Find the standard matrix for each linear transformation. Give on.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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![### Linear Transformations and Their Standard Matrices
In this section, we will find the standard matrix for each given linear transformation and provide some explanations for better understanding.
#### Problem 2:
**Find the standard matrix for each linear transformation. Give some explanation.**
---
**a.** \( T : \mathbb{R}^2 \to \mathbb{R}^2 \) *rotates points about the origin by 180° (or \(\pi\) radians).*
**Explanation:**
The standard matrix for a rotation by \( \theta \) around the origin is given by:
\[ \begin{pmatrix}
\cos(\theta) & -\sin(\theta) \\
\sin(\theta) & \cos(\theta)
\end{pmatrix} \]
For a rotation by 180° or \(\pi\) radians:
\[ \cos(\pi) = -1 \]
\[ \sin(\pi) = 0 \]
Therefore, the standard matrix for this transformation is:
\[ \begin{pmatrix}
-1 & 0 \\
0 & -1
\end{pmatrix} \]
---
**b.** \( T : \mathbb{R}^2 \to \mathbb{R}^2 \) *reflects points about the line \( y = -x \).*
**Explanation:**
The reflection matrix about the line \( y = -x \) can be derived using the formula for reflection across a line \( y = mx \):
\[ \begin{pmatrix}
\frac{1-m^2}{1+m^2} & \frac{2m}{1+m^2} \\
\frac{2m}{1+m^2} & \frac{m^2-1}{1+m^2}
\end{pmatrix} \]
For \( y = -x \), the slope \( m \) is -1:
\[ \begin{pmatrix}
\frac{1-(-1)^2}{1+(-1)^2} & \frac{2(-1)}{1+(-1)^2} \\
\frac{2(-1)}{1+(-1)^2} & \frac{(-1)^2-1}{1+(-1)^2}
\end{pmatrix} = \begin{pmatrix}
0 & -1 \\
-1 &](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F54d39984-15d1-49e5-8ad0-848d7106f454%2F863ad08f-6af2-4fe0-857f-46b49d69a83a%2Fj26swkj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Linear Transformations and Their Standard Matrices
In this section, we will find the standard matrix for each given linear transformation and provide some explanations for better understanding.
#### Problem 2:
**Find the standard matrix for each linear transformation. Give some explanation.**
---
**a.** \( T : \mathbb{R}^2 \to \mathbb{R}^2 \) *rotates points about the origin by 180° (or \(\pi\) radians).*
**Explanation:**
The standard matrix for a rotation by \( \theta \) around the origin is given by:
\[ \begin{pmatrix}
\cos(\theta) & -\sin(\theta) \\
\sin(\theta) & \cos(\theta)
\end{pmatrix} \]
For a rotation by 180° or \(\pi\) radians:
\[ \cos(\pi) = -1 \]
\[ \sin(\pi) = 0 \]
Therefore, the standard matrix for this transformation is:
\[ \begin{pmatrix}
-1 & 0 \\
0 & -1
\end{pmatrix} \]
---
**b.** \( T : \mathbb{R}^2 \to \mathbb{R}^2 \) *reflects points about the line \( y = -x \).*
**Explanation:**
The reflection matrix about the line \( y = -x \) can be derived using the formula for reflection across a line \( y = mx \):
\[ \begin{pmatrix}
\frac{1-m^2}{1+m^2} & \frac{2m}{1+m^2} \\
\frac{2m}{1+m^2} & \frac{m^2-1}{1+m^2}
\end{pmatrix} \]
For \( y = -x \), the slope \( m \) is -1:
\[ \begin{pmatrix}
\frac{1-(-1)^2}{1+(-1)^2} & \frac{2(-1)}{1+(-1)^2} \\
\frac{2(-1)}{1+(-1)^2} & \frac{(-1)^2-1}{1+(-1)^2}
\end{pmatrix} = \begin{pmatrix}
0 & -1 \\
-1 &
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