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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.

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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.

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"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. –  T. Bongers 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
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