Hello Students! I'm Dr Gajendra Purohit and you're watching my youtube channel where I upload videos on Engg mathematics & BSc If you're preparing for any competitive exam where higher mathematics is asked then my youtube channel will be very helpful Today I'll be starting a new topic that is vector spaces which is a series of linear algebra I'll be discussing all these topics Also I've another channel where i've started a new series on CSIR NET If you're attempting for life sciences, physics, mathematics etc then you can watch my videos to strengthen your Part A i.e. general aptitude Click on the i tab to watch my videos Vector spaces is an important concept of linear algebra, linear algebra id based on vector spaces And before learning this you must be well aware of the topic Group Theory with concepts of Group, Ring and Field I've uploaded many videos on the same, so click on i tab to watch my playlist on group theory for better understanding Internal and External composition are two very important concepts here Let V be any set , only one set in internal composition When operation is applied between two vector elements then it should return another vector For eg here when we take operation on two elements it returns back a vector Normally we talk about vector addition here Here V and F are two sets where F is the field and V is the vector The mapping goes from f: VF --->V We take an element from the field and an element from the vector 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 know that R is the subfield of C here C(Q) will be a vector space here R(Q) will be a vector space But here Q(Z) will never be a vector space Why Q(Z) is not a vector space? here Z should be a field and it's not a field that is why it is not a vector space Also the multiplicative inverse does not exist here Example 1 To a proof in a question if it is vector space or no we use some simple tricks 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 so it is not a vector space This question is usually asked in the exams here we need to prove that the n tuples of the elements of any field f is a vector space over field F here we need to prove that (V,+) will be a abelian group Then we need to satisfy the scalar multiplication property and then we need to check the common properties of scalar multiplication & vector addition so this is how we'll proof it n tuple elements are (a1, a2, a3... an) the n tuple is denoted by Vn(F) or Vn So alpha + beta will be added element wise first we'll check if Vn(F) is an abelian group or no We take 3 n tuples here and add them This property can easily be proven and will be associative For next property we take 2 n tuples for commutative addition We add alpha + beta or beta + alpha, it'll be the same and hence it is commutative The additive identity will be 0=(0, 0, ...0) and if we add this to our n tuples it'll give the same and when we take the inverse of this n tuple which will be -a1, -a2, -a3... -an and when this is added to n tuple it'll give identity that is 0=(0, 0, 0...0) so the inverse element also exists here we take alpha & beta as vectors and a as scalar first we add alpha & beta and then multiply the scalar a so we take a as common and we get aalpha + abeta We have alpha we'll prove that aalpha + balpha we have (ab)alpha so ab gets multiplied inside the bracket We take b out and we get a(b*alpha) 1 needs to be the unit element of the field and for field we know that it will satisfy the multiplicative identity So we'll always get 1.alpha = alpha so we have proved everything here that is abelian group, scalar multiplication, distributive property first we apply this operation now we add these and these values are equal to 0 the values we get should be same but it's not same here so the property cannot be satisfied and hence it'll not be vector space if it would have been (a+c, b+d) then it could be vector space I'll soon be uploading many such videos on vector space about subspace & its properties