Answer to Question #123261 in Linear Algebra for Jose Sammut

Question

Prove that a set of n linearly independent vectors in a n–dimensional
vector space V span V

Answer 1

Given: we have 'n' linearly independent vectors and so they span a subset of V. 

Since the dimension of the vector space is 'n' therefore the maximum number of linearly independent vectors it can have is 'n'. 

So if there is an element that is not obtainable from these 'n' vectors then the vector space would be 'n+1' dimensional which is in contradiction to the given information. So the set of 'n' linearly independent vectors span the vector space of n dimension.

Hence proved.