2. Determine whether the set S = {(2,1,0), (-1,0,1), (8,3,-2) } is linearly independent or linearly dependent. -1 C 3 J 8 ㄱ linearly independent Does the set S form a basis for R³? Yes
2. Determine whether the set S = {(2,1,0), (-1,0,1), (8,3,-2) } is linearly independent or linearly dependent. -1 C 3 J 8 ㄱ linearly independent Does the set S form a basis for R³? Yes
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 2: Linear Independence and Basis**
**Question:**
Determine whether the set \( S = \{ (2,1,0), (-1,0,1), (8,3,-2) \} \) is linearly independent or linearly dependent.
**Solution:**
The solution involves analyzing the matrix formed by the vectors in \( S \):
\[
\begin{bmatrix}
2 & -1 & 8 \\
1 & 0 & 3 \\
0 & 1 & -2 \\
\end{bmatrix}
\]
Illustration shows these vectors being checked for linear independence.
Steps:
1. Write the vectors as columns of a matrix.
2. Reduce the matrix to row echelon form.
In the diagram, there are markings and a cross-out indicating analysis, with a question mark and a subtraction noted (-7).
**Conclusion:**
The vectors are found to be "linearly independent."
**Additional Question:**
Does the set \( S \) form a basis for \( \mathbb{R}^3 \)?
**Conclusion:**
Yes, it does. This is confirmed by the note indicated as "yes."
The handwritten notes indicate analysis leading to these conclusions confidently, confirming that the vectors span \( \mathbb{R}^3 \) and are independent, hence forming a basis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faea2220e-5b79-4f4c-ba0b-cd87c88bb3af%2Fa66f86eb-e733-4fe5-8da9-6eeb98bffe04%2F330gi8n_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 2: Linear Independence and Basis**
**Question:**
Determine whether the set \( S = \{ (2,1,0), (-1,0,1), (8,3,-2) \} \) is linearly independent or linearly dependent.
**Solution:**
The solution involves analyzing the matrix formed by the vectors in \( S \):
\[
\begin{bmatrix}
2 & -1 & 8 \\
1 & 0 & 3 \\
0 & 1 & -2 \\
\end{bmatrix}
\]
Illustration shows these vectors being checked for linear independence.
Steps:
1. Write the vectors as columns of a matrix.
2. Reduce the matrix to row echelon form.
In the diagram, there are markings and a cross-out indicating analysis, with a question mark and a subtraction noted (-7).
**Conclusion:**
The vectors are found to be "linearly independent."
**Additional Question:**
Does the set \( S \) form a basis for \( \mathbb{R}^3 \)?
**Conclusion:**
Yes, it does. This is confirmed by the note indicated as "yes."
The handwritten notes indicate analysis leading to these conclusions confidently, confirming that the vectors span \( \mathbb{R}^3 \) and are independent, hence forming a basis.
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 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,
