This is a very basic question that I feel that I should know the answer to but haven't been able to think through clearly.
In linear algebra, we learn that the basis of a finite dimensional vector space can be thought of as 'co-ordinates' of that space. And, we model what we intuitively understand as the euclidean plane, using the vector space $\mathbb{R^2}$ equipped with the standard inner product and metric etc. The underlying space is taken to be independent of the choice of basis, that is we understand that properties that are inherent to the space are those that will be invariant under change of basis.
Now, $\mathbb{R^2}$ comes with a canonical basis: this can be understood as saying that given any arbitrary two dimensional vector space and any basis, the vectors $v_i$ of the basis under the co-ordinate map maps to $e_i$. Since, we have also introduced inner products and thus a notion of parallel, the intuitive picture we now have of the co-odrinate grid is a criss-cross of lines. and the 'co-ordinates' are called, imaginatively, 'rectangular co-ordinates'.
In school, we also learn about polar 'co-ordinates' of the plane. The associated picture is of concentric circles and rays fanning out of the origin. However, these 'co-ordinates' do not fit within the 'Basic Linear Algebra' framework (since among other things, $0$ has no unique representation and the functions that change the variables are not linear). The one way of seeing the transformation of the 'rulings' of the plane is to consider $\mathbb{R^2}$ as $\mathbb{C}$ and the change as the map $\exp: \mathbb C \to \mathbb C$.
What is the framework in which the notion of 'co-ordinates' subsumes both these pictures (likewise for cylindrical, spherical, etc in dimension(?) three). My second and connected question: is there a linear algebraic connection for using two numbers to represent points in polar co-ordinate, i.e. is it because the vector space dimension of $\mathbb{E^2}$ is two. My third question is: am I confusing different concepts of 'dimension' here?
Added: Thanks for all the replies. They are all great but given my continuing dissatisfaction, either I have not really understood the answers or haven't been able to communicate the question properly (or perhaps and quite likely, I don't know what I want to ask).
Now, when we think of a (topological) manifold, we think of some object that is locally euclidean, i.e., we already have a 'handle' on $\mathbb{E^n}$ and want to use our knowledge of being able to do things, such as calculus, onto the new object . In my question, I am looking at ways we can 'handle' $\mathbb{E^2}$ itself, so invoking manifolds, seems a bit like putting the cart before the horse. I want to say something like this: The point of polar co-ordinates is to represent the plane by looking at it as $S^1\times \mathbb{R}$, where $S^1$ is a basic object like $\mathbb R$.