Consider the set R^(+) of positive real numbers under the following operations: Is R^(+) a vector space under these operations? Prove or disprove. Solution Some people doesn\'t include 0 in R+, I don\'t know if you are or not ! We will treat both case. I\'ll write P for plus operation to avoid any confusion with normal plus and . for scalar multiplication We defined xPy=x*y and k.x = x^k (notice the .) I\'ll take the order of axioms required as same than wikipedia (http://en.wikipedia.org/wiki/Vector_space) I\'ll write them all, even if some fails, so you can see which one fails. Associativity of addition : uP(vPw) = u(vw) = (uv)w = (uPv)Pw => OK Commutativiy of addition : uPv = uv = vu = vPu => OK Identity element of addition: uP1 = u*1=u , so 1 is the identity element for addition => OK Inverse element of addition : Suppose u>0 then uP(1/u) = u*(1/u) = 1 So here it works ONLY if your R+ doesn\'t contain 0, otherwise this axioms fails with u=0 Compatilibility of scalar multiplication with field mutliplication a.(b.v)=(b.v)^a = (v^b)^a = v^(ab)=(ab).v => OK Identity element of scalar multplication : 1.v = v^1 = v => OK Distributivity of scalar multplication with respect to vector addition : a.(uPv) = (uv)^a = u^a*v^a = u^aPv^A=(a.u)P(a.v) => OK Distributivity of scalar multiplication with respect to field addition : (aPb).v = v^(aPb)=v^(ab) but (a.v)P(b.v) = (v^a)P(v^b)=v^a*v^b = v^(a+b) => FAIL So distributivity of scalar multiplication with respect to field addition fails (and also inverse element of addition if R+ is defined as not containing 0 in your class definition).

1. Consider the set R^(+) of positive real numbers under the following operations:
Is R^(+) a vector space under these operations? Prove or disprove.
Solution
Some people doesn't include 0 in R+, I don't know if you are or not ! We will treat both case.
I'll write P for plus operation to avoid any confusion with normal plus and . for scalar
multiplication
We defined xPy=x*y and k.x = x^k (notice the .)
I'll take the order of axioms required as same than wikipedia
(http://en.wikipedia.org/wiki/Vector_space)
I'll write them all, even if some fails, so you can see which one fails.
Associativity of addition : uP(vPw) = u(vw) = (uv)w = (uPv)Pw => OK
Commutativiy of addition : uPv = uv = vu = vPu => OK
Identity element of addition: uP1 = u*1=u , so 1 is the identity element for addition => OK
Inverse element of addition : Suppose u>0 then uP(1/u) = u*(1/u) = 1
So here it works ONLY if your R+ doesn't contain 0, otherwise this axioms fails with u=0

2. Compatilibility of scalar multiplication with field mutliplication a.(b.v)=(b.v)^a = (v^b)^a =
v^(ab)=(ab).v => OK
Identity element of scalar multplication : 1.v = v^1 = v => OK
Distributivity of scalar multplication with respect to vector addition : a.(uPv) = (uv)^a = u^a*v^a
= u^aPv^A=(a.u)P(a.v) => OK
Distributivity of scalar multiplication with respect to field addition :
(aPb).v = v^(aPb)=v^(ab) but (a.v)P(b.v) = (v^a)P(v^b)=v^a*v^b = v^(a+b) => FAIL
So distributivity of scalar multiplication with respect to field addition fails (and also inverse
element of addition if R+ is defined as not containing 0 in your class definition)