Similarity

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.

Sources

• Seth Warner: Modern Algebra (1965): $\S 28$: Example $28.3$