Homothety/Adjoint endomorphism/Example

For a homothety ${\displaystyle {}V\rightarrow V}$ with scaling factor ${\displaystyle {}s\in {\mathbb {K} }}$ on a ${\displaystyle {}{\mathbb {K} }}$-vector space ${\displaystyle {}V}$, endowed with an inner product, the homothety with scaling factor ${\displaystyle {}{\overline {s}}}$ is the adjoint mapping. Indeed, we have

${\displaystyle {}\left\langle sv,w\right\rangle =s\left\langle v,w\right\rangle ={\overline {\overline {s}}}\left\langle v,w\right\rangle =\left\langle v,{\overline {s}}w\right\rangle \,.}$