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back/index.tex; %% | @@ -1268,6 +1270,388 @@ gelten. und auf \textbf{c')}, \textbf{b')}, \textbf{d')} »Ja«. \end{rem} +%% ******************************************************************************** +%% FILE: body/linear-systems.tex +%% ******************************************************************************** + +\chapter[Lineare Gleichungssysteme]{Lineare Gleichungssysteme} + \label{ch:lgs} + +%% AUFGABE 1 +\let\altsectionname\sectionname +\def\sectionname{Aufgabe} +\section[Aufgabe 1]{} + \label{sec:9} +\let\sectionname\altsectionname + + \begin{exer*} + Bestimmen Sie eine Basis des Lösungsraums von dem Gleichungssystem, + ${A\mathbf{x}=\zerovector}$, + wobei $A$ die Matrix $\begin{smatrix} + 2 &-4 &0 &6 &2\\ + 3 &-6 &1 &13 &2\\ + -7 &14 &-1 &-32 &-9\\ + \end{smatrix}$ + über dem Körper ${K=\reell}$ ist. + \end{exer*} + + \begin{soln*} + Um die \uline{allgemeine Form} der Lösungen + zu ${A\mathbf{x}=\zerovector}$ zu bestimmen, + reduzieren wir zunächst $A$ auf Zeilenstufenform + und normalisieren: + + \begin{mathe}[mc]{rcl} + A &\xrightarrow{\substack{ + Z_{2} \mapsfrom 2\cdot Z_{2} - 3\cdot Z_{1}\\ + Z_{3} \mapsfrom 2\cdot Z_{3} + 7\cdot Z_{1}\\ + }} & + \begin{matrix}{rrrrr} + 2 &-4 &0 &6 &2\\ + 0 &0 &2 &8 &-2\\ + 0 &0 &-2 &-22 &-4\\ + \end{matrix} + \\ + &\xrightarrow{\substack{ + Z_{3} \mapsfrom Z_{3} + Z_{2}\\ + }} & + \begin{matrix}{rrrrr} + 2 &-4 &0 &6 &2\\ + 0 &0 &2 &8 &-2\\ + 0 &0 &0 &-14 &-6\\ + \end{matrix} + \\ + &\xrightarrow{\substack{ + Z_{1} \mapsfrom 7\cdot Z_{1} + 3\cdot Z_{3}\\ + Z_{2} \mapsfrom 7\cdot Z_{2} + 4\cdot Z_{3}\\ + }} & + \begin{matrix}{rrrrr} + 14 &-28 &0 &0 &-4\\ + 0 &0 &14 &0 &-38\\ + 0 &0 &0 &-14 &-6\\ + \end{matrix} + \\ + &\xrightarrow{\substack{ + Z_{1} \mapsfrom Z_{1}:2\\ + Z_{2} \mapsfrom Z_{2}:2\\ + Z_{3} \mapsfrom Z_{3}:-2\\ + }} & + \begin{matrix}{ccccc} + \boxed{7} &-14 &0 &0 &-2\\ + 0 &0 &\boxed{7} &0 &-19\\ + 0 &0 &0 &\boxed{7} &3\\ + \end{matrix} + .\\ + \end{mathe} + + Aus der Zeilenstufenform ergibt sich, + in + ${A\mathbf{x}=\zerovector}$ + die Variablen $x_{2}, x_{5}$ frei sind + und + + \begin{mathe}[mc]{rcr0lcr0l} + \eqtag[eq:1:ex:1:ch:lgs] + x_{4} &= & && &-(3/7)&x_{5}\\ + x_{3} &= & && &(19/7)&x_{5}\\ + x_{1} &= &2&x_{2} &+ &(2/7)&x_{5}.\\ + \end{mathe} + + Eine Basis des Lösungsraums, + ${\ker(A)=\{\mathbf{x}\in\reell^{5}\mid A\mathbf{x}=\zerovector\}}$, + lässt sich nach \cite[Satz~5.3.8]{sinn2020} finden, + indem wir die Lösungen berechnen, + für die genau eine der freien Unbekannten auf einen Wert ungleich $0$ + und alle anderen auf $0$ gesetzt werden.