Proof similarity transformations are bijective. I'm trying to prove that similarity transformations from $\mathbb{R}^2$ to $\mathbb{R}^2$ are bijective, I've already proven its injectivity by contradiction, so I'm really just stuck on the surjectivity.
Let's define similarity transformations, we call a function $\psi : \mathbb{R}^2 \longrightarrow  \mathbb{R}^2$ a similarity transformation if it is not constant and if there exists an $\alpha \in \mathbb{R}_{>0}$ such that for all $x, y \in \mathbb{R}^2$ so that $x \neq y$  we have $\alpha \cdot d(x,y) = d(\psi(x), \psi(y))$.
I proved that a similarity transformation is injective by contradiction, a contradiction follows from the definition of non-injectivity and the fact that $\alpha$ is bigger than zero.
Any help would be greatly appreciated!

Hint:
Prove $\psi$ is an affine map, and the associated linear map is injective. Hence, as it is an endomorphism in finite dimension, this linear map is bijective.
