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