2. Consider the following vectors 3 V1 = Vz = -2 V3 -4 2 a) Show that V1, V2 are linearly independent. b) Show that V1, V2, V3 are linearly dependent.
2. Consider the following vectors 3 V1 = Vz = -2 V3 -4 2 a) Show that V1, V2 are linearly independent. b) Show that V1, V2, V3 are linearly dependent.
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 and Dependence of Vectors
#### Given Vectors:
Consider the following vectors:
\[
\mathbf{v}_1 = \begin{bmatrix} 1 \\ 0 \\ 2 \end{bmatrix}, \quad \mathbf{v}_2 = \begin{bmatrix} 0 \\ -2 \\ 0 \end{bmatrix}, \quad \mathbf{v}_3 = \begin{bmatrix} 3 \\ -4 \\ 6 \end{bmatrix}
\]
#### Tasks:
a) Show that \(\mathbf{v}_1\) and \(\mathbf{v}_2\) are linearly independent.
b) Show that \(\mathbf{v}_1\), \(\mathbf{v}_2\), and \(\mathbf{v}_3\) are linearly dependent.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9fc36e3b-ed84-42cc-af8c-7e597dbc05f9%2F515093e6-1022-40f2-9c1d-ae20beb8651a%2F8jik0d8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Linear Independence and Dependence of Vectors
#### Given Vectors:
Consider the following vectors:
\[
\mathbf{v}_1 = \begin{bmatrix} 1 \\ 0 \\ 2 \end{bmatrix}, \quad \mathbf{v}_2 = \begin{bmatrix} 0 \\ -2 \\ 0 \end{bmatrix}, \quad \mathbf{v}_3 = \begin{bmatrix} 3 \\ -4 \\ 6 \end{bmatrix}
\]
#### Tasks:
a) Show that \(\mathbf{v}_1\) and \(\mathbf{v}_2\) are linearly independent.
b) Show that \(\mathbf{v}_1\), \(\mathbf{v}_2\), and \(\mathbf{v}_3\) are linearly dependent.
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