Your task is to determine if each given set of vectors is LI or LD. Select True if the set is Linearly independent and false if it is not. The reduced row echelon form of a matrix with columns, and is given. (a) Vi 9 V2 = 2 and v3 = -2 31 9 rref 1 2 -2 (21 5 (10 4 0 1 -3 (00 0
Your task is to determine if each given set of vectors is LI or LD. Select True if the set is Linearly independent and false if it is not. The reduced row echelon form of a matrix with columns, and is given. (a) Vi 9 V2 = 2 and v3 = -2 31 9 rref 1 2 -2 (21 5 (10 4 0 1 -3 (00 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
![Your task is to determine if each given set of vectors is LI or LD. Select True if the set is Linearly independent and False if it is not. The reduced row echelon form of a matrix with columns, and is given.
(a)
\[
\mathbf{v}_1 = \begin{pmatrix} 3 \\ 1 \\ 2 \end{pmatrix}, \quad \mathbf{v}_2 = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix} \quad \text{and} \quad \mathbf{v}_3 = \begin{pmatrix} 9 \\ -2 \\ 5 \end{pmatrix}
\]
\[
\begin{pmatrix} 3 & 1 & 9 \\ 1 & 2 & -2 \\ 2 & 1 & 5 \end{pmatrix}
\xrightarrow{\text{rref}}
\begin{pmatrix} 1 & 0 & 4 \\ 0 & 1 & -3 \\ 0 & 0 & 0 \end{pmatrix}
\]
The set \(\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}\) is a linearly independent set. _______
- ○ True
- ○ False
Explanation:
This problem involves checking if a set of vectors is linearly independent. It starts by listing the vectors \(\mathbf{v}_1\), \(\mathbf{v}_2\), and \(\mathbf{v}_3\). The vectors are combined into a matrix, and the row reduction (reduced row echelon form) of this matrix is provided. The final matrix has a zero row which indicates linear dependence, so the answer is "False".](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9c49f70f-02c0-4076-b0a5-564b9a6f1d8d%2F15cf3107-9d17-4cc4-b65a-e1132a045f4a%2Fsdp2bx_processed.png&w=3840&q=75)
Transcribed Image Text:Your task is to determine if each given set of vectors is LI or LD. Select True if the set is Linearly independent and False if it is not. The reduced row echelon form of a matrix with columns, and is given.
(a)
\[
\mathbf{v}_1 = \begin{pmatrix} 3 \\ 1 \\ 2 \end{pmatrix}, \quad \mathbf{v}_2 = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix} \quad \text{and} \quad \mathbf{v}_3 = \begin{pmatrix} 9 \\ -2 \\ 5 \end{pmatrix}
\]
\[
\begin{pmatrix} 3 & 1 & 9 \\ 1 & 2 & -2 \\ 2 & 1 & 5 \end{pmatrix}
\xrightarrow{\text{rref}}
\begin{pmatrix} 1 & 0 & 4 \\ 0 & 1 & -3 \\ 0 & 0 & 0 \end{pmatrix}
\]
The set \(\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}\) is a linearly independent set. _______
- ○ True
- ○ False
Explanation:
This problem involves checking if a set of vectors is linearly independent. It starts by listing the vectors \(\mathbf{v}_1\), \(\mathbf{v}_2\), and \(\mathbf{v}_3\). The vectors are combined into a matrix, and the row reduction (reduced row echelon form) of this matrix is provided. The final matrix has a zero row which indicates linear dependence, so the answer is "False".
![Your task is to determine if each given set of vectors is LI or LD. Select True if the set is linearly independent and False if it is not. The reduced row echelon form of a matrix with columns is given.
Vectors:
\[
\mathbf{v}_1 = \begin{pmatrix} 2 \\ 1 \\ 3 \end{pmatrix}, \quad \mathbf{v}_2 = \begin{pmatrix} -2 \\ -5 \\ 2 \end{pmatrix}, \quad \mathbf{v}_3 = \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}
\]
Matrix and its reduced row echelon form (rref):
\[
\begin{pmatrix} 1 & -2 & 1 \\ 2 & -5 & 2 \\ 3 & 2 & -1 \end{pmatrix} \rightarrow \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}
\]
The set \(\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}\) is a linearly independent set. _________
- ○ True
- ○ False](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9c49f70f-02c0-4076-b0a5-564b9a6f1d8d%2F15cf3107-9d17-4cc4-b65a-e1132a045f4a%2F4bo446_processed.png&w=3840&q=75)
Transcribed Image Text:Your task is to determine if each given set of vectors is LI or LD. Select True if the set is linearly independent and False if it is not. The reduced row echelon form of a matrix with columns is given.
Vectors:
\[
\mathbf{v}_1 = \begin{pmatrix} 2 \\ 1 \\ 3 \end{pmatrix}, \quad \mathbf{v}_2 = \begin{pmatrix} -2 \\ -5 \\ 2 \end{pmatrix}, \quad \mathbf{v}_3 = \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}
\]
Matrix and its reduced row echelon form (rref):
\[
\begin{pmatrix} 1 & -2 & 1 \\ 2 & -5 & 2 \\ 3 & 2 & -1 \end{pmatrix} \rightarrow \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}
\]
The set \(\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}\) is a linearly independent set. _________
- ○ True
- ○ False
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