Similarity_Mapping_is_Linear_Operator

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 $G$'''.