Lineare Abbildung/Polynomring und Spektrum/Fakt/Beweis

Es seien ${\displaystyle {}{\mathfrak {m}}_{v}={\left\{F\in K[V]\mid F(v)=0\right\}}}$ und ${\displaystyle {}{\mathfrak {m}}_{w}={\left\{G\in K[W]\mid G(w)=0\right\}}}$ die zu den Vektoren ${\displaystyle {}v}$ bzw. ${\displaystyle {}w}$ gehörigen maximalen Ideale. Die Aussage folgt aus

${\displaystyle {}{\begin{aligned}\varphi ^{*}({\mathfrak {m}}_{v})&=\varphi ^{-1}({\mathfrak {m}}_{v})\\&={\left\{G\in K[W]\mid \varphi (G)\in {\mathfrak {m}}_{v}\right\}}\\&={\left\{G\in K[W]\mid \varphi (G)(v)=0\right\}}\\&={\left\{G\in K[W]\mid {\left(G\circ \psi \right)}(v)=0\right\}}\\&={\left\{G\in K[W]\mid G(\psi (v))=0\right\}}\\&={\mathfrak {m}}_{\psi (v)}.\end{aligned}}}$