From: boost (boost_at_[hidden])
Date: 2002-01-23 02:28:09
On Wednesday 23 January 2002 00:07, hubert_holin wrote:
> 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 :-) ).
Well, we 'mad physicists' even use Determinants involving grassmann algebras
( anticommuting beasts ). But linear Algebra get's a "little bit more
involved" then, since you have to introduce some kind of ordering to
keep track of commutations. But if a matrix libraries already takes
care of not changing orders that would be fine.
Note, that I'm not propsing to write an LA-Package for theses things,
just having "+", "-", "*", and operations like transpose() would be fine
for playing with some ideas.
> 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).
SU(2) would be more fruitful more me, but anyway this refers to the
'fun part', aka 'not so important'.
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