Find the null space associated with the equations: 2x+z+w = 0 7x-6y–z–w-v = 0 -5x+4y+w = 0
Find the null space associated with the equations: 2x+z+w = 0 7x-6y–z–w-v = 0 -5x+4y+w = 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
![**Problem Statement:**
Find the null space associated with the equations:
1. \(2x + z + w = 0\)
2. \(7x - 6y - z - w - v = 0\)
3. \(-5x + 4y + w = 0\)
To solve this problem, transform these equations into a matrix equation and find the basis for the null space. The null space represents all possible solutions \((x, y, z, w, v)\) to the equations.
**Matrix Representation:**
The system can be written in matrix form \(A \mathbf{x} = \mathbf{0}\), where:
\[
A = \begin{bmatrix}
2 & 0 & 1 & 1 & 0 \\
7 & -6 & -1 & -1 & -1 \\
-5 & 4 & 0 & 1 & 0
\end{bmatrix}
\]
\(\mathbf{x} = \begin{bmatrix} x \\ y \\ z \\ w \\ v \end{bmatrix}\)
\(\mathbf{0} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}\)
The null space is the set of all vectors \(\mathbf{x}\) such that \(A \mathbf{x} = \mathbf{0}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F80a13d8c-5188-4010-ba23-710cc489cc25%2Ff54c9458-0fc3-4e8a-81ee-d65cc843c61f%2Fmp65bjz_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the null space associated with the equations:
1. \(2x + z + w = 0\)
2. \(7x - 6y - z - w - v = 0\)
3. \(-5x + 4y + w = 0\)
To solve this problem, transform these equations into a matrix equation and find the basis for the null space. The null space represents all possible solutions \((x, y, z, w, v)\) to the equations.
**Matrix Representation:**
The system can be written in matrix form \(A \mathbf{x} = \mathbf{0}\), where:
\[
A = \begin{bmatrix}
2 & 0 & 1 & 1 & 0 \\
7 & -6 & -1 & -1 & -1 \\
-5 & 4 & 0 & 1 & 0
\end{bmatrix}
\]
\(\mathbf{x} = \begin{bmatrix} x \\ y \\ z \\ w \\ v \end{bmatrix}\)
\(\mathbf{0} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}\)
The null space is the set of all vectors \(\mathbf{x}\) such that \(A \mathbf{x} = \mathbf{0}\).
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 6 steps with 7 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,
