Re: Halmos Lin Algebra Problem - clarification
- From: ceckhard@xxxxxxxxx
- Date: 29 Apr 2006 17:34:25 -0700
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?
Yes. In this case the zero element is the real number 1. If a>0, it's
inverse is 1/a.
.
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