This video introduces vector spaces, a key linear algebra concept. The instructor explains the necessary conditions for a set to be a vector space, including internal and external composition properties, and provides examples to illustrate whether a given set qualifies as a vector space or not. The video also touches upon group theory and its relation to vector spaces. And the product is a vector element v will be vector space over field f. These are some properties that needs to be satisfied. The internal composition of in v should be abelian in plus first property is that is, should be closed. Second property is that is, should be abelian. Third property is that is, should be associative. Fourth property is that the identity should be 0. Fifth property is that is should be inversible. All the properties of abelian are applied here, product of an element of f and v should give a vector. The v should be closed with respect to the scalar multiplication for eg-Q(z) is never vector space will be proved using this property. The first property is about abelian which is about addition. Next property was on vector multiplication. In this property multiplication & addition is clubbed. We know that here c(r) will be a vector space. we take the points that satisfy the equation as both are vectors, and the addition should be closed, the sum of the vectors should be a vector. We take the sum, and we get -36, which is a vector and should satisfy the equation. But -36 is not equal to 0. So it is not a vector space, because the additive vector does not belong to v, the points satisfy the equation. But this sum does not satisfy the equation.