Endomorphism space/Conjugation/Isomorphism/Exercise

Let ${}V$ be a ${}K$-vector space, and let

$g\colon V\longrightarrow V$

denote an isomorphism. Show that the mapping

$\Psi _{g}\colon \operatorname {End} _{}{\left(V\right)}\longrightarrow \operatorname {End} _{}{\left(V\right)},f\longmapsto gfg^{-1},$

is an isomorphism of vector spaces, and that, moreover, the identities

${}\Psi _{g}(f_{1}\circ f_{2})=\Psi _{g}(f_{1})\circ \Psi _{g}(f_{2})\,$

and

${}\Psi _{g}(\operatorname {Id} _{V})=\operatorname {Id} _{V}\,$

hold.

Create a solution