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It is not hard to see that $(\mathbb R^2,+)$ with this product

$ {r\cdot(x,y)=(rx,ry) } $

is vector space over field $\mathbb R$.

I'm looking for another product that $(\mathbb R^2,+)$ is vector space over $\mathbb R$. I know

$n*(x,y)=\underbrace{(x,y)\oplus (x,y) \oplus \cdots \oplus (x,y)}_n=(nx,ny)$ but I have no idea for arbitrary element of $\mathbb R$. Any Suggestion. Thanks

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    You are trying to find other definitions for scalar multiplications?2012-06-23
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    Aren't the two products equivalent, at least for natural $n$?2012-06-23
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    @AsafKaragila yes I'm trying find another definitions for scalar multiplications2012-06-23
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    You point out that $n\cdot (x,y) = (n\cdot x, n\cdot y)$ for $n\in \Bbb N$. It's worth pointing out that you can extend this argument to show that $q\cdot (x,y) = (q\cdot x, q\cdot y)$ for $q\in \Bbb Q$. And, if you have just a bit more structure (like say, considering inner product space structures of $(\Bbb R^2,+)$ over $\Bbb R$), you can extend the argument further to any $r\in \Bbb R$.2012-06-23

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