Suppose $f$ is a bijection $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$ where $n\geq2$. If $f$ maps 1-D flat to 1-D flat, is $f$ an affine map?
Given two vectors $\boldsymbol{u}$ and $\boldsymbol{v}$, $c\boldsymbol{u}+(1-c)\boldsymbol{v}$ represents an 1-D flat, where $c$ is a variable scalar. Since $f$ maps 1-D flats to 1-D flats, $$ \forall c\in\mathbb{R}, \exists c'\in\mathbb{R}, f(c\boldsymbol{u}+(1-c)\boldsymbol{v}) = c'f(\boldsymbol{u})+(1-c')f(\boldsymbol{v}) $$ I think I can prove that $f$ is affine once I prove that $c=c'$, but I don't know how to prove $c=c'$.