Find the standard matrix for the linear transformation T: R2 R² that reflects points about the origin 3π and then rotates points about the origin by radians. 4
Find the standard matrix for the linear transformation T: R2 R² that reflects points about the origin 3π and then rotates points about the origin by radians. 4
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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![**Problem Statement:**
Find the standard matrix for the linear transformation \( T: \mathbb{R}^2 \to \mathbb{R}^2 \) that reflects points about the origin and then rotates points about the origin by \(\frac{3\pi}{4}\) radians.
**Solution Approach:**
To solve this problem, two key operations are involved:
1. **Reflection about the Origin:**
- The matrix for reflecting points about the origin in \(\mathbb{R}^2\) is:
\[
R = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}
\]
2. **Rotation by \(\frac{3\pi}{4}\) Radians:**
- The matrix for rotating points about the origin by an angle \(\theta\) is:
\[
M(\theta) = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix}
\]
- Here, \(\theta = \frac{3\pi}{4}\), so:
\[
M\left(\frac{3\pi}{4}\right) = \begin{bmatrix} \cos\left(\frac{3\pi}{4}\right) & -\sin\left(\frac{3\pi}{4}\right) \\ \sin\left(\frac{3\pi}{4}\right) & \cos\left(\frac{3\pi}{4}\right) \end{bmatrix} = \begin{bmatrix} -\frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \end{bmatrix}
\]
**Standard Matrix Calculation:**
The standard matrix for the transformation \( T \) is obtained by multiplying the matrices for reflection and rotation: \( T = M\left(\frac{3\pi}{4}\right) \times R \).
Thus, the standard matrix for \( T \) is:
\[
T = \begin{bmatrix} -\frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F18574973-f25e-4ab8-b7d6-6007b5b87fc4%2Fbc2aef9f-d91e-4fb4-ab28-424519d6b2c1%2F7lp0x2f_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the standard matrix for the linear transformation \( T: \mathbb{R}^2 \to \mathbb{R}^2 \) that reflects points about the origin and then rotates points about the origin by \(\frac{3\pi}{4}\) radians.
**Solution Approach:**
To solve this problem, two key operations are involved:
1. **Reflection about the Origin:**
- The matrix for reflecting points about the origin in \(\mathbb{R}^2\) is:
\[
R = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}
\]
2. **Rotation by \(\frac{3\pi}{4}\) Radians:**
- The matrix for rotating points about the origin by an angle \(\theta\) is:
\[
M(\theta) = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix}
\]
- Here, \(\theta = \frac{3\pi}{4}\), so:
\[
M\left(\frac{3\pi}{4}\right) = \begin{bmatrix} \cos\left(\frac{3\pi}{4}\right) & -\sin\left(\frac{3\pi}{4}\right) \\ \sin\left(\frac{3\pi}{4}\right) & \cos\left(\frac{3\pi}{4}\right) \end{bmatrix} = \begin{bmatrix} -\frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \end{bmatrix}
\]
**Standard Matrix Calculation:**
The standard matrix for the transformation \( T \) is obtained by multiplying the matrices for reflection and rotation: \( T = M\left(\frac{3\pi}{4}\right) \times R \).
Thus, the standard matrix for \( T \) is:
\[
T = \begin{bmatrix} -\frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}
![Suppose \( T: \mathbb{R}^3 \to \mathbb{R}^2 \) is a linear transformation with \( T(x_1, x_2, x_3) = (-10x_2, -5x_1 + 11x_2 - 7x_3) \). Find the (standard) matrix \( A \) such that \( T(\mathbf{x}) = A\mathbf{x} \).
\[
\begin{bmatrix}
& & \\
& & \\
\end{bmatrix}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F18574973-f25e-4ab8-b7d6-6007b5b87fc4%2Fbc2aef9f-d91e-4fb4-ab28-424519d6b2c1%2F3uzxnxh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Suppose \( T: \mathbb{R}^3 \to \mathbb{R}^2 \) is a linear transformation with \( T(x_1, x_2, x_3) = (-10x_2, -5x_1 + 11x_2 - 7x_3) \). Find the (standard) matrix \( A \) such that \( T(\mathbf{x}) = A\mathbf{x} \).
\[
\begin{bmatrix}
& & \\
& & \\
\end{bmatrix}
\]
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