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?


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*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...

A hint: Show that $(0,0)$ has no additive inverse. This is tricky: the additive identity is $(1,0)$, since:
$(x,y) + (1,0) = (x\cdot 1 - y\cdot 0, x\cdot 0 + y \cdot 1) = (x,y)$ and
$(1,0) + (x,y) = (1\cdot x - 0 \cdot y, 1 \cdot y + 0\cdot x) = (x,y)$
So you need to show there is NO $(x,y)$ with:
$(0,0) + (x,y) = (1,0)$.
