:** Consider a point \( (x, y) \).
2. **Reflection Process:** Reflecting over the line \( y = x \) swaps its coordinates to \( (y, x) \).
3. **Matrix Representation:** As a matrix operation, this is expressed by multiplying the point vector with the transformation matrix.
### Application:
The matrix is then used to perform the reflection operation on any vector in the 2D plane \( \mathbb{R}^2 \).
### Conclusion:
Understanding the application of this matrix can be an essential tool for various real-world graphics and engineering problems involving geometric transformations.
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