1. The problem statement, all variables and given/known data 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. 3. 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?