Let $A:{\mathbb{R}}^2\to {\mathbb{R}}^2$ be a linear map. Let $\theta (-,-):{\mathbb{R}}^2\times {\mathbb{R}}^2\to [0,2\pi ]$ be a map that returns the angle between these two vectors. Then $A$ is called angle preserving 1;; $\theta (A{v}_1,A{v}_2)=\theta (v_1,v_2)$ for all $v_1,v_2\in {\mathbb{R}}^2$ nonzero and $det(A)>0$ In particular, this means that a conformal mapping is always injective and thus bijective.