4. Consider V₁ = V₂ = (a) Show that V₁, V2, V3 are linearly independent. (b) Find the dimension of Span(V₁, V2, V3). (c) Is any vector (x, y, z) € R³ in Span(V₁, V2, V3)? If so, express the vector as a linear combination of V1, V2, V3.
4. Consider V₁ = V₂ = (a) Show that V₁, V2, V3 are linearly independent. (b) Find the dimension of Span(V₁, V2, V3). (c) Is any vector (x, y, z) € R³ in Span(V₁, V2, V3)? If so, express the vector as a linear combination of V1, V2, V3.
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
![## Problem 4
Consider the vectors:
\[
\mathbf{v}_1 = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}, \quad \mathbf{v}_2 = \begin{bmatrix} 5 \\ 6 \\ 4 \end{bmatrix}, \quad \mathbf{v}_3 = \begin{bmatrix} -3 \\ -7 \\ -1 \end{bmatrix}
\]
### Tasks:
(a) Show that \(\mathbf{v}_1\), \(\mathbf{v}_2\), \(\mathbf{v}_3\) are linearly independent.
(b) Find the dimension of \(\text{Span}(\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3)\).
(c) Is any vector \((x, y, z) \in \mathbb{R}^3\) in \(\text{Span}(\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3)\)? If so, express the vector as a linear combination of \(\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff6c1c23e-1f73-4b65-bca5-bb8e7ffd22c6%2F9a8a9d48-5f6c-4bf8-b70e-d1f5ab15fdcb%2Fpudwlqc_processed.png&w=3840&q=75)
Transcribed Image Text:## Problem 4
Consider the vectors:
\[
\mathbf{v}_1 = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}, \quad \mathbf{v}_2 = \begin{bmatrix} 5 \\ 6 \\ 4 \end{bmatrix}, \quad \mathbf{v}_3 = \begin{bmatrix} -3 \\ -7 \\ -1 \end{bmatrix}
\]
### Tasks:
(a) Show that \(\mathbf{v}_1\), \(\mathbf{v}_2\), \(\mathbf{v}_3\) are linearly independent.
(b) Find the dimension of \(\text{Span}(\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3)\).
(c) Is any vector \((x, y, z) \in \mathbb{R}^3\) in \(\text{Span}(\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3)\)? If so, express the vector as a linear combination of \(\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\).
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