How to prove a linear space? Hi I would like to prove the following:
Is the set $V=R^2$ with:
$ +: V \times V \to V$: $((x,y),(x',y')$ $\to(x \cdot x' -y \cdot y', x \cdot y'+y \cdot x')$
$ \cdot:\mathbb{R}  \times V \to V$: $(a,(x,y)) \to (a \cdot x, a \cdot y)$
a linear space?


*

*property: I would start with $x,y,z$ with $x=(x_1,x_2),y=(y_1,y_2),z=(z_1,z_2)$
in order to show that $(x+y)+z$ or  $(z+y)+x$ 


Here I just dont know how to proceed...

As an alternative to the answer by David Wheeler, here is a way to see it that just requires you to do a simple computation.
If this was a vector space, then we would have $v + v = 1\cdot v + 1\cdot v = (1+1)\cdot v = 2\cdot v$ for all $v\in V$ by the distributive rule.
But if $v = (1,1)$ then $2\cdot v = (2,2)$ while $v + v = (1\cdot 1 - 1\cdot 1, 1\cdot 1 + 1\cdot 1) = (0,2)$, so the distributive rule fails to hold.
