I don't know whether this is a question at all but I would like to know why an element $(x,y)$ of $\mathbb R^2:=\mathbb R\times\mathbb R$ corresponds exactly to the point $(x,y$) in the Cartesian coordinate system (as in coordinates determined by the ordered basis $\{i,j\}$).
This doesn't happen if we use polar coordinates, for example $(1,1)\in\mathbb R^2$ corresponds to $(\sqrt 2,\pi/4)$. What is the crucial property missing from other systems and how can I prove this?
$\mathbb{R}^n$ is defined to be the set of $n$-tuples of real numbers.
So there's a canonical basis $\{e_1,e_2,...,e_n\}$ of $\mathbb{R}^n$ (seen as a vector space) given by $$e_1 = (1,0,0,...,0)$$ $$e_2 = (0,1,0,...,0)$$ $$\vdots$$ $$e_n = (0,0,0,...,1)$$ Now consider the dual basis $\{e^1,e^2,...,e^n\}$ s.t. $e^i(e_j)=\delta^i_j$.
The dual basis gives you natural coordinates $\{x^1,...,x^n\}$ on $\mathbb{R}^n$ given by $x^i(v):=e^i(v)$ for $i=1,2,...,n$ and $v\in \mathbb{R}^n$.
Then you can play with other systems of coordinates.
In the end, the canonical one is simply due to the fact that $\mathbb{R}^n$ are $n$-tuples of real numbers.