Find the orthogonal projection of v = projy(v) = [8] -25 H -7 onto the subspace V of R³ spanned by 24 Q-O -4 and 0
Find the orthogonal projection of v = projy(v) = [8] -25 H -7 onto the subspace V of R³ spanned by 24 Q-O -4 and 0
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
100%
![### Orthogonal Projection in Vector Spaces
In this section, we will discuss how to find the orthogonal projection of a vector onto a subspace.
**Problem Statement:**
Given a vector \( \mathbf{v} \) and a subspace \( V \) of \( \mathbb{R}^3 \), find the orthogonal projection of \( \mathbf{v} \) onto \( V \).
**Given Data:**
- Vector \( \mathbf{v} \):
\[
\mathbf{v} = \begin{bmatrix}
-25 \\
-7 \\
24
\end{bmatrix}
\]
- Subspace \( V \) of \( \mathbb{R}^3 \), spanned by:
\[
\begin{bmatrix}
1 \\
-4 \\
-2
\end{bmatrix}
\quad \text{and} \quad
\begin{bmatrix}
2 \\
0 \\
7
\end{bmatrix}
\]
**Objective:**
To find \( \text{proj}_V \mathbf{v} \):
\[
\text{proj}_V \mathbf{v} = \begin{bmatrix}
\boxed{} \\
\boxed{} \\
\boxed{}
\end{bmatrix}
.
\]
**Solution Outline:**
1. **Identify the Basis Vectors of the Subspace:**
Let \( \mathbf{u}_1 = \begin{bmatrix}
1 \\
-4 \\
-2
\end{bmatrix} \)
and \( \mathbf{u}_2 = \begin{bmatrix}
2 \\
0 \\
7
\end{bmatrix} \).
2. **Compute the Projection of \( \mathbf{v} \) onto the Span of \( \mathbf{u}_1 \) and \( \mathbf{u}_2 \):**
The orthogonal projection \( \text{proj}_V \mathbf{v} \) is the sum of the projections of \( \mathbf{v} \) onto each of the basis vectors \( \mathbf{u}_1 \) and \( \mathbf{u}_2 \):
\[
\text{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffb2133c9-e1e5-4d56-9c72-044227328930%2Fabcdb115-1dbb-4558-9321-325c0e870199%2Fmxumg1v_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Orthogonal Projection in Vector Spaces
In this section, we will discuss how to find the orthogonal projection of a vector onto a subspace.
**Problem Statement:**
Given a vector \( \mathbf{v} \) and a subspace \( V \) of \( \mathbb{R}^3 \), find the orthogonal projection of \( \mathbf{v} \) onto \( V \).
**Given Data:**
- Vector \( \mathbf{v} \):
\[
\mathbf{v} = \begin{bmatrix}
-25 \\
-7 \\
24
\end{bmatrix}
\]
- Subspace \( V \) of \( \mathbb{R}^3 \), spanned by:
\[
\begin{bmatrix}
1 \\
-4 \\
-2
\end{bmatrix}
\quad \text{and} \quad
\begin{bmatrix}
2 \\
0 \\
7
\end{bmatrix}
\]
**Objective:**
To find \( \text{proj}_V \mathbf{v} \):
\[
\text{proj}_V \mathbf{v} = \begin{bmatrix}
\boxed{} \\
\boxed{} \\
\boxed{}
\end{bmatrix}
.
\]
**Solution Outline:**
1. **Identify the Basis Vectors of the Subspace:**
Let \( \mathbf{u}_1 = \begin{bmatrix}
1 \\
-4 \\
-2
\end{bmatrix} \)
and \( \mathbf{u}_2 = \begin{bmatrix}
2 \\
0 \\
7
\end{bmatrix} \).
2. **Compute the Projection of \( \mathbf{v} \) onto the Span of \( \mathbf{u}_1 \) and \( \mathbf{u}_2 \):**
The orthogonal projection \( \text{proj}_V \mathbf{v} \) is the sum of the projections of \( \mathbf{v} \) onto each of the basis vectors \( \mathbf{u}_1 \) and \( \mathbf{u}_2 \):
\[
\text{
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 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,