\footnote{ + Im \cite[Satz~5.3.8]{sinn2020} wird beschrieben, dass man $1$ verwendet, + aber man kann die Elemente einer Basis + mit beliebigen Werten ungleich $0$ multiplizieren, + ohne zu ändern, + dass die Menge eine Basis ist. + } + Hier gilt + + \begin{mathe}[mc]{rcrcl} + x_{2}=1,\,x_{5}=0 + &\Longrightarrow + &\mathbf{x} + &\eqcrefoverset{eq:1:ex:1:ch:lgs}{=} + &\begin{svector} 2\\ 1\\ 0\\ 0\\ 0\\\end{svector}\\ + x_{2}=0,\,x_{5}=7 + &\Longrightarrow + &\mathbf{x} + &\eqcrefoverset{eq:1:ex:1:ch:lgs}{=} + &\begin{svector} 2\\ 0\\ 19\\ -3\\ 7\\\end{svector}\\ + \end{mathe} + + Darum ist \fbox{$\bigg\{\begin{svector} 2\\ 1\\ 0\\ 0\\ 0\\\end{svector}, \begin{svector} 2\\ 0\\ 19\\ -3\\ 7\\\end{svector}\bigg\}$} + eine Basis des Lösungsraums. + \end{soln*} + + \begin{rem*} + Es ist empfehlenwert hier zu überprüfen, + dass $A\mathbf{x}$ wirklich gleich $\zerovector$ für alle Basiselemente gilt. + \end{rem*} + + \begin{exer*} + Bestimmen Sie den Spaltenraum von $A$ aus der letzten Aufgabe (noch über dem Körper $K=\reell$). + \end{exer*} + + \begin{soln*} + Bezeichne mit $a^{(j)}\in\reell^{3}$ für $j\in\{1,2,\ldots,5\}$ + die Spalten von $A$. + Aus der o.\,s. Zeilenstufenform, + + \begin{mathe}[mc]{c} + \begin{matrix}{ccccc} + \boxed{7} &-14 &0 &0 &-2\\ + 0 &0 &\boxed{7} &0 &-19\\ + 0 &0 &0 &\boxed{7} &3\\ + \end{matrix}, + \end{mathe} + + geht hervor, dass Spalten $1$, $3$, $4$ aus $A$ eine Basis + des Spaltenraums bilden.\footnote{ + Die allgemeine Begründung ist wie folgt: + Aus der Zeilenstufenform folgt, dass Spalten $1$, $3$, $4$ aus $A$ + linear unabhängig sind und dass die übrigen von diesen linear abhängig sind. + D.\,h. $\cal{A}:=\{a^{(1)},\,a^{(3)},\,a^{(4)}\}$ ist linear unabhängig + und $a^{(2)},a^{(5)}\in\vectorspacespan\cal{A}$. + Da $\range(A)=\vectorspacespan\{a^{(1)},\,a^{(2)},\,a^{(3)},\,a^{(4)},\,a^{(5)}\}$, + ist folglich $\cal{A}$ eine Basis für $\range(A)$. + } + Also ist + + \begin{mathe}[mc]{c} + \boxed{\bigg\{ + \begin{vector} 2\\ 3\\ -7\\\end{vector}, + \begin{vector} 0\\ 1\\ -1\\\end{vector}, + \begin{vector} 6\\ 13\\ -32\\\end{vector} + \bigg\}} + \end{mathe} + + eine Basis des Spaltenraums. + \end{soln*} + + \textbf{Zur Kontrolle:} Aus der letzten Teilaufgabe erhielten wir + eine Basis des Lösungsraums der Länge $2$, + d.\,h. $\dim(\ker(A))=2$, + und hier wurde eine Basis des Spaltenraums der Länge $3$, + d.