Similarity Mapping is Linear Operator

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

Let $\beta \in K$.

Then the mapping:
 * $s_\beta: G \to G$ defined by $\map {s_\beta} {\mathbf x} = \beta \mathbf x$

is a linear operator on $G$.

If $\beta \ne 0$ then $s_\beta$ is an automorphism of $G$, and $\paren {s_\beta}^{-1} = s_{\beta^{-1} }$

The linear operators $s_\beta$, where $\beta \ne 0$, are called similarities of $G$.

Proof
Since $\map \beta {\mathbf x + \mathbf y} = \beta \mathbf x + \beta \mathbf y$ and $\map \beta {\lambda \mathbf x} = \map \lambda {\beta \mathbf x}$, the fact of $s_\beta$ being a linear operator is immediately apparent.

We have:
 * $\map {\paren {s_{\beta^{-1} } \circ s_\beta} } {\mathbf x} = \map {\beta^{-1} } {\beta \mathbf x} = \mathbf x = \map \beta {\beta^{-1} \mathbf x} = \map {\paren {s_\beta \circ s_{\beta^{-1} } } } {\mathbf x}$

which proves the second bit.

Also known as
An older term for a similarity is similitude.