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 [itex]e_1[/itex] and [itex]e_2[/itex] Now, we know that every vector in R^2 can also be uniquely represented by unique polar coordinates ([itex]r[/itex], [itex]\Theta[/itex]), 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.
I'm afraid my explanation is very coherent - sorry. Usually, coordinates are associated with manifolds. Sometimes, a manifold is also a vector space. In this case, coodinates are sometime associated with a basis for the vector space, but they don't have to be. The surface of the Earth is a 2-dimensional manifold that has lattitude and longitude as one coordinate system. The surface of the Earth is not a vector space, but it can, locally, be approximated by a vector space. The surface at any point can be approximated by a 2-dimensional plane that is tangent to the surface at that point. These tangent spaces are vector spaces, and coordinate systems give rise to particular bases for these vector spaces. In general, an n-dimensional (topological) manifold is something that, for a myopic observer, looks like a piece of R^n, i.e., any point of the manifold is contained in an open neighbourhoood from which which there is a continous map (with continuous inverse) onto an open subset of R^n. The image (under any such map) of any point p of an n-dimensional manifold is an element of R^n - the coordinates of p. R^2 is both a manifold and a vector space. (r, theta) coordinates use the manifold structure of R^2, but not the vector space structure. Cartesian coorinates are related of R^2 to both the manifold and vector space structures of R^2. (r, theta) coordinates do give rise, at point, to tangent vectors, though.
that makes a lot of sense, thanks. not sure why I didn't think of that :) LOL, now that I think about it, I remember that I was once asked to derive the radial and tangent basis vectors,a s a function of time, for a rotating reference frame on a physics quiz. Somehow never made the connection! :)