Dave Rusin <email@example.com> 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? > > 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.
The first time, I spent in a linear algebra course (Kaiserslautern, 1986) introduced sets, groups, rings and fields, modules and then vector spaces. (in that order) From what I have observed, this must have been quite the standard sequence at that time.
Of course, nobody insisted on a full blown treatment of group theory or module theory before introducing vector spaces. The introductions provided the definitions, examples and those results, variants of which would be met later in the theory of vector spaces. And certainly the definition of a module is not more complicated than the one of a vector space.
From the students point of view, the advantages were:
- one could proceed from definitions with few operations and axioms to those with more. Remembering the axioms for a vector space was a no-brainer, because we had seen most of them before.
- those concepts common to all of the above situations could be introduced for each of these situations. This also did build confidence. By the time we met vector spaces, the notion of image and kernel as well as the proof of the first homomorphism theorem was completely obvious because we had seen it for sets, groups, rings and modules.
- more examples could be given. It would be a bit strange if you could not even talk about Z^2 because it is not a vector space.
The important point is to know, when to stop pursuing full generality. But nowadays nobody needs to be afraid of modules in a linear algebra course, unless it is only a MMC (Matrix Manipulation Course).