# Which of the axioms for a vector space are satisfied by $(\Bbb R^n,+,\cdot)$?

On $\Bbb R^n$ define two operations

$\alpha+\beta=\alpha-\beta$;

$c\cdot \alpha=-c\alpha$

The operation on the right are the usual ones.Which of the axioms for a vector space are satisfied by ($\Bbb R^n,+,\cdot)$?

• What is it you don't understand about this problem? – Git Gud Apr 6 '13 at 13:49
• What axiom system? For me, there is only one axiom: $(V,+,\cdot)$ is a $k$-vector space if $(V,+)$ is an abelian group on which $k$ acts via $\cdot$. – Hagen von Eitzen Apr 6 '13 at 13:51

I guess even with parenthesis it is a bit complicated I gonna use the symbol $\boxplus$ for your addition defined by $a\boxplus b=a-b$.
It is surely not associative, as $$(\alpha \boxplus\beta)\boxplus \gamma=(\alpha-\beta) \boxplus \gamma=\alpha-\beta - \gamma$$ but on the other hand $$\alpha \boxplus(\beta\boxplus\gamma)=\alpha \boxplus (\beta-\gamma) = \alpha-\beta+\gamma$$ just calculate on after the other. There are a lot of axioms, and you learn the most if you do it yourself.