Content deleted Content added
→Fields: Response |
|||
Line 23:
A vector space is a module over a division ring -- not over a field. This article errs because the set of quaternions is a division ring but not a field. To speak of a vector space over the quaternions you can't define vector spaces only over fields. [[User:Jalongi|Jalongi]] ([[User talk:Jalongi|talk]]) 07:11, 19 July 2009 (UTC)
:Maybe I'm misunderstanding what you're saying or I missed something in the article but it seems correct to me. The only time the article mentions quaternions is as a vector space over the reals which is a field. The sources I've checked all define a vector space to be over a field, do you have a reference defining it over a division ring? Off the top of my head I don't see that any of the theory would break if you defined it that way but maybe there just aren't a lot of applications.--[[User:RDBury|RDBury]] ([[User talk:RDBury|talk]]) 22:37, 19 July 2009 (UTC)
::Hungerford's Algebra (which seems to be a popular reference and graduate text) defines a vector space to be a unitary module over a division ring. I misread the article's example involving the quaternions. The quaternions are vector space over the real numbers, but according to Hungerford's definition we could use quaternions as scalars and still have a vector space. One part of the theory that breaks down under the more general definition is that a matrix over a division ring does not have a well-defined rank. It has a left-rank and a right-rank, I think. We would just have to be careful about commutativity in proofs. Thanks for catching my error.
== Function spaces and generalized coordinate spaces ==
| |||