\,h. $\rank(A)=\dim(\range(A))=3$. + Wir sehen dass $\dim(\ker(A))+\rank(A)=5=\dim(\reell^{5})$, + 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. + } + + \begin{rem*} + Falls in der letzten Aufgabe der Körper etwas anderes wäre, + so hätten wir die Zeilenstufenform erneut für diesen Körper berechnen müssen. + \end{rem*} + +%% AUFGABE 2 +\clearpage +\let\altsectionname\sectionname +\def\sectionname{Aufgabe} +\section[Aufgabe 2]{} + \label{sec:10} +\let\sectionname\altsectionname + + \begin{exer*} + Bestimmen Sie eine Basis des Lösungsraums von dem Gleichungssystem, + ${A\mathbf{x}=\zerovector}$, + wobei $A$ die Matrix $\begin{smatrix} + 2 &-4 &0 &6 &2\\ + 3 &-6 &1 &13 &2\\ + -7 &14 &-1 &-32 &-9\\ + \end{smatrix}$ + über dem Körper ${K=\mathbb{F}_{7}}$ ist. + \end{exer*} + + \begin{soln*} + Um die \uline{allgemeine Form} der Lösungen + zu ${A\mathbf{x}=\zerovector}$ zu bestimmen, + reduzieren wir $A$ auf Zeilenstufenform + und normalisieren. + Bei/vor jedem Schritt ersetzen wir Werte durch kanonische Darstellungen von Zahlen modulo $7$. + + \begin{mathe}[mc]{rcl} + A &= & + \begin{matrix}{rrrrr} + 2 &3 &0 &6 &2\\ + 3 &1 &1 &6 &2\\ + 0 &0 &6 &3 &5\\ + \end{matrix} + \end{mathe} + + Die Zeilenstufenform wird wie folgt berechnet: + + \begin{mathe}[mc]{rcl} + A &\xrightarrow{\substack{ + Z_{2} \mapsfrom Z_{2}+2\cdot Z_{1}\\ + }} & + \begin{matrix}{rrrrr} + 2 &3 &0 &6 &2\\ + 0 &0 &1 &4 &6\\ + 0 &0 &6 &3 &5\\ + \end{matrix} + \\ + &\xrightarrow{\substack{ + Z_{3} \mapsfrom Z_{3} + Z_{2} + }} & + \begin{matrix}{rrrrr} + 2 &3 &0 &6 &2\\ + 0 &0 &1 &4 &6\\ + 0 &0 &0 &0 &4\\ + \end{matrix} + \\ + &\xrightarrow{\substack{ + Z_{1} \mapsfrom 2\cdot Z_{1} - Z_{3}\\ + Z_{2} \mapsfrom 2\cdot Z_{2} - 3\cdot Z_{3}\\ + }} & + \begin{matrix}{rrrrr} + 4 &6 &0 &5 &0\\ + 0 &0 &2 &1 &0\\ + 0 &0 &0 &0 &4\\ + \end{matrix} + \\ + &\xrightarrow{\substack{ + Z_{1} \mapsfrom 2\cdot Z_{1}\\ + Z_{2} \mapsfrom 4\cdot Z_{2}\\ + Z_{3} \mapsfrom 2\cdot Z_{3}\\ + }} & + \begin{matrix}{rrrrr} + \boxed{1} &5 &0 &3 &0\\ + 0 &0 &\boxed{1} &4 &0\\ + 0 &0 &0 &0 &\boxed{1}\\ + \end{matrix} + .\\ + \end{mathe} + + (Beachte, dass $2^{-1}=4$ und $4^{-1}=2$ in $\mathbb{F}_{7}$, weil $2\cdot 4=8\equiv 1\mod 7$. + Darum wurden im letzten Schritt die Zeilen jeweils mit $2$ oder $4$ multipliziert.) + + Aus der Zeilenstufenform ergibt sich, + in + ${A\mathbf{x}=\zerovector}$ + die Variablen $x_{2},x_{4}$ frei sind + und + + \begin{mathe}[mc]{rcr0lcr0l} + x_{5} &= &0\\ + x_{3} &= & && &-4&x_{4}\\ + x_{1} &= &-5&x_{2} &- &3&x_{4}.\\ + \end{mathe} + + Eine Basis des Lösungsraums, + ${\ker(A)=\{\mathbf{x}\in\mathbb{F}_{7}^{5}\mid A\mathbf{x}=\zerovector\}}$, + lässt sich nach \cite[Satz~5.3.8]{sinn2020} finden, + indem wir die Lösungen berechnen, + für die genau eine der freien Unbekannten auf einen Wert ungleich $0$ + und alle anderen auf $0$ gesetzt werden. + Hier gilt + + \begin{mathe}[mc]{rcrclcl} + x_{2}=1,\,x_{4}=0 + &\Longrightarrow + &\mathbf{x} + &= &\begin{svector} -5\\ 1\\ 0\\ 0\\ 0\\\end{svector} + &= &\begin{svector} 2\\ 1\\ 0\\ 0\\ 