On 28 Apr 2004 22:51:35 GMT, Anonymous wrote: > George:
>>I was taught (33 years ago in the University of Birmingham, UK) that a >>vector space is >>(1) a set of elements (the vectors) which form an abelian group under >>addition; >>(2) etc.... >>The phrase "commutative group" or "additive group" might have been used >>instead of "abelian group"--you'll forgive me if I can't quite remember! >>
> I think you just made me remember what an abelian group is. It's closed under > its operation, it's associative, commutative, it's got an inverse, and it's got > the identity element in it. There may be some other condition, since aren't > all the above needed for any group?
All except the "commutative" part. If the group operation happens to be commutative, then the group is abelian.
>I remember something about an abelian > group forming a table with a diagonal, or something....
Nope. "Abelian group" simply means "commutative group", and nothing more.
> So, basically, "abelian group" is just an easier way of saying "vector space". In approximately the same sense that "steering wheel" is just an easier way of saying "automobile". You have omitted two of the three things needed in order to have a vector space:
(1) A collection of vectors V, which form an abelian group with respect to vector addition (this is the only part you got right),
(2) A scalar field F,
(3) An operation called scalar multiplication, which binds V and F together. The operation is associative and satisfies the distributive laws.
-- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.