In this video we will see that how can we define linear combination of a set of vectors, and then we will also see how can you find out linear span of a set of vectors. Linear combination of vectors, how can we define it? Let v be a real vector space and let S, which is consisting of v_1, v_2 up to v_k is a subset of vectors in V. Then our vector V of the form v is = Alpha_1v_1 + Alpha_2v_2 up to Alpha_kv_k, where all Alphas are real numbers is said to be a linear combination of the vectors in the sentence. Basically you have a set S; this S is given to you, and you have some vector v, which can be expressed as sum of Alpha_1 times v_1 + Alpha_2 times v_2 up to Alpha_k times v_k. These Alpha_1, Alpha_2, we have two if they exist, then we can say that this v is nothing but a linear combination of the elements of v_1, v_2 up to v_k. For example, see, you have a set S which is consisting of these two elements; 1, 1 and 4, 3. Now, if you see this vector; - 2, - 1. This - 2, - 1 can be expressed as two times the first element and - 1 times the second elements. You can verify. This is basically what? This is (2, 2). This is - (4, 3). Now when you simplify it, it is nothing but 2- 4 and it is 2-3, which is basically (-2 -1) the same. There exist Alpha_1, Alpha_2. This is Alpha_1, this is Alpha_2. Alpha_1 is 2 and Alpha_2 is - 1 such that this vector V can be expressed as linear combination of these two vectors so we say that this element is a linear combination of v_1 and v_2. Now, take another example. Say these two elements are in R^3 and this vector 1, 1, - 1 if you verify, then it cannot be expressed as linear combination of these two elements. Why it cannot be expressed? If it can be expressed as linear combination of these two vectors, then there must exist some Alpha_1. Alpha_2 such that this v can be expressed as Alpha_1v_1 + Alpha_2v_2. It is 1, 1, - 1. Alpha_1 times the first element + Alpha_2 times the second element. If we are able to find Alpha_1, Alpha_2 such that this equality hold, then we say that this can be expressed as linear combination of these two elements, otherwise it cannot be expressed. Let us try to simplify it. This is basically 1, 1, - 1; left-hand side is same and this is Alpha_1 + Alpha_2 and then this is Alpha_1 and then this is Alpha_2. These two are equal this means the first element is equal to the first element, the 2nd element = the 2nd element, and so on. This means Alpha_1 + Alpha_2 is 1. Alpha_1 is 1 and Alpha_2 is - 1. From this Alpha_1 + Alpha_2 is 0 which is not 1. This means there does not exist any Alpha_1 and Alpha_2 such that this can be expressed as linear combination of these two elements. Now what do you mean by linear span? Say S which is a collection of v_1, v_2 up to v_k is a subset of a vector space V. Now linear span of S is a set of all linear combinations of elements of the set S. What does it mean? It means that it is basically Alpha_1v_1 + Alpha_2v_2 up to Alpha_k v_k where Alpha_i belongs to R. See here, v_1,v_2, v_k are fixed, now you vary Alphas: Alpha_1, Alpha_2 up to Alpha_k. If you vary Alpha you will get many elements of this set, and the collection of all those elements we call as a span of the set S. We denote it by this square bracket, square bracket means span of the elements of the set S. Now take this example. The linear span of the set these two elements; 1, 0 and 0,1 is the entire R^2. Why it is R^2? Because if you take any vector a, b in R^2 then a, b in R^2 can be expressed as a times the first element and b times the second element. Say you take any element 2, 3. Any element 2, 3 can be written as 2 times the first element and 3 times the second element. We can see that if you take the collection of all linear combination of these two elements, it will constitute R^2. The span of these two elements is = R^2. Now what is span of these two elements? Let us try to see that this span is = this. If you want to find all the span of these two elements; so take any element of v belongs to this. This implies if v belongs to the span of this, this means v can be written as linear combination of these two elements. Now suppose v is some of a, b, c in R^3. This a, b, c is basically = Alpha times this + Beta times this which is Alpha, this is - Beta, and this 2Alpha + Beta. This implies Alpha is nothing but a, - Beta is b, and 2Alpha + Beta equals to c. If you substitute Alpha = a here and Beta = - b here, this is the condition. This implies 2a - b - c = 0. This means the span of these two vectors is nothing but collection of all those a, b, c in R^3 such that this equality holds. The span of these two vectors is basically this subspace of R^3. In this lecture we have seen that if a set of vectors given to you, how can you see that a given vector can be expressed as linear combination of those vectors or not. Further, we have also seen that what do you mean by linear span of a given set?
