TITLE: Question about basis and finite dimensional vector space
QUESTION [11 upvotes]: I have seen the statement "Every finite dimensional vector space has a basis." (Here on page 5)
I'm confused about what this tells me. It seems to tell me nothing: by definition, the dimension of a vector space is the number of elements in a basis of it. Then saying a vector space is finite dimensional is the same as saying that it has a basis.
Or are there any other definitions of dimension than the number of basis elements?

REPLY [4 votes]: In the document you're linking to, the following is used as the definition of finitely dimensional space.

If a vector space $V$ is spanned by a finite number of vectors, we say that it is finite dimensional.

So we do not know a priori that every finite-dimensional space (w.r.t. this definition) has a basis, we need to show this.
Only after you show that every such space has a finite basis and that any two bases have the same number elements, you can use $V$ has a basis consisting of finitely many elements as an equivalent definition of finite-dimensional space.

I think that the definition of finite-dimensional space as the space having a finite spanning set is used quite often in introductory texts on linear algebra. (Among other things, you can speak about finite-dimensional spaces before defining the notion of basis.)
IIRC I have even seen some texts where basis is assumed to have at least one element, so if you use that definition, then $\{0\}$ is an example of finite dimensional space which does not have a basis. (It is finite dimensional because it is spanned by the finite set $\{0\}$.)
Of course, if you've already mastered linear algebra, you will use any of equivalent definitions of finite-dimensional space almost automatically, details like this are important only in the introductory course, when the students encounter these notions for the first time.

EDIT: Now I noticed that this was explained already in Mark Bennet's comment above.