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Topic: Failing Linear Algebra:
Replies: 54   Last Post: Jan 10, 2007 12:47 PM

 Messages: [ Previous | Next ]
 Marc Olschok Posts: 409 Registered: 12/6/04
Re: Failing Linear Algebra:
Posted: Apr 29, 2004 8:29 AM

Dave Rusin <rusin@vesuvius.math.niu.edu> 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).

Marc

Date Subject Author
4/22/04 Guest
4/22/04 Michael N. Christoff
1/10/07 Gerry Myerson
1/10/07 Jonathan Miller
1/10/07 Guest
1/10/07 David C. Ullrich
1/10/07 Acid Pooh
1/10/07 Guest
4/23/04 Brian Borchers
4/27/04 Guest
1/10/07 maky m.
4/26/04 David Ames
1/10/07 Guest
1/10/07 Michael Stemper
1/10/07 maky m.
4/23/04 Porker899
4/27/04 Guest
1/10/07 Abraham Buckingham
1/10/07 Mitch Harris
1/10/07 Guest
1/10/07 Grey Knight
1/10/07 Guest
1/10/07 Toni Lassila
1/10/07 Thomas Nordhaus
1/10/07 George Cox
4/28/04 Dave Rusin
4/28/04 George Cox
4/28/04 George Cox
4/29/04 Marc Olschok
4/29/04 Mitch Harris
4/29/04 Robert Israel
4/28/04 Guest
4/29/04 Guest
1/10/07 Dave Rusin