- #1
kwal0203
- 69
- 0
Homework Statement
Determine whether this set equipped with the given operations is a vector space. For those that are not vector spaces identify the axiom that fails.
Set = V = all pairs of real real numbers of the form (x,y) where x>=0, with the standard operations on R^2.
The Attempt at a Solution
This set is not a vector space because it is not closed under scalar multiplication I.e. -1*(1,1)=(-1,-1) which is not in V as x<0 and because there is not always a vector in V such that u+(-u)=(-u)+u=0 I.e. when u=(1,1) then -u=(-1,-1) which is not in V as again x<0.
My question is why does axiom 8 hold which states:
(K+m)u=ku+km
I.e. if k=-1, m=-1, u=(1,1) ----> (-1+-1)u=(-1,-1)+(-1,-1)=(-2,-2) which is not in V as x<0.
Does axiom 8 not require the solution to be in the set V?