Re: Halmos Lin Algebra Problem - clarification

On Sun, 30 Apr 2006 00:16:23 GMT, Guy Corrigall wrote:
I'm ploughing through Halmos Problem Book. He defines a vector space as a
commutative *group* V of vectors, a field F of scalars and a scalar
multiplication rule.

In Problem 21(4) he declares "Let F be R (the reals) and V the *set * Rplus
of all real positive numbers..." He then defines a peculiar addition rule in
V (a+b = ab) and a scalar multiplication rule ab = b exp a and invites the
reader to establish whether this rule makes V a vector space, by checking
through all the axioms. His answer is "yes", the addition and scalar
multiplication rules 'make' V a vector space.

But - how could V be a vector space, whatever the scalar multiplication
rule? Surely Rplus is not a group (no zero, no additive inverses)? Does
Halmos' vector addition rule make the set V a comutative group?

The group axioms do not mention a "zero". The requirement is that there
be an identity element, and Rplus has one.




--
Dave Seaman
U.S. Court of Appeals to review three issues
concerning case of Mumia Abu-Jamal.
<http://www.mumia2000.org/>
.