I was trying to solve this exercise:
Decide whether or not the set $\Bbb R^2$, with addition defined by
$$(x,y) + (a,b) = (x+a+1, y+b)$$
and with scalar multiplication $$r\cdot(x,y) = (rx+r-1, ry)$$
is a (real) vector space.
So, I try to prove via its space's properties (associatity, conmutativity, etc) and it works. Even preservation of scale. But when I try to prove for zero $(0\cdot v=0)$ It doesn't work for me, I mean $0\cdot(x, y) = (-1, 0)$ which is clearly not $(0,0)$
But book's answer is: it is an space, so I must be wrong. Now, thinking about it, I'm starting to believe that zero vector of vector spaces isn't necessary $(0,0,\dots,0)$. And then for this example $(-1,0)$ is the zero of the space, and it can be found taking any space element and doing $x - x$ ($x$ minus itself).
Am I right? Or am I omitting something more obvious. Thanks.