The problem involves defining a set V = R with a non-standard vector addition defined as a + b = ab and scalar multiplication as za = a^z. The task is to demonstrate that V satisfies the properties of a vector space.
The discussion is ongoing, with participants exploring the implications of the definitions provided. Some have suggested that the additive identity cannot depend on the specific element of V, while others question the existence of an additive inverse for certain elements, particularly zero. There is no explicit consensus on whether V can be classified as a vector space under the given definitions.
Participants note potential constraints regarding the definition of the additive identity and inverse, as well as the implications of including zero in the set V. There is also mention of the possibility of restricting V to positive real numbers to satisfy vector space properties.
While there is a single additive identity (and it is NOT 0), each member of V has a different additive inverse. The definition of "additive identity" (I will call it "O" for the moment) is that "a+ O= a". Here, "addition" of vectors is defined as the product of the numbers: "a+ b= ab" above. Therefore, "a+ O"= aO= a. What number is O?jack_bauer said:Homework Statement
Define V =R with vector addition a+b=ab and scalar multiplication za=a^z.
Show that V is a vector space.
Homework Equations
a+b=ab, za=a^z
The Attempt at a Solution
I was able to check all the axioms but one, the additive inverse axiom where for all v in V there exists a... -v in V such that v+(-v)= 0. So far I have this: a+0=a(0)=0. In this case I believe 0 is the additive inverse.