Dave Rusin wrote: > > In article <408EC9A3.DB73D12C@SPAMbtinternet.com.invalid>, > George Cox <george_coxANTI@SPAMbtinternet.com.invalid> wrote: > > [Vector space.] > > >What is the precise > >definition? If the precise definition has words like "abelian group" in > >it, then what is the precise meaning of them? And so on. > > Yeah, sure. A vector space is a module over a field. That's absolutely > true and probably given that way in Bourbaki somewhere, but even in > selective institutions I can't believe students learn about modules > before vector spaces. Is there really a source anywhere that pretends to > _introduce_ vector spaces in terms of abelian groups?
I was taught (33 years ago in the University of Birmingham, UK) that a vector space is (1) a set of elements (the vectors) which form an abelian group under addition; (2) etc.... The phrase "commutative group" or "additive group" might have been used instead of "abelian group"--you'll forgive me if I can't quite remember!
The course was the first algebra course in the first year of a BA.
How do you/US universities/today's commonly used texts define vector space?
> > Hmm, I suppose some here would argue that it messes up students' > understanding of modules if they are first tainted by facts learned > for vector spaces, which then have to be unlearned in the general case. > > dave
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