Endomorphism/Dilation/Every vector neq 0 is eigenvector/Exercise

Let ${}K$ be a field, and let ${}\varphi \colon V\rightarrow V$ be an endomorphism on a ${}K$-vector space ${}V$. Show that ${}\varphi$ is a homothety if and only if every vector ${}v\in V$, ${}v\neq 0$, is an eigenvector

of ${}\varphi$.

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