Question

Prove that every finite dimensional vector space has a basis.

Answer 1

Let W be nonzero finitedimensional space. We can find some nonzero u1 in W. Then space
W1=(WSpan(u1))U{0} is again finite dimensional vector space. If it is nonzero,
then we can implement the above construction and gain u2, that is linearly
independent to u1.
By mathematical induction, w can continue the process, until we get Wn={0}, and
it always occurs as W was finite dimensional.
So, the constructed above set u1,u2,...,un forms a linearly independent set and
is maximal with this property, so it is a basis.