# Linear after change of variables

Many interesting non-linear functions of several variables like $\|x\|^2 =\sum\limits_{k=1}^n x_k^2$ become linear in "different variables". Is there a characterization of all (continuous, differentiable,...) functions $f:\mathbb R^n \to \mathbb R^m$ such that there exist $h:\mathbb R\to\mathbb R$ and a linear map $A:\mathbb R^n\to \mathbb R^m$ with $f(x_1,\ldots,x_n)= A(h(x_1),\ldots,h(x_n))$ for all $x=(x_1,\ldots,x_n)\in \mathbb R^n$?

• I see, I misunderstud a little the question, I was thinking that $h$ is a diffeo of $\mathbb R^n$, while you allow only diffeos of the form $\otimes^n h$ – user126154 Mar 25 '14 at 10:19