Affine mapping/Linear part/Homomorphism/Exercise

Let ${}E$ be an affine space over the ${}K$-vector space ${}V$. Show that the mapping that assigns to an affine mapping

$\psi \colon E\longrightarrow E$

its linear part ${}\psi _{0}$, satisfies the following properties.

1. ${}{\left(\operatorname {Id} _{E}\right)}_{0}=\operatorname {Id} _{V}\,,$

2. ${}(\psi \circ \varphi )_{0}=\psi _{0}\circ \varphi _{0}\,.$

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