Definition talk:Vector Space
Doesn't K need to be a field for the definition of "vector space" to hold? --Grambottle 05:40, 18 December 2008 (UTC)
Not according to 1965: Seth Warner: Modern Algebra. Depends on how important it is for the underlying structure to be commutative. I believe it need not be - and that may be important in the development of tensors. It may be useful to allow the elements of $K$ to be matrices. Not sure yet. Only just started studying this stuff.--prime mover (talk) 06:32, 18 December 2008 (UTC)
This is another thing where I've never seen it defined anyway BUT: V an abelian group, F a field. Then V is a vector space over F if ...
Note both Wikipedia and MathWorld have this defintion, as well as a wealth of textbooks.
Just a question: is this Warner: Modern Algebra your main reference? I've noticed a lot of really odd notation in your articles, and that book is rather old, which might explain why I perceive your notation as weird. --Grambottle
Yes, Warner's the main references I've been using for that aspect of things. It's extremely thorough and his approach is sound. He comes from the direction of defining something as widely as possible. (Although in most cases your underlying structure of your vector space is a field, his view is that it doesn't necessarily have to be.) All other treatments I've seen (my library is fairly extensive but I'm limited to what I've been able to find in bookstores and charity shops) seem to start from the assumption that your underlying structure is a number field.
Another source is Knuth, who makes every attempt to be up-to-date in his notation and has done some genuine notational innovation. If my notation's weird then maybe that's what you're noticing. Do we want to go through and change everything? If you have different ideas for notation of different things, feel free to add a "notes" or "comment" section to the definition which introduces it - I monitor everything - and we can see what's what. --prime mover (talk) 07:15, 18 December 2008 (UTC)
Cool. Be weary about using notation from just a single reference and "notational innovation", especially with an oddball like that Warner guy. "Notational innovation" typically strays away from convention, a lot of time for the worse (though sometimes it does turn out better). I'll take that book out of the UW library when I get back in January and check him out. In the meantime, if I see anything funny, I'll post it on the discussion page for the most part. I'll flat-out edit something you've done only if I'm incredibly confident in what I'm doing.
And I'm guessing by "number field" you just mean "field" :P Any field can be used to construct a vector space, once given a sufficient abelian group. --Grambottle
Yes I know any field can be used etc., but, as I say, a lot of the sources I've got access to are either elementary (and therefore don't really "do" abstract algebra) or as written with the emphasis on applied maths and engineering. Therefore they assume that the field is (usually) $\R$ for which they assume the usual field axioms. Warner is the most thorough of all the other sources I've got. --prime mover (talk) 07:56, 18 December 2008 (UTC)
One of the absolute best references I've come across which seems to do very well with conventions is:
Advanced Modern Algebra, Rotman (http://books.google.ca/books?id=BU9qAAAACAAJ&dq=advanced+modern+algebra+rotman)
Definitely the most comprehensive algebra book I've seen.--Grambottle 22:08, 18 December 2008 (UTC)