Endomorphisms/Bijective/Line to same line/Scalar/Exercise

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

$\varphi ,\psi \colon V\longrightarrow V$

denote automorphisms, such that, for every linear subspace ${}U\subseteq V$, the identity ${}\varphi (U)=\psi (U)$ holds. Show that ${}\varphi =a\psi$ with some ${}a\in K$.

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