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Interpret $\mathbb{R}^3$ as coordinate system.

I know coordinate system involves a $x$-axis, a $y$-axis, and a $z$ axis. I also know that $\mathbb{R}^3=\{(x,y,z): x,y,z\in\mathbb{R}\}$. This expressed as a set. But, I have no idea how to express it as a coordinate system. Any help is appreciated. Thank you.

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    I don't think it's clear what your requirements are to be a coordinate system. I think most people think of $\mathbb R^3$ as the set of ordered triples of real numbers. What else is there? – 2017-02-13
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    I guess how we know that (x,y,z) is a point from $\Re^3$. – 2017-02-13
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    Well you just told me how above. You wrote that $x,y,z \in \mathbb R$. Each coordinate belongs to $\mathbb R$, so the triple belongs to $\mathbb R^3$. Sounds exactly like what you want. – 2017-02-13
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    But, it is not expressed as coordinate system. That's what I am having trouble of. – 2017-02-13
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    Well, why not? What is it that's missing? It's not clear where you're stuck. – 2017-02-13
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    Exactly what is a coordinate system? – 2017-02-13
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    I think of coordinates as a way of parameterizing a space so that each point refers to a unique point in the space. Surely this is true, since you can parameterize $3D$ space in exactly the way you've described - using rectangular coordinates and the $x,y,z$ axes. – 2017-02-13
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    To have axes is the same as having *basis vectors*. A coordinate system on $\mathbb{R}^3$ is a set of basis vectors ${\bf e}_1,{\bf e}_2,{\bf e}_3$ such that any point in the set can be written uniquely on the form $(x,y,z) = a{\bf e}_1 + b{\bf e}_2 + c {\bf e}_3$. We then call $[a,b,c]$ the coordinates of the point $(x,y,z)$ with respect to the given basis. The standard basis of $\mathbb{R}^3$ is ${\bf e}_1 = (1,0,0)$, ${\bf e}_2 = (0,1,0)$ and ${\bf e}_3 = (0,0,1)$ for which $(x,y,z) = x{\bf e}_1 + y{\bf e}_2 + z {\bf e}_3$ and the coordinates $[x,y,z]$ are the same as the point itself. – 2017-02-13

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