# Answer to Question #23679 in Linear Algebra for Jacob Milne

Question #23679

Prove that every finite dimensional vector space has a basis.

Expert's answer

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.

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.

## Comments

## Leave a comment