Question

Let V be a vector space of dimension n > 0, show that
(a) Any set of n linearly independent vectors in V forms a basis.

(b) Any set of n vectors that span V forms a basis.

Answer 1

(a) Any set of n linearly independent vectors in V forms a basis.

Suppose that S = {v1,v2,…,vn} are n linearly independent vectors. Take any non-zero vector v∈V. As V is of dimension n, the vectors v,v1,…,vn satisfy a linear dependency.


where not all the coefficients α,α1,…,αn are 0. In fact, α≠0 as S is linearly independent. Therefore, one can write

which proves that S is a basis for V

(b) Any set of n vectors that span V forms a basis

Suppose that S = {v1,v2,…,vn} span V.

If vectors are not linearly independent.

v1 can be written as

which shows dimension of S is having dimension less than n. (given dim(V) = n) which is clearly false. so, the vectors of set S are linearly independent.
Hence from (a) , any set of n linearly independent vectors forms basis. So, S forms basis.
hence proved.