Matrices/Inner automorphisms/Base change/Example

For a fixed invertible matrix ${\displaystyle {}B\in \operatorname {GL} _{n}\!{\left(K\right)}}$, the conjugation

${\displaystyle \kappa _{B}\colon \operatorname {GL} _{n}\!{\left(K\right)}\longrightarrow \operatorname {GL} _{n}\!{\left(K\right)},M\longmapsto BMB^{-1},}$

is just the mapping that assigns, to a describing matrix ${\displaystyle {}M}$ of a linear mapping with respect to a basis, the describing matrix with respect to a new basis.