Similarity Mapping is Linear Operator

Theorem
Let $G$ be a vector space over a field $\struct {K, + \times}$.

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
To prove that $s_\beta$ is a linear operator it is sufficient to demonstrate that:
 * $(1): \quad \forall \mathbf x, \mathbf y \in G: \map {s_\beta} {\mathbf x + \mathbf y} = \map {s_\beta} {\mathbf x} + \map {s_\beta} {\mathbf y}$
 * $(2): \quad \forall \mathbf x \in G: \forall \lambda \in K: \map {s_\beta} {\lambda \mathbf x} = \lambda \map {s_\beta} {\mathbf x}$

Indeed:

and:

Hence the result.