Let V denote the set of ordered pairs of real numbers. For $(a_1, a_2), (b_1,b_2) \in V, (a_1, a_2)+(b_1,b_2)=(a_1+b_1, a_2b_2)$.
I am trying to prove that a set V is (or is not) a vector space and I am stuck at proving the additive identity works and at showing the additive inverse doesn't. I have found the zero element to be (0,1) but I'm not sure how to work that into a proof. I am also not sure how to logically prove that if $a=(a_1,0)$, there is no inverse.