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