I am solving a problem in which it's given that V = $\mathbb R$ and F = $\mathbb R$ and the definition of vector addition and scalar multiplication is given. The problem is to check which axioms of vector space are satisfied and which are not.
I am getting unique additive identity as $ \frac{-v}{2}$ $\forall$ $v\in V$.
My doubt is that has the additive identity to be same for $\forall$ $v\in V$, like the one in the usual addition (which is 0) or it can be different but unique for different elements of V like the additive inverse in usual addition (which is $-v$ $\forall$ $v\in V$).