TITLE: Alternate Definition of a Finite Dimensional Vector Space and why is this not used?
QUESTION [0 upvotes]: My textbook defines a finite dimensional vector space as follows: 

The vector space $V(F)$ is said to be finite dimensional or finitely generated if there exists a finite subset $S$ of $V(F)$ such that $V=L(S).$

I was wondering why didn't the author use the following definition, which I think is equivalent: 

The vector space $V(F)$ is said to be finite dimensional or finitely generated if the basis of the vector space is finite. 

Is there any specific reason or is it just author's choice?

REPLY [2 votes]: Perhaps the reason is that, at that point, the first author has not proven that every basis of a vector space has the same cardinal. Note that, in the second definition, they talk about the basis. That would not be a correct definition if we didn't know about that result.

REPLY [1 votes]: we can prove that every finitely generated vector space has a basis and then prove that they all have the same size.
It wuld be a lot more awkward to first start only with vector spaces that have a basis.
If you want to get a grasp of what the above looks like you may want to learn some module theory, modules with finite basis are called free modules of finite dimension.
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