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--8-9--8-- 7 1. Are the vectors u =| and w= 10 independent? 4 12 2 and 3 3 independent? 5 2. Are the vectors 4 3. What is the maximum number of independent vectors we can have in R" ?

--8-9--8-- 7 1. Are the vectors u =| and w= 10 independent? 4 12 2 and 3 3 independent? 5 2. Are the vectors 4 3. What is the maximum number of independent vectors we can have in R" ?

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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How do I solve the first 3 questions?

**Homework** 1. Are the vectors \( \mathbf{u} = \begin{pmatrix} 1 \\ 0 \\ 4 \end{pmatrix}, \mathbf{v} = \begin{pmatrix} -2 \\ -1 \\ 3 \end{pmatrix}, \) and \( \mathbf{w} = \begin{pmatrix} 7 \\ 10 \\ 12 \end{pmatrix} \) independent? 2. Are the vectors \( \begin{pmatrix} 1 \\ 2 \\ 3 \\ 4 \end{pmatrix} \) and \( \begin{pmatrix} -1 \\ 3 \\ 5 \\ 0 \end{pmatrix} \) independent? 3. What is the maximum number of independent vectors we can have in \( \mathbb{R}^n \)? 4. Show that if the vectors \( \mathbf{v_1} = \begin{pmatrix} 2 \\ 3 \\ 1 \end{pmatrix}, \mathbf{v_2} = \begin{pmatrix} 4 \\ 6 \\ 0 \end{pmatrix}, \mathbf{v_3} = \begin{pmatrix} 10 \\ 15 \\ -1 \end{pmatrix} \) are dependent, any one of them must be a linear combination of the other two.
Transcribed Image Text:**Homework** 1. Are the vectors \( \mathbf{u} = \begin{pmatrix} 1 \\ 0 \\ 4 \end{pmatrix}, \mathbf{v} = \begin{pmatrix} -2 \\ -1 \\ 3 \end{pmatrix}, \) and \( \mathbf{w} = \begin{pmatrix} 7 \\ 10 \\ 12 \end{pmatrix} \) independent? 2. Are the vectors \( \begin{pmatrix} 1 \\ 2 \\ 3 \\ 4 \end{pmatrix} \) and \( \begin{pmatrix} -1 \\ 3 \\ 5 \\ 0 \end{pmatrix} \) independent? 3. What is the maximum number of independent vectors we can have in \( \mathbb{R}^n \)? 4. Show that if the vectors \( \mathbf{v_1} = \begin{pmatrix} 2 \\ 3 \\ 1 \end{pmatrix}, \mathbf{v_2} = \begin{pmatrix} 4 \\ 6 \\ 0 \end{pmatrix}, \mathbf{v_3} = \begin{pmatrix} 10 \\ 15 \\ -1 \end{pmatrix} \) are dependent, any one of them must be a linear combination of the other two.
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