Are the vectors V₁ = 2 1 1 V₂ = 1₂V3 3 4 1 linearly independent? -9
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
![**Are the vectors \( v_1 \), \( v_2 \), \( v_3 \) linearly independent?**
Vectors:
\[ v_1 = \begin{bmatrix}2 \\ 1 \\ -1\end{bmatrix} \]
\[ v_2 = \begin{bmatrix}1 \\ 1 \\ 3\end{bmatrix} \]
\[ v_3 = \begin{bmatrix}4 \\ 1 \\ -9\end{bmatrix} \]
To determine if \( v_1 \), \( v_2 \), and \( v_3 \) are linearly independent, you can form a matrix with these vectors as columns and row reduce it to see if the determinant is non-zero or if the row-reduced echelon form yields a matrix with a pivot in every row. If either of these conditions is true, the vectors are linearly independent; otherwise, they are linearly dependent.
Set up the matrix:
\[ A = \begin{bmatrix}2 & 1 & 4 \\ 1 & 1 & 1 \\ -1 & 3 & -9\end{bmatrix} \]
Row reduce the matrix \( A \) to its echelon form or compute its determinant for further analysis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff869c4ac-8e58-4431-846d-0202389eaebe%2F84f126d2-313b-45df-8b22-29821f7fb328%2Fceiklb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Are the vectors \( v_1 \), \( v_2 \), \( v_3 \) linearly independent?**
Vectors:
\[ v_1 = \begin{bmatrix}2 \\ 1 \\ -1\end{bmatrix} \]
\[ v_2 = \begin{bmatrix}1 \\ 1 \\ 3\end{bmatrix} \]
\[ v_3 = \begin{bmatrix}4 \\ 1 \\ -9\end{bmatrix} \]
To determine if \( v_1 \), \( v_2 \), and \( v_3 \) are linearly independent, you can form a matrix with these vectors as columns and row reduce it to see if the determinant is non-zero or if the row-reduced echelon form yields a matrix with a pivot in every row. If either of these conditions is true, the vectors are linearly independent; otherwise, they are linearly dependent.
Set up the matrix:
\[ A = \begin{bmatrix}2 & 1 & 4 \\ 1 & 1 & 1 \\ -1 & 3 & -9\end{bmatrix} \]
Row reduce the matrix \( A \) to its echelon form or compute its determinant for further analysis.
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,
