Find the matrices of the following linear transformations from R³ to R³. The orthogonal projection onto the y-axis: 393
Find the matrices of the following linear transformations from R³ to R³. The orthogonal projection onto the y-axis: 393
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
more details,
![**Title: Understanding Linear Transformations and Orthogonal Projections in \( \mathbb{R}^3 \)**
**Introduction:**
This lesson focuses on finding matrices that represent specific linear transformations in three-dimensional space, denoted as \( \mathbb{R}^3 \). We will explore the concept of orthogonal projection onto the y-axis.
**Problem Statement:**
Find the matrices of the following linear transformations from \( \mathbb{R}^3 \) to \( \mathbb{R}^3 \).
**Task:**
- Determine the matrix representing the orthogonal projection onto the y-axis.
**Matrix Representation:**
To represent an orthogonal projection onto the y-axis in \( \mathbb{R}^3 \), consider the following structure of a 3x3 matrix:
\[
\begin{bmatrix}
0 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 0
\end{bmatrix}
\]
**Explanation:**
The above matrix transformation projects any vector in \( \mathbb{R}^3 \) onto the y-axis. This means that the output vector will have its x and z components set to zero, while the y component remains unchanged.
**Conclusion:**
Using this matrix, any vector \(\begin{bmatrix} x \\ y \\ z \end{bmatrix}\) in \( \mathbb{R}^3 \) will be transformed to \(\begin{bmatrix} 0 \\ y \\ 0 \end{bmatrix}\), effectively projecting it onto the y-axis. Understanding these transformations is essential for applications in geometry, computer graphics, and data science.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F584ffa08-94ad-4eb4-a87d-fd258db9f802%2F89c60429-9f7d-4156-85d0-eff5c0f4fa72%2Fg24uzfq_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Understanding Linear Transformations and Orthogonal Projections in \( \mathbb{R}^3 \)**
**Introduction:**
This lesson focuses on finding matrices that represent specific linear transformations in three-dimensional space, denoted as \( \mathbb{R}^3 \). We will explore the concept of orthogonal projection onto the y-axis.
**Problem Statement:**
Find the matrices of the following linear transformations from \( \mathbb{R}^3 \) to \( \mathbb{R}^3 \).
**Task:**
- Determine the matrix representing the orthogonal projection onto the y-axis.
**Matrix Representation:**
To represent an orthogonal projection onto the y-axis in \( \mathbb{R}^3 \), consider the following structure of a 3x3 matrix:
\[
\begin{bmatrix}
0 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 0
\end{bmatrix}
\]
**Explanation:**
The above matrix transformation projects any vector in \( \mathbb{R}^3 \) onto the y-axis. This means that the output vector will have its x and z components set to zero, while the y component remains unchanged.
**Conclusion:**
Using this matrix, any vector \(\begin{bmatrix} x \\ y \\ z \end{bmatrix}\) in \( \mathbb{R}^3 \) will be transformed to \(\begin{bmatrix} 0 \\ y \\ 0 \end{bmatrix}\), effectively projecting it onto the y-axis. Understanding these transformations is essential for applications in geometry, computer graphics, and data science.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
