Vector space/Finite-dimensional/Inner product/Adjoint endomorphism/Properties/Fact/Proof/Exercise

Let ${}V$ be a finite-dimensional ${}{\mathbb {K} }$-vector space, endowed with an inner product ${}\left\langle -,-\right\rangle$. Show that the adjoint endomorphism fulfills the following properties (here, ${}\varphi ,\psi$ denote endomorphisms).

1. ${}{\left(\varphi +\psi \right)}^{\hat {}}={\hat {\varphi }}+{\hat {\psi }}\,.$

2. ${}{\left(s\varphi \right)}^{\hat {}}={\overline {s}}{\hat {\varphi }}\,.$

3. ${}{\left({\hat {\varphi }}\right)}^{\hat {}}=\varphi \,.$

4. ${}{\left(\varphi \circ \psi \right)}^{\hat {}}={\hat {\psi }}\circ {\hat {\varphi }}\,.$

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