if you could show steps as well not just answer Consider the following vectors:11Vị =;V2 =1;V3 =4(a) Determine if these vectors are linearly independent or dependent.(b) Is is possible to express1V =2-3as a linear combination of v1, v2, and v3?

Given vector are v1=121, v2=132 and v3=104. (a) We have determine whether the these vectors are…

Answered: Consider the following vectors: 1 1 2…

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

if you could show steps as well not just answer

Consider the following vectors:
1
1
Vị =
;V2 =
1
;V3 =
4
(a) Determine if these vectors are linearly independent or dependent.
(b) Is is possible to express
1
V =
2
-3
as a linear combination of v1, v2, and v3?

Expert Solution

Step 1

Given vector are $v_{1} = [\begin{matrix} 1 \\ 2 \\ 1 \end{matrix}], v_{2} = [\begin{matrix} 1 \\ 3 \\ 2 \end{matrix}] a n d v_{3} = [\begin{matrix} 1 \\ 0 \\ 4 \end{matrix}]$ .

(a) We have determine whether the these vectors are linearly independent or dependent.

First write the given vectors in matrix form, which is given below

Let $A = [\begin{matrix} 1 & 1 & 1 \\ 2 & 3 & 0 \\ 1 & 2 & 4 \end{matrix}]$

Now,

$\begin{matrix} \det (A) = 1 (12 - 0) - 1 (8 - 0) + 1 (4 - 3) \\ = 12 - 8 + 1 \\ = 5 \end{matrix}$

Hence, $\det (A) = 5$

Since, $\det (A) \neq 0$ that is matrix $A = [\begin{matrix} 1 & 1 & 1 \\ 2 & 3 & 0 \\ 1 & 2 & 4 \end{matrix}]$ has rank $3$ .

This implies the the given vectors $v_{1} = [\begin{matrix} 1 \\ 2 \\ 1 \end{matrix}], v_{2} = [\begin{matrix} 1 \\ 3 \\ 2 \end{matrix}] a n d v_{3} = [\begin{matrix} 1 \\ 0 \\ 4 \end{matrix}]$ are linearly independent.