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From: hubert_holin (Hubert.Holin_at_[hidden])
Date: 20020122 18:07:35
Paris (U.E.), le 23/01/2001
Bonsoir
One can extend the usual definition of vector spaces in
several ways.
On the one hand, one can certainely use noncommutative base
fields. Unfortunately, most of Linear Algebra breaks down in such a
context (play around with the notion of determinants to see what can
happen, even with mildmanered beings such as quaternions :) ).
It is far more fruitfull to use a structure not based upon a
field, but upon a (commutative) ring, say Z/4Z. These structures are
usually known as modules and have a rich theory (and applications).
A bientot!
Hubert Holin
Hubert.Holin_at_[hidden]
P.S.: Sory I can't participate more, but I am in over my head with
various things at the moment, and the Boost list is reaching saturation
volume.
I am not a mad scientist, I am a mad mathematician!
 In boost_at_y..., Peter Schmitteckert (boost) <boost_at_s...> wrote:
> Salut,
>
> On Friday 18 January 2002 23:48, jhrwalter wrote:
> >  In boost_at_y..., Peter Schmitteckert (boost) <boost_at_s...> wrote:
> [...]
> > > BTW, is one allowed to use "noncommutative" scalars for ublas,
> > > or is it implied that "a * b == b * a" ?
> >
> > To get usual linear algebra functionality, one should assume that the
> > scalars form the approximation of a field. But I currently do not see
> > any obvious reason, why one couldn't try to use a noncommutative
> > multiplication for scalars. But then the approximations of vector
> > space laws and linear operator properties may not hold.
>
> A reason that one can't use noncommutative scalars could be
> (compiler) optimizations that change the order of operands.
>
>
> Best wishes,
> Peter
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