Similarity Mapping is Linear Operator

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

Let $\beta \in K$.

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

is a linear operator on $G$.

Proof
Because:
 * $\map \beta {\mathbf x + \mathbf y} = \beta \mathbf x + \beta \mathbf y$
 * $\map \beta {\lambda \mathbf x} = \map \lambda {\beta \mathbf x}$

the fact of $s_\beta$ being a linear operator is immediately apparent.