0\\\end{svector}\\ + x_{2}=0,\,x_{4}=1 + &\Longrightarrow + &\mathbf{x} + &= &\begin{svector} -3\\ 0\\ -4\\ 1\\ 0\\\end{svector} + &= &\begin{svector} 4\\ 0\\ 3\\ 1\\ 0\\\end{svector}\\ + \end{mathe} + + Darum ist \fbox{$\bigg\{\begin{svector} 2\\ 1\\ 0\\ 0\\ 0\\\end{svector}, \begin{svector} 4\\ 0\\ 3\\ 1\\ 0\\\end{svector}\bigg\}$} + eine Basis des Lösungsraums. + \end{soln*} + + \begin{rem*} + Wiederum wird empfohlen, zu kontrollieren, + dass $A\mathbf{x}=\zerovector$ für alle Basiselemente gilt. + Beachte hier, dass wir modulo $7$ berechnen sollen.\footnote{ + In \textbf{octave}, \textbf{python}, \textit{etc.} + benutzt man \texttt{\%} für Moduloberechnungen. + In \textbf{octave} kann auch direkt + \texttt{(A \* [2, 1, 0, 0, 0].') \% 7} + eingeben. + In \textbf{python} kann man + \texttt{(np.matmul(A, [2, 1, 0, 0, 0]) \% 7} + eingeben (wenn man vorher \textit{numpy} als \textit{np} konventionsgemäß importiert). + } + Bei den o.\,s. Lösungen kommen + $% + A\cdot \begin{svector} 2\\ 1\\ 0\\ 0\\ 0\\\end{svector} + =\begin{svector} 0\\ 0\\ 0\\\end{svector} + $ + und + $% + A\cdot \begin{svector} 4\\ 0\\ 3\\ 1\\ 0\\\end{svector} + =\begin{svector} 14\\ 28\\ -63\\\end{svector} + =\begin{svector} 0\\ 0\\ 0\\\end{svector} + $ + raus, sodass wir erleichtert sein können, + dass unsere Basiselemente richtig sind. + (Ob die Größe der Basis stimmt, ist aber nicht damit überprüft. + Da müssen wir einfach prüfen, dass die Zeilenstufenform richtig berechnet wurde, + um die Anzahl der Stufen und freien Variablen zu bestätigen.) + \end{rem*} + + \begin{exer*} + Bestimmen den Spaltenraum von $A$ aus der letzten Aufgabe (noch über dem Körper $K=\mathbf{F}_{7}$). + \end{exer*} + + \begin{soln*} + Bezeichne mit $a^{(j)}\in\reell^{3}$ für $j\in\{1,2,\ldots,5\}$ + die Spalten von $A$. + Aus der o.\,s. Zeilenstufenform, + + \begin{mathe}[mc]{c} + \begin{matrix}{ccccc} + \boxed{1} &5 &0 &3 &0\\ + 0 &0 &\boxed{1} &4 &0\\ + 0 &0 &0 &0 &\boxed{1}\\ + \end{matrix}, + \end{mathe} + + geht hervor, dass Spalten $1$, $3$, $5$ aus $A$ eine Basis + des Spaltenraums bilden. + Also ist + + \begin{mathe}[mc]{c} + \boxed{\bigg\{ + \begin{vector} 2\\ 3\\ -7\\\end{vector}, + \begin{vector} 0\\ 1\\ -1\\\end{vector}, + \begin{vector} 2\\ 2\\ -9\\\end{vector} + \bigg\}} + \end{mathe} + + eine Basis des Spaltenraums. + \end{soln*} + + \textbf{Zur Kontrolle:} + 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. + } + + \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. + \end{rem*} + %% ******************************************************************************** %% FILE: back/index.tex %% ********************************************************************************