Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
expand_more
expand_more
format_list_bulleted
Bartleby Related Questions Icon
Related questions
Question
? Solve
![画像:**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}}](https://content.bartleby.com/qna-images/question/18574973-f25e-4ab8-b7d6-6007b5b87fc4/bc2aef9f-d91e-4fb4-ab28-424519d6b2c1/7lp0x2f_thumbnail.jpeg){kind=link}
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}}
![画像: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 \( T(\mathbf{x}) = A\mathbf{x} \). \[ \begin{bmatrix} & & \\ & & \\ \end{bmatrix} \]](https://content.bartleby.com/qna-images/question/18574973-f25e-4ab8-b7d6-6007b5b87fc4/bc2aef9f-d91e-4fb4-ab28-424519d6b2c1/3uzxnxh_thumbnail.jpeg){kind=link}
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 \( T(\mathbf{x}) = A\mathbf{x} \).
\[
\begin{bmatrix}
& & \\
& & \\
\end{bmatrix}
\]
Expert Solution
Check MarkThis question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
bartleby
Step by stepSolved in 4 steps with 9 images
Knowledge Booster
Background pattern image
Similar questions
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
- Text book imageAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, IncorporatedText book imageNumerical Methods for EngineersAdvanced MathISBN:9780073397924Author:Steven C. Chapra Dr., Raymond P. CanalePublisher:McGraw-Hill EducationText book imageIntroductory Mathematics for Engineering Applicat...Advanced MathISBN:9781118141809Author:Nathan KlingbeilPublisher:WILEY
- Text book imageMathematics For Machine TechnologyAdvanced MathISBN:9781337798310Author:Peterson, John.Publisher:Cengage Learning,Text book imageText book image
Text book image
Advanced Engineering Mathematics
Advanced Math
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Wiley, John & Sons, Incorporated
Text book image
Numerical Methods for Engineers
Advanced Math
ISBN:9780073397924
Author:Steven C. Chapra Dr., Raymond P. Canale
Publisher:McGraw-Hill Education
Text book image
Introductory Mathematics for Engineering Applicat...
Advanced Math
ISBN:9781118141809
Author:Nathan Klingbeil
Publisher:WILEY
Text book image
Mathematics For Machine Technology
Advanced Math
ISBN:9781337798310
Author:Peterson, John.
Publisher:Cengage Learning,
Text book image
Text book image