Suppose there exists a set in R2 such that (x,y) +(x',y') = (x+x', y+y') and k(x,y) = (kx,0)
One of the required properties for a set to be a vector space is that there exists an additive inverse -u in the set that when added to a vector in the vector space u equals the additive identity, which I see to be (0,0)
I believe that there isn't such an additive inverse, but I'm not completely sure. Could someone clarify?
The additive inverse of $(x, y)$ is $(-x, -y)$. Note that this is not the same as the element
$$(-1) \cdot (x, y) = (-x, 0).$$
This, by the way, implies that we do not have a vector space with these operations.