5. Let H = {(a - 3b, b-a, a, b): a, b E R}. Show that H is a subspace of R¹.
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
Practice Question
Number 5.
![**Subspace Verification Example**
**Problem Statement:**
Let \( H = \{(a-3b, b-a, a, b) : a, b \in \mathbb{R}\} \). Show that \( H \) is a subspace of \( \mathbb{R}^4 \).
**Solution Steps:**
To demonstrate that \( H \) is a subspace of \( \mathbb{R}^4 \), we need to verify that it satisfies three main properties:
1. **The Zero Vector is in \( H \)**:
We need to show that the zero vector \( \vec{0} = (0, 0, 0, 0) \) is an element of \( H \).
Evaluate \( (a-3b, b-a, a, b) \) for \( a = 0 \) and \( b = 0 \):
\[
(0-3 \cdot 0, 0-0, 0, 0) = (0, 0, 0, 0)
\]
The zero vector is indeed in \( H \).
2. **Closed under Addition**:
For any two vectors \( \vec{u} = (a - 3b, b - a, a, b) \) and \( \vec{v} = (c - 3d, d - c, c, d) \) in \( H \), their sum should also be in \( H \).
Consider \( \vec{u} + \vec{v} \):
\[
(a - 3b, b - a, a, b) + (c - 3d, d - c, c, d) = ((a - 3b) + (c - 3d), (b - a) + (d - c), a + c, b + d)
\]
We need to verify that this can be written in the form \( (e - 3f, f - e, e, f) \):
Let \( e = a + c \) and \( f = b + d \):
\[
(a + c - 3(b + d), (b + d) - (a + c), a + c, b + d) = (e](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F13d3a07d-24ab-4d07-ab36-ffae21b1955b%2F77357a09-fa82-4245-afca-be0aab9f7189%2Flulccs_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Subspace Verification Example**
**Problem Statement:**
Let \( H = \{(a-3b, b-a, a, b) : a, b \in \mathbb{R}\} \). Show that \( H \) is a subspace of \( \mathbb{R}^4 \).
**Solution Steps:**
To demonstrate that \( H \) is a subspace of \( \mathbb{R}^4 \), we need to verify that it satisfies three main properties:
1. **The Zero Vector is in \( H \)**:
We need to show that the zero vector \( \vec{0} = (0, 0, 0, 0) \) is an element of \( H \).
Evaluate \( (a-3b, b-a, a, b) \) for \( a = 0 \) and \( b = 0 \):
\[
(0-3 \cdot 0, 0-0, 0, 0) = (0, 0, 0, 0)
\]
The zero vector is indeed in \( H \).
2. **Closed under Addition**:
For any two vectors \( \vec{u} = (a - 3b, b - a, a, b) \) and \( \vec{v} = (c - 3d, d - c, c, d) \) in \( H \), their sum should also be in \( H \).
Consider \( \vec{u} + \vec{v} \):
\[
(a - 3b, b - a, a, b) + (c - 3d, d - c, c, d) = ((a - 3b) + (c - 3d), (b - a) + (d - c), a + c, b + d)
\]
We need to verify that this can be written in the form \( (e - 3f, f - e, e, f) \):
Let \( e = a + c \) and \( f = b + d \):
\[
(a + c - 3(b + d), (b + d) - (a + c), a + c, b + d) = (e
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,
