Anonymous wrote: > George: > >>Then you're wrong. This waffle is next to useless. What is the precise >>definition? If the precise definition has words like "abelian group" in >>it, then what is the precise meaning of them? > > I never learned a definition of "vector space" that had "abelian group" in it. > The only time we spoke of abelian groups was in algebraic structures. If I > remember correctly, an abelian group is a group that has an invertible table.
What is an "invertible table" ? By the way, a group is abelian if the group operation is commutative.
You can group the axioms in the definition for a vectorspace as follows
(a) those axioms only involving addition (b) those axioms only involving multiplication with scalars (c) the mixed distributivity laws
those axioms in (a) are those for an abelian group. In most linear algebra courses, the definition of an (abelian) group should at least be mentioned because
- the axioms from (a) can be remembered easier. - the definition of a field also includes them for the field addition. - a lot of concepts from the theory of vectorspaces have counterparts in the theory of abelian groups; this gives a chance for learning through (modified) repetition.