# Basis of a vector space

1. Dec 5, 2007

### mathboy

Maximal subspace

Problem: Prove that every vector space V has maximal subspace, i.e. a proper subspace that is not properly contained in a proper subspace of V.

I let A be the collection of all proper subspaces of V, but I can't prove that every totally ordered subcollection of A has an upper bound in A. The problem that the union of proper subspaces is not necessarily a proper subspace of V. What do I do now?

Last edited: Dec 5, 2007
2. Dec 5, 2007

### morphism

But the union of a chain of subspaces is a subspace.

3. Dec 5, 2007

### JasonRox

Think basis elements.

4. Dec 5, 2007

### andytoh

But it has to be a proper subspace of V.

For example { span{1}, span{1,x}, span{1,x,x^2}, span{1,x,x^2,x^3}, ..... } is a chain of proper subspaces of R[x], but its union is all of R[x], which is not a proper subspace of R[x].

Last edited: Dec 5, 2007
5. Dec 5, 2007

### andytoh

JasonRox's idea is good, take a basis of V and delete one element. The span of that would have to be a maximal subspace.

But I'm assuming that mathboy wants to use Zorn's lemma. In that case choose any v in V, and let A be the collection of all subspaces not containing v. This time the upper bound of any chain will be a proper subspace. The maximal element of A would be a maximal subspace of V.

Last edited: Dec 5, 2007
6. Dec 5, 2007

### morphism

Oops! I should learn to read! Thanks for pointing that out.

7. Dec 5, 2007

### Office_Shredder

Staff Emeritus
I don't think the problem implies there IS a basis of V (unless it turns out all vector spaces have a basis, and I just don't know that yet)

8. Dec 5, 2007

### JasonRox

Is V finite-dimensional? Is the book assuming that?

Do you know what finite-dimensional is?

9. Dec 5, 2007

### andytoh

Every vector space V has a basis, whether it is finite-dimensional or not. In mathboy's problem V can be infinite-dimensional and the result is still true.

If you want to prove that V has a basis if V is infinite-dimensional, you would have to use Zorn's lemma as well. Ultimately, mathboy's problem rests on Zorn's Lemma.

My approach to mathboy's problem is: Choose any v in V, and let A be the collection of all subspaces not containing v and then use Zorn's lemma. But I'm trying to figure out if there is a better partially ordered set to use, because my A seems a little clumsy (though I believe it would still get the job done).

10. Dec 5, 2007

### JasonRox

Of course I know this!

Ok, a vector space has a basis {v_1,...}, now delete one vector from there and span that that set. What do you get?

Voila!

11. Dec 5, 2007

### mathboy

Thanks guys. I forgot to say that I have to use Zorn's Lemma. But I know how to proceed now. I will use the collection of all proper subspaces that does not contain some fixed v in V.