 In this video, we're going to provide the solution for question number five for exam one for math 2270 in which case we're asked which of the following sets of vectors are linearly independent Do pay attention to the language right here if the question we're asking which of the following sets of vectors are literally dependent That would affect my final choice But either way I do have to go through and check is s independent or dependent is ts independent or dependent is you? Independent or dependent so let's look at choice s Now I'm looking at choice s right here I do notice that s contains the zero vector any set that contains the zero vector is automatically linearly Dependent so therefore I know set s is not Independent a dependent set since I'm looking for independence. I'm going to put an x on s there Now let's look at set t set t. There are two vectors and so if a set of vectors is linearly Dependent it means one of the vectors is a little combination of the other Well for two vectors the only way one to be a combination of the other is that they're scaling multiples of each other So could these vectors be skill or multiples if you call this vector v for a moment Then how do you form this one looking at the first entry? How do you go from one to negative one? You'd have to times up by negative one. So is this equal to negative v, right? Well, you take one times negative one that gives you a negative one But if you take two times negative one that should give you negative two that doesn't quite work out here So this tells us that these vectors are not skill or multiples of each other and therefore this set of vectors would be linearly dependent a set of two vectors is only It's only dependent if and only if one vector is skill or multiple the other and that didn't happen here Then when you look at choice you notice that we have four vectors inside of our three We have too many vectors if we thought of this as a system of linear equations Then we would have three equations and four variables We would be an undetermined system one of the variables would have to be a free variable and Because it's if there's a free variable in the system that would give you multiple solutions That would give you non-trivial solutions to the homogeneity system if you have too many vectors Then you are automatically linearly dependent and so both set s and you are literally dependent Set t is linearly independent and as we're looking for independent sets that means the correct response would be choice B