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b/docs/zusatz.tex index cab1fa4..6f8a99d 100644 --- a/docs/zusatz.tex +++ b/docs/zusatz.tex @@ -1636,20 +1636,19 @@ gelten. Aus der letzten Teilaufgabe erhielten wir $\dim(\ker(A))=2$, und hier wurde nebenbei gezeigt, dass $\rank(A)=\dim(\range(A))=3$. Also gilt $\dim(\ker(A))+\rank(A)=5=\dim(\reell^{5})$, - sodass die Dimensionsformel für lineare Abbildungen erfüllt ist.\footnote{ - Das heißt nicht, dass unsere berechneten Basen deswegen richtig ist. - Dies ist lediglich zu kontrollieren, - dass unsere Basen »nicht offensichtlich falsch« sind. - } + sodass die Dimensionsformel für lineare Abbildungen erfüllt ist. \begin{rem*} - Die hier verwendete Matrix, $A$, war »die gleiche« wie in Aufgabe 1. - Nur war der Körper anders. - Wir sehen, dass wir nicht einfach so die über $\reell$ berechnete Zeilenstufenform - in dieser Aufgabe übernehmen durften. - Wenn man die Berechnung der Zeilenreduktion als Zwischenschritt betrachtet, - so erkennt man, dass dieser Zwischenschritt beim Wechseln des Körper - der Sicherheit halber nochmals durchgeführt werden muss. + Die hier verwendete Matrix, $A$, war »die gleiche« wie in Aufgabe 1, + nur mit einem anderen Körper. + Wir sehen, dass wir die über $\reell$ berechnete Zeilenstufenform + in dieser Aufgabe nicht einfach so übernehmen durften. + D.\,h. wenn im 1. Teil einer Aufgabe man eine Basis des Lösungsraums von $A$ über $\reell$ bestimmen soll, + und dann im 2. Teil eine Basis des Spaltenraums von $A$ über $\reell$ + und dann im 3. Teil eine Basis des Spaltenraums von $A$ über bspw. $\mathbb{F}_{7}$, + dann kann man im 1.+2. Teil dieselbe Zeilenstufenform gebrauchen, + aber im 3. Teil berechnet man die Zeilenstufenform am liebsten ganz von vorne in dem neuen Körper, + um einer Nummer sicher zu gehen. \end{rem*} %% ********************************************************************************