3. Let u 1 V= 0 1 1₁~= | W = a and x = b с 0 (a) Are vectors U, V, and w linearly independent? (b) Do u, v', and w span the entire R³? (c) Do u, v', and w form a basis of R³? ex to
3. Let u 1 V= 0 1 1₁~= | W = a and x = b с 0 (a) Are vectors U, V, and w linearly independent? (b) Do u, v', and w span the entire R³? (c) Do u, v', and w form a basis of R³? ex to
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
![### Linear Algebra Problem Set
Consider the following vectors in \(\mathbb{R}^3\):
\[
\mathbf{u} = \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}, \quad \mathbf{v} = \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}, \quad \mathbf{w} = \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}, \quad \text{and} \quad \mathbf{x} = \begin{bmatrix} a \\ b \\ c \end{bmatrix}.
\]
#### Tasks
(a) Are vectors \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{w}\) linearly independent?
(b) Do \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{w}\) span the entire \(\mathbb{R}^3\)?
(c) Do \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{w}\) form a basis of \(\mathbb{R}^3\)?
---
In this exercise, explore the concepts of linear independence, spanning sets, and basis in the context of vector spaces.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F801d34df-dcab-45aa-85e0-2349cce424a4%2Fefaa589c-81ae-47bf-b07a-c09d81ad4a57%2Fp6rwruo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Linear Algebra Problem Set
Consider the following vectors in \(\mathbb{R}^3\):
\[
\mathbf{u} = \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}, \quad \mathbf{v} = \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}, \quad \mathbf{w} = \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}, \quad \text{and} \quad \mathbf{x} = \begin{bmatrix} a \\ b \\ c \end{bmatrix}.
\]
#### Tasks
(a) Are vectors \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{w}\) linearly independent?
(b) Do \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{w}\) span the entire \(\mathbb{R}^3\)?
(c) Do \(\mathbf{u}\), \(\mathbf{v}\), and \(\mathbf{w}\) form a basis of \(\mathbb{R}^3\)?
---
In this exercise, explore the concepts of linear independence, spanning sets, and basis in the context of vector spaces.
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