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 $$s_\beta \left({\vec x}\right) = \beta \vec x$$ is a linear operator on $$G$$.

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

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

Proof

 * Since $$\beta \left({\vec x + \vec y}\right) = \beta \vec x + \beta \vec y$$ and $$\beta \left({\lambda \vec x}\right) = \lambda \left({\beta \vec x}\right)$$, the fact of $$s_\beta$$ being a linear operator is immediately apparent.


 * We have $$\left({s_{\beta^{-1}} \circ s_\beta}\right) \left({\vec x}\right) = \beta^{-1} \left({\beta \vec x}\right) = \vec x = \beta \left({\beta^{-1} \vec x}\right) = \left({s_\beta \circ s_{\beta^{-1}}}\right) \left({\vec x}\right)$$

which proves the second bit.

Comment
An older term for a similarity is similitude.