According to Linear Algebra, all linear transformations from $R^2$ to $R^2$ can be represented by a 2-by-2 matrix $A \in R^{2x2}$. Essentially, four real numbers (i.e., the entries of the table) are all that is required to represent all linear transformation in this space.
Note: Probably complex numbers also need to be accounted for in the above.
So what is the case for all transformations from $R^2$ to $R^2$ (both linear and non-linear) ? Is it possible to use $n$ numbers to represent all these transformations, as was done with 4 numbers for linear transformations?
Even if we restrict to analytic transformation(not to mention bigger spaces, such as smooth transformations, or continous transformations) , this cannot be achieved for any $n$, since the space of transformations in this case is infinite dimensional, u can see this by considering the following (linearly independent) family : $f_{n, m} (x, y) =x^{m} y^{n} $