I was reading my text and just had some questions about transformations:
(1) Are all line preserving transformations linear transformations? Why?
I want to say yes... but I feel like the answer is no, haha.
(2) Are all linear transformations angle preserving transformations?
(3) Define a transformation $t: z \mapsto t(z)$ that is not a Möbius transformation.
Is that anything that is not one-to-one? So can I just define $z \mapsto\lvert z\rvert$?
Thanks!
(1) If by "line-preserving" you mean there is some line $L=\left\{z_0+tz_1|t\in\mathbb{R}\right\}$ such that $f(L)=L$, then the answer is no. Consider the line $\left\{t|t\in\mathbb{R}\right\}$ and the map $f:z\mapsto z+1$. This is clearly not linear over $\mathbb{C}$, but does preserve a line.
(2) No. Consider the linear map $f:z\mapsto 0$. This pretty clearly doesn't preserve angles, if you even try to define the "angle" between $0$ and itself.
(3) Is this different than (2)? Yes, $f:z\mapsto \left|z\right|$ would work, if you call that a transformation (I guess it depends on your definition of a "transformation").
Sorry if this is to basic. Hope it helps.