Determine true or false. are linearly independent. True False 1 2 ··· 2 0 -3 The vectors
Determine true or false. are linearly independent. True False 1 2 ··· 2 0 -3 The vectors
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
![### Linear Independence of Vectors
**Determine true or false.**
The vectors
\[
\begin{pmatrix}
1 \\
2 \\
-3
\end{pmatrix}, \quad
\begin{pmatrix}
2 \\
0 \\
-1
\end{pmatrix}, \quad
\begin{pmatrix}
0 \\
4 \\
-5
\end{pmatrix}
\]
are linearly independent.
**Options:**
- ◯ True
- ◯ False
**Explanation:**
To determine whether the given vectors are linearly independent, we can set up a matrix with these vectors as columns and perform row reduction to see if we get a pivot (leading 1) in every column. If there are pivots in every column, the vectors are linearly independent; otherwise, they are linearly dependent.
This concept is crucial in linear algebra and has applications in various fields including computer science, physics, and economics. Linear independence ensures that no vector in the set can be written as a linear combination of the others, indicating that they span a space with maximum dimension given the number of vectors.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc353cc0f-0c4d-4d5b-a3e6-e746dbb90f94%2F2a0c13ed-552f-48d0-8f5d-be38972e40d9%2Fsm7r78q_processed.png&w=3840&q=75)
Transcribed Image Text:### Linear Independence of Vectors
**Determine true or false.**
The vectors
\[
\begin{pmatrix}
1 \\
2 \\
-3
\end{pmatrix}, \quad
\begin{pmatrix}
2 \\
0 \\
-1
\end{pmatrix}, \quad
\begin{pmatrix}
0 \\
4 \\
-5
\end{pmatrix}
\]
are linearly independent.
**Options:**
- ◯ True
- ◯ False
**Explanation:**
To determine whether the given vectors are linearly independent, we can set up a matrix with these vectors as columns and perform row reduction to see if we get a pivot (leading 1) in every column. If there are pivots in every column, the vectors are linearly independent; otherwise, they are linearly dependent.
This concept is crucial in linear algebra and has applications in various fields including computer science, physics, and economics. Linear independence ensures that no vector in the set can be written as a linear combination of the others, indicating that they span a space with maximum dimension given the number of vectors.
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