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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 ($\Bbb R^n,+,\cdot)$?

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    $\begingroup$ What is it you don't understand about this problem? $\endgroup$ – Git Gud Apr 6 '13 at 13:49
  • $\begingroup$ 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 $\cdot$. $\endgroup$ – Hagen von Eitzen Apr 6 '13 at 13:51
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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.

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    $\begingroup$ @GitGud thanks i fixed it, and is used another symbol for the addition i hope that makes it clearer a bit. $\endgroup$ – Dominic Michaelis Apr 6 '13 at 13:55

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