September 23, 2011 10 Comments
So how should the lazy mathematician write down a vector space? We want to make sure we write down enough to determine the whole space, so we want to make sure whatever vectors we write, they span the vector space. But we don’t want to write down anything that we could have already determined from other vectors we wrote down, so we want to make sure what we write is linearly independent. This is so important, it gets its own name.
A basis for a vector space is a linearly independent spanning set. That is, is a basis for if and is linearly independent.
Verify for yourself that each of these are proposed bases are both linearly independent and span the entire space.
- is a basis for as a vector space over the field .
- is a basis for over the field . So is .
In the second example, you saw that vector spaces can have more than one basis. This is totally fine, and a lot of linear algebra is about looking for bases with nice properties.
So is it true that every vector space has a basis? It seems true, but maybe its hard to prove. Here are some key facts about this question:
- If finitely many vectors can span the entire vector space, it has a basis. Just take a finite spanning set, and remove vectors until they are linearly independent.
- The statement “Every vector space has a basis” is equivalent to the axiom of choice!
If you don’t think this is cool, you are wrong. Tomorrow I’ll prove that the axiom of choice implies that every vector space has a basis. The other direction isn’t really of interest to us right now. The proof is an application of Zorn’s lemma, another statement equivalent to the axiom of choice.
If you’re one of those who is wary about using the axiom of choice, I have two things to say. First, chill out. It’s useful. Second, we’ll mostly be dealing with the first kind of “finitely-spanned” vector spaces, so we don’t need the axiom of choice.