>>11275564a vector in R^n gives you an operator on functions, the directional derivative, and the vector is fully determined by how it differentiates functions. the "standard basis" vectors correspond precisely to the operators of partial derivatives. so to answer your question
>Partial derivatives = co-ordinate bases?yes
the point is that definition of vectors as "bunch of numbers" literally works only in R^n. definition as certain differential operators works in a much more general setting.
