31 5.4. u = -5 -F 0 -1 2 2 pendent in R4? If not find an explicit non-trivial linear relationship between them. V = -1 and w = Are the vectors u, v, w linearly inde-
31 5.4. u = -5 -F 0 -1 2 2 pendent in R4? If not find an explicit non-trivial linear relationship between them. V = -1 and w = Are the vectors u, v, w linearly inde-
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
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![### Linear Independence of Vectors
**Question 5.4**: Given the vectors
\[ \mathbf{u} = \begin{bmatrix} 3 \\ -1 \\ 2 \\ 2 \end{bmatrix}, \mathbf{v} = \begin{bmatrix} 1 \\ -5 \\ 0 \\ 4 \end{bmatrix}, \text{ and } \mathbf{w} = \begin{bmatrix} 4 \\ 1 \\ 3 \\ 1 \end{bmatrix}, \]
are the vectors \(\mathbf{u}, \mathbf{v}, \mathbf{w}\) linearly independent in \(\mathbb{R}^4\)? If not, find an explicit non-trivial linear relationship between them.
**Explanation**:
- The columns represent the components of each vector in \(\mathbb{R}^4\).
- To determine if the vectors \(\mathbf{u}, \mathbf{v}, \mathbf{w}\) are linearly independent, we need to check if there exists a non-trivial linear combination of these vectors that equals zero:
\[ a\mathbf{u} + b\mathbf{v} + c\mathbf{w} = \mathbf{0} \]
where \(\mathbf{0}\) is the zero vector in \(\mathbb{R}^4\), and \(a, b, c\) are scalars.
To check for linear independence:
1. Set up the augmented matrix:
\[
\left[\begin{array}{ccc|c}
3 & 1 & 4 & 0 \\
-1 & -5 & 1 & 0 \\
2 & 0 & 3 & 0 \\
2 & 4 & 1 & 0
\end{array}\right]
\]
2. Perform row operations to reduce this matrix to row echelon form (REF) or reduced row echelon form (RREF).
3. If the matrix reduces to a row with all zeros in the vector columns but a non-zero entry in the augmented column (i.e., an inconsistency), the vectors are linearly dependent. If not, we conclude they are independent.
If the vectors are not independent, the values of \(a, b, c\) will provide an explicit non-trivial linear relationship between \(\mathbf{u}, \mathbf](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe5147bf2-ddd5-4450-99d9-d8f97b210b53%2F0f1d4060-d110-4392-9f83-020bf6a3b52b%2F9q7x9si_processed.png&w=3840&q=75)
Transcribed Image Text:### Linear Independence of Vectors
**Question 5.4**: Given the vectors
\[ \mathbf{u} = \begin{bmatrix} 3 \\ -1 \\ 2 \\ 2 \end{bmatrix}, \mathbf{v} = \begin{bmatrix} 1 \\ -5 \\ 0 \\ 4 \end{bmatrix}, \text{ and } \mathbf{w} = \begin{bmatrix} 4 \\ 1 \\ 3 \\ 1 \end{bmatrix}, \]
are the vectors \(\mathbf{u}, \mathbf{v}, \mathbf{w}\) linearly independent in \(\mathbb{R}^4\)? If not, find an explicit non-trivial linear relationship between them.
**Explanation**:
- The columns represent the components of each vector in \(\mathbb{R}^4\).
- To determine if the vectors \(\mathbf{u}, \mathbf{v}, \mathbf{w}\) are linearly independent, we need to check if there exists a non-trivial linear combination of these vectors that equals zero:
\[ a\mathbf{u} + b\mathbf{v} + c\mathbf{w} = \mathbf{0} \]
where \(\mathbf{0}\) is the zero vector in \(\mathbb{R}^4\), and \(a, b, c\) are scalars.
To check for linear independence:
1. Set up the augmented matrix:
\[
\left[\begin{array}{ccc|c}
3 & 1 & 4 & 0 \\
-1 & -5 & 1 & 0 \\
2 & 0 & 3 & 0 \\
2 & 4 & 1 & 0
\end{array}\right]
\]
2. Perform row operations to reduce this matrix to row echelon form (REF) or reduced row echelon form (RREF).
3. If the matrix reduces to a row with all zeros in the vector columns but a non-zero entry in the augmented column (i.e., an inconsistency), the vectors are linearly dependent. If not, we conclude they are independent.
If the vectors are not independent, the values of \(a, b, c\) will provide an explicit non-trivial linear relationship between \(\mathbf{u}, \mathbf
![### Linear Dependence in Vector Spaces
**Exercise 5.4**
**Q:** Are the given vectors independent or dependent?
**A:** No, they are dependent. The linear equation representing the dependence is:
\[3u - v - 2w = 0\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe5147bf2-ddd5-4450-99d9-d8f97b210b53%2F0f1d4060-d110-4392-9f83-020bf6a3b52b%2Fdun89fs_processed.png&w=3840&q=75)
Transcribed Image Text:### Linear Dependence in Vector Spaces
**Exercise 5.4**
**Q:** Are the given vectors independent or dependent?
**A:** No, they are dependent. The linear equation representing the dependence is:
\[3u - v - 2w = 0\]
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