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From: hubert_holin (Hubert.Holin_at_[hidden])
Date: 2002-01-22 18:07:35

Paris (U.E.), le 23/01/2001


                One can extend the usual definition of vector spaces in
several ways.

                On the one hand, one can certainely use non-commutative 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 mild-manered 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

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

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 "non-commutative" 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 non-commutative scalars could be
> (compiler) optimizations that change the order of operands.
> Best wishes,
> Peter

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