Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I've somehow managed to go through 3 years of my Maths degree without truly understanding what the term "finitely generated vector space" means.

Generated by what? Generating what? And for what? I have a very vague idea of what this term means and usually gloss over it when doing proofs but I think I just want a simple break-down of the meaning of this.

share|cite|improve this question

It means that the vector space has a finite generating set - that is, that a vector space $V$ has got a finite subset $\mathcal{B}$ such that $V = \text{span}\{b \in \mathcal{B}\}$. If the vector space is over a field, then this is equivalent to having a finite basis.

share|cite|improve this answer
"If the vector space is over a field", as opposed to what? – Michael Albanese Sep 15 '13 at 19:05
@MichaelAlbanese I've seen modules over division rings called vector spaces as well. – user61527 Sep 15 '13 at 19:07
I've never seen that. I've heard modules explained as vector spaces over rings, but only in a purely informal sense. – Michael Albanese Sep 15 '13 at 19:08

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.