Endomorphism/Restriction to eigenspace/Dilation/Exercise

Let ${}K$ be a field, and let ${}\varphi \colon V\rightarrow V$ be an endomorphism on a ${}K$-vector space ${}V$. Let ${}\lambda \in K$ and let

${}U=\operatorname {Eig} _{\lambda }{\left(\varphi \right)}\,$

be the corresponding eigenspace. Show that ${}\varphi$ can be restricted to a linear mapping

$\varphi {|}_{U}\colon U\longrightarrow U,v\longmapsto \varphi (v),$

and that this mapping is the homothety with scale factor ${}\lambda$.

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