2. (a) (b) (c) Determine if the vectors are linearly independent. Justify your answer. and HNO 8 09-0 and 22 and 5 -3 5 3
2. (a) (b) (c) Determine if the vectors are linearly independent. Justify your answer. and HNO 8 09-0 and 22 and 5 -3 5 3
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![## Problem 2
### Determine if the vectors are linearly independent. Justify your answer.
#### (a)
\[ \begin{bmatrix} 11 \\ 2 \end{bmatrix}, \begin{bmatrix} -3 \\ 1 \end{bmatrix}, \text{ and } \begin{bmatrix} 0 \\ 1 \end{bmatrix} \]
#### (b)
\[ \begin{bmatrix} -1 \\ 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}, \text{ and } \begin{bmatrix} -1 \\ 2 \\ 3 \end{bmatrix} \]
#### (c)
\[ \begin{bmatrix} 2 \\ 2 \\ -3 \\ 4 \end{bmatrix} \text{ and } \begin{bmatrix} 5 \\ -3 \\ 5 \\ 1 \end{bmatrix} \]
### Explanation
To determine if the vectors are linearly independent, you need to form a matrix using the vectors as columns and compute the rank of the matrix. Here's a brief overview of what you'll need to do in each case:
#### Case (a):
You form the matrix:
\[ \begin{bmatrix}
11 & -3 & 0 \\
2 & 1 & 1
\end{bmatrix} \]
Check if the columns are linearly independent.
#### Case (b):
You form the matrix:
\[ \begin{bmatrix}
-1 & 1 & -1 \\
1 & 0 & 2 \\
2 & -1 & 3
\end{bmatrix} \]
Check if the columns are linearly independent.
#### Case (c):
You form the matrix:
\[ \begin{bmatrix}
2 & 5 \\
2 & -3 \\
-3 & 5 \\
4 & 1
\end{bmatrix} \]
Check if the columns are linearly independent.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F254fd25e-9eb6-4bee-adcd-079d468c7958%2F51a67454-c964-40b4-9c1b-d7024117c7b0%2Fcm7kd4h_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Problem 2
### Determine if the vectors are linearly independent. Justify your answer.
#### (a)
\[ \begin{bmatrix} 11 \\ 2 \end{bmatrix}, \begin{bmatrix} -3 \\ 1 \end{bmatrix}, \text{ and } \begin{bmatrix} 0 \\ 1 \end{bmatrix} \]
#### (b)
\[ \begin{bmatrix} -1 \\ 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}, \text{ and } \begin{bmatrix} -1 \\ 2 \\ 3 \end{bmatrix} \]
#### (c)
\[ \begin{bmatrix} 2 \\ 2 \\ -3 \\ 4 \end{bmatrix} \text{ and } \begin{bmatrix} 5 \\ -3 \\ 5 \\ 1 \end{bmatrix} \]
### Explanation
To determine if the vectors are linearly independent, you need to form a matrix using the vectors as columns and compute the rank of the matrix. Here's a brief overview of what you'll need to do in each case:
#### Case (a):
You form the matrix:
\[ \begin{bmatrix}
11 & -3 & 0 \\
2 & 1 & 1
\end{bmatrix} \]
Check if the columns are linearly independent.
#### Case (b):
You form the matrix:
\[ \begin{bmatrix}
-1 & 1 & -1 \\
1 & 0 & 2 \\
2 & -1 & 3
\end{bmatrix} \]
Check if the columns are linearly independent.
#### Case (c):
You form the matrix:
\[ \begin{bmatrix}
2 & 5 \\
2 & -3 \\
-3 & 5 \\
4 & 1
\end{bmatrix} \]
Check if the columns are linearly independent.
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