In this video, we will learn that a set of vectors is given to you, then how we will check that whether the sector is linearly dependent or independent. So first of all linear dependence of vectors, now suppose a finite set of vectors v1, v2, up to vn of a real vector space v is given to you, right? And you want to verify that it Is linearly dependent or independent, so it is said to be linearly dependent if there exist scalers c1, c2, cn, not all zero in R, such that, this equal to zero, right. So what does it mean? It means that if we call it linear combination, right, you take c1v1 + c2v2 + …. + cnvn = 0. And if it implies that not all c1’s are 0, then this means the set of vectors is linearly dependent. So, this set is given to you, this set consisting of three elements 2,1; 1,2; -1,1; and all are in R2. So, we have to verify that the set is linearly dependent. So how to verify it? So, you take c1 times the first element. First element is (2,1) + c2 (1,2) + c3 (-1,1) = (0,0) Now, this further implies, this is (2c1, c1) + (c2, 2c2) + (-c3, c3) = (0,0) This implies, when you add them, the left-hand side will be what (2c1 + c2 – c3, c1+ 2c2 + c3) = (0,0) So, this implies 2c1 + c2 – c3 = 0 and C1 + 2c2 + c3 = 0. Now when you solve them, you add them, so this will be 3c1 + 3c2 = 0, which implies c1 = -c2, right? This is first condition, and when you put c1 = -c2 in any one of the equation, then you will get 2c1 – c1 – c3 = 0 and imply c3 = 0, right? So, if you take say, c1 = 1, there are many solutions of this system of equations, say you take c1 = 1, then you c2 = -1, and c3 = 1. So, there is a solution of this equation where c1, c2, c3 are not all 0, so this implies that this set is linearly dependent. On the other hand, if you see linearly independent, so linearly independent means that if you take c1v1 + c2v2 … + cnvn = 0, so it implies only c1, c2, c3, all should be 0, right, so that means linearly independent set. So, let us take one example, suppose you take this set consisting of two elements of R3 and you are to verify that the set is linearly independent. So you take alpha times the first element, beta times the second element right, put it equal to 0,0,0. So, this implies alpha equal to 0 right from the first condition, 2 alpha plus beta equal to 0 from the second condition, alpha minus beta equal to 0 from the third condition. And these three conditions imply since alpha is 0, so from the second condition beta is 0, and alpha 0, beta 0, satisfying third condition. So, this implies alpha equal to beta equal to 0. So that means this equation has only one solution which is alpha equal to zero and beta equal to zero, so hence we can say that this sector is linearly independent set. So, we have seen that if a set is given to you, then how you can verify that a given set is linearly dependent or independent. So, on the other hand, we can also say like this, that if a set of vectors is linearly independent, that means we cannot express any element of the set in terms of or as a linear combination of the remaining elements, right? If the set is linearly dependent, then we can always express a given element of a set, there will exist one element of a set which can be expressed as a linear combination of the remaining elements. In this video, we have seen that, what are linearly independent and dependent Vectors. Further, we have also seen that if a set of vectors is given to you, how will you verify whether that set is linearly independent or dependent.
