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Determine whether the following set of vectors is linearly independent o dependlent_Linearly independent Linearly dependent...

Question

Determine whether the following set of vectors is linearly independent o dependlent_Linearly independent Linearly dependent

Determine whether the following set of vectors is linearly independent o dependlent_ Linearly independent Linearly dependent



Answers

determine whether the given set of vectors is linearly independent or linearly dependent in $\mathbb{R}^{n} .$ In the case of linear dependence, find a dependency relationship. $$\{(3,6,9)\}$$.

The question states determine whether this set of three vectors is linearly independent in two by two matrix space with riel coefficients. Um, no, it is not linearly independent because three, if it's in two by two space, you can only have two matrices that would be independent. Ah, third maitresse make six would have to be I dependent. So because there are three vectors and the space is only two dimensional space, um, they are not.

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

We're ready to determine whether these two factors are linearly independent in two by two matrix space with rial terms. Um, the only way that they would be dependent of each other is if they are proportional to each other. So if I take every element of matrix to and multiply by negative too, do I get every element of matrix one while negative one times negative two is too. That's good. One times negative too, is not three. So yes, these are in de pen.

Determine whether the given set of vectors and in this case, they are functions are linearly independent regarding second order functions with real coefficients. Well, I noticed that these air actually both first order functions. So one of them is one minus X and one of them is one plus X. They would need to be proportional to each other. So one minus X would have to eat equals some constant times one plus x. There is no such constant. Um, I guess we could even one minus X equals C plus X. We could keep going with this. Ah, one equal C. Okay. How that happened? Oh, no, that doesn't make sense. You add acts on both sides. So then, um see is one minus x squared. The point is that C there is no constant. This would be like a function, but there is no constant that's going to relate one minus x 21 plus x So these are independent


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