Polynomring/Lineare Abbildung/Jacobiideal/Fakt/Beweis

Beweis

Die Kettenregel, angewendet auf

$K^{n}{\stackrel {L}{\longrightarrow }}K^{m}{\stackrel {F}{\longrightarrow }}K,$

liefert

${}\left({\frac {\partial {\left(F\circ L\right)}}{\partial X_{1}}},\,\ldots ,\,{\frac {\partial {\left(F\circ L\right)}}{\partial X_{n}}}\right)=\left({\frac {\partial F}{\partial Y_{1}}},\,\ldots ,\,{\frac {\partial F}{\partial Y_{m}}}\right)\circ L\,.$

Somit ist

${}{\frac {\partial {\left(F\circ L\right)}}{\partial X_{i}}}\in {\left({\frac {\partial F}{\partial Y_{1}}}\circ L,\ldots ,{\frac {\partial F}{\partial Y_{m}}}\circ L\right)}=L(J_{F})\,$

und

${}J_{F\circ L}\subseteq L(J_{F})\,.$

Zur bewiesenen Aussage