Similarity Mapping is Automorphism

Theorem
Let $G$ be a vector space over a field $K$.

Let $\beta \in K$.

Let $s_\beta: G \to G$ be the similarity on $G$ defined as:
 * $\forall \mathbf x \in G: \map {s_\beta} {\mathbf x} = \beta \mathbf x$

If $\beta \ne 0$ then $s_\beta$ is an automorphism of $G$.

Proof
By definition, a vector space automorphism on $G$ is a vector space isomorphism from $G$ to $G$ itself.

By definition, a vector space isomorphism is a mapping $\phi: V \to W$ such that:


 * $(1): \quad \phi$ is a bijection
 * $(2): \quad \forall \mathbf x, \mathbf y \in V: \map \phi {\mathbf x + \mathbf y} = \map \phi {\mathbf x} +' \map \phi {\mathbf y}$
 * $(3): \quad \forall \mathbf x \in V: \forall \lambda \in K: \map \phi {\lambda \mathbf x} = \lambda \map \phi {\mathbf x}$

It has been established in Similarity Mapping is Linear Operator that $s_\beta$ is a linear operator on $G$.

Hence $(2)$ and $(3)$ follow by definition of linear operator.

It remains to prove bijectivity.