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Determine by inspection whether the vectors are linearly independent. Justify each answer.$\left[\begin{array}{r}{1} \\ {4} \\ {-7}\end{array}\right],\left[\begin{array}{r}{-2} \\ {5} \\ {3}\end{array}\right],\left[\begin{array}{l}{0} \\ {0} \\ {0}\end{array}\right]$

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Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 7

Linear Independence

Introduction to Matrices

McMaster University

Idaho State University

Lectures

01:32

In mathematics, the absolu…

01:11

03:37

Determine by inspection wh…

02:21

02:10

02:25

01:46

03:31

In Exercises $1-4,$ determ…

03:48

02:43

08:02

Determine all values of th…

01:15

determine whether the give…

in this, uh, in this video, we're gonna be solving problem number 20 from section 1.7, which is based on their independence. So here were given three back. Here's one for negative seven negative. 253000 On these there are column workers and were asked to determine just by looking at homes and no work. Uh, if they're literally independent or dependent so and easier to solve. This is by looking at the Book of the theorems and the serum. Nine states that if a set of column acres contains the zero vector, then the set is clearly linearly dependent. So we have the answer. The set is linearly dependent. And here's why. Um so for a set to be literally dependent are literally independent, the vector equation or the major situation X equals or that we learned back in section one point for Has equal has to have only the triple solution was in this case is 000 because we have three vectors. That means there's gonna be three variables that we're gonna be stalling for if you were to set this up as a system of equations. So if you were to set up a matrix augmented matrix to solve this. Well, uh, for X equals zero, we would set it up like this. I'm sorry. Sorry for my handwriting, but these these two are zero backers. Um, so here, right off the bat, we know that this zero accurate causes a free variable. For that, they're variable to be free. So the other two variables will be written in, uh, terms of the third variable Where, uh, that they're variable. It can be anything on the number line can be any value number line. And, uh, the other two variables will be written in terms of the three. Very also be able change. Based on what value The free variable is, uh, chosen to be so, since this has way more solutions with just a trivial solution. Uh, this set of factors is only really

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