# Vague question about polar coordinate basis

by jbusc
Tags: basis, coordinate, polar, vague
 P: 212 I hae a kind of strange, vague question. We know that any vector in R^2 can be uniquely represented by unique cartesian coordinates (x, y). If we wish to be more rigorous in our definition of "coordinates" we consider them to be the coefficients of the linear combination of the standard basis vectors $e_1$ and $e_2$ Now, we know that every vector in R^2 can also be uniquely represented by unique polar coordinates ($r$, $\Theta$), except for the zero vector. Does this mean that we can consider those "coordinates" to be coefficients with respect to some basis vectors? I would think no, but it seems odd that one coordinate system can be considered to have a "basis" and the other cannot. My only linear algebra text I have is Strang, who is vague on coordinate representations in general, and barely brings up polar coordinates at all.