Let S be a linearly independent set in a finite dimensional

) Let S be a linearly independent set in a finite dimensional vector space V . Prove that S can be enlarged to give a basis for V .

Solution

We know that basis of V means any linearly independent set which can generate all the elements of V.
Given that S is a linearly independent set.
If S is basis of V then there is nothing to prove.
If S is not basis of V then there exists some vectors say a1,a2,...ak which are in V but not in S.
then we can find basis of vectors a1,a2,...,ak and combine that with linearly independent set S to generate all the vectors of V.
In other words S can be enlarged to give a basis for V .