### Linear Transformation Matrix Problem **Problem Statement:** Which of the matrices below represents the linear transformation from \( \mathbb{R}^2 \) to \( \mathbb{R}^2 \) which rotates points an angle of \( \frac{\pi}{4} \) counterclockwise and then reflects across the y-axis. **Options:** **A:** \[ \begin{pmatrix} \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ -\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{pmatrix} \] **B:** \[ \begin{pmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{pmatrix} \] **C:** \[ \begin{pmatrix} -\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{pmatrix} \] **D:** \[ \begin{pmatrix} -\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ -\frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \end{pmatrix} \] **Explanation:** To find which matrix represents the transformation, we need to combine the rotation and reflection transformations. The matrix for rotating points by \( \frac{\pi}{4} \) counterclockwise is: \[ R_{\frac{\pi}{4}} = \begin{pmatrix} \cos \frac{\pi}{4} & -\sin \frac{\pi}{4} \\ \sin \frac{\pi}{4} & \cos \frac{\pi}{4} \end{pmatrix} = \begin{pmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{pmatrix} \]
### Linear Transformation Matrix Problem **Problem Statement:** Which of the matrices below represents the linear transformation from \( \mathbb{R}^2 \) to \( \mathbb{R}^2 \) which rotates points an angle of \( \frac{\pi}{4} \) counterclockwise and then reflects across the y-axis. **Options:** **A:** \[ \begin{pmatrix} \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ -\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{pmatrix} \] **B:** \[ \begin{pmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{pmatrix} \] **C:** \[ \begin{pmatrix} -\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{pmatrix} \] **D:** \[ \begin{pmatrix} -\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ -\frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \end{pmatrix} \] **Explanation:** To find which matrix represents the transformation, we need to combine the rotation and reflection transformations. The matrix for rotating points by \( \frac{\pi}{4} \) counterclockwise is: \[ R_{\frac{\pi}{4}} = \begin{pmatrix} \cos \frac{\pi}{4} & -\sin \frac{\pi}{4} \\ \sin \frac{\pi}{4} & \cos \frac{\pi}{4} \end{pmatrix} = \begin{pmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{pmatrix} \]
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 Matrix Problem
**Problem Statement:**
Which of the matrices below represents the linear transformation from \( \mathbb{R}^2 \) to \( \mathbb{R}^2 \) which rotates points an angle of \( \frac{\pi}{4} \) counterclockwise and then reflects across the y-axis.
**Options:**
**A:**
\[
\begin{pmatrix}
\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\
-\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2}
\end{pmatrix}
\]
**B:**
\[
\begin{pmatrix}
\frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\
\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2}
\end{pmatrix}
\]
**C:**
\[
\begin{pmatrix}
-\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\
\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2}
\end{pmatrix}
\]
**D:**
\[
\begin{pmatrix}
-\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\
-\frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2}
\end{pmatrix}
\]
**Explanation:**
To find which matrix represents the transformation, we need to combine the rotation and reflection transformations. The matrix for rotating points by \( \frac{\pi}{4} \) counterclockwise is:
\[
R_{\frac{\pi}{4}} = \begin{pmatrix}
\cos \frac{\pi}{4} & -\sin \frac{\pi}{4} \\
\sin \frac{\pi}{4} & \cos \frac{\pi}{4}
\end{pmatrix} = \begin{pmatrix}
\frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\
\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2}
\end{pmatrix}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3ed4e6f2-ec2f-4b59-835f-0c83565bb723%2F442d3221-6866-4ab1-842f-16b69f429a9b%2F4g3my9d_processed.png&w=3840&q=75)
Transcribed Image Text:### Linear Transformation Matrix Problem
**Problem Statement:**
Which of the matrices below represents the linear transformation from \( \mathbb{R}^2 \) to \( \mathbb{R}^2 \) which rotates points an angle of \( \frac{\pi}{4} \) counterclockwise and then reflects across the y-axis.
**Options:**
**A:**
\[
\begin{pmatrix}
\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\
-\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2}
\end{pmatrix}
\]
**B:**
\[
\begin{pmatrix}
\frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\
\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2}
\end{pmatrix}
\]
**C:**
\[
\begin{pmatrix}
-\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\
\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2}
\end{pmatrix}
\]
**D:**
\[
\begin{pmatrix}
-\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\
-\frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2}
\end{pmatrix}
\]
**Explanation:**
To find which matrix represents the transformation, we need to combine the rotation and reflection transformations. The matrix for rotating points by \( \frac{\pi}{4} \) counterclockwise is:
\[
R_{\frac{\pi}{4}} = \begin{pmatrix}
\cos \frac{\pi}{4} & -\sin \frac{\pi}{4} \\
\sin \frac{\pi}{4} & \cos \frac{\pi}{4}
\end{pmatrix} = \begin{pmatrix}
\frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\
\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2}
\end{pmatrix}
\]
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