Let v ∈ V {\displaystyle {}v\in V} . Then v ∈ Eig λ ( φ ) {\displaystyle {}v\in \operatorname {Eig} _{\lambda }{\left(\varphi \right)}} if and only if φ ( v ) = λ v {\displaystyle {}\varphi (v)=\lambda v} , and this is the case if and only if λ v − φ ( v ) = 0 {\displaystyle {}\lambda v-\varphi (v)=0} holds, which means ( λ ⋅ Id V − φ ) ( v ) = 0 {\displaystyle {}{\left(\lambda \cdot \operatorname {Id} _{V}-\varphi \right)}(v)=0} .
