From 6d8d4d24dd400cf8c0617f57c5c22163cd6bc619 Mon Sep 17 00:00:00 2001
From: raj_mathe <raj.dahya@math.uni-leipzig.de>
Date: Mon, 11 Jan 2021 12:48:25 +0100
Subject: [PATCH] master > master: quizzes 7--8

---
 docs/loesungen.pdf | Bin 552019 -> 558684 bytes
 docs/loesungen.tex | 265 ++++++++++++++++++++++++++++++++++++++++-----
 2 files changed, 238 insertions(+), 27 deletions(-)

diff --git a/docs/loesungen.pdf b/docs/loesungen.pdf
index 736a4b98e57e80f41ead5d5ed1293c72d6b6aecb..ebbe8325692abda0765af4f59c18423fddb43da3 100644
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diff --git a/docs/loesungen.tex b/docs/loesungen.tex
index 5ea7186..e1c79cd 100644
--- a/docs/loesungen.tex
+++ b/docs/loesungen.tex
@@ -76,6 +76,10 @@
 %%            — body/quizzes/quiz5.tex;
 %%            |
 %%            — body/quizzes/quiz6.tex;
+%%            |
+%%            — body/quizzes/quiz7.tex;
+%%            |
+%%            — body/quizzes/quiz8.tex;
 %%        |
 %%        — back/index.tex;
 %%            |
@@ -1386,7 +1390,7 @@
 
     \noindent
     \hraum{\footnotesize Raj Dahya}\hraum\\
-    \hraum{\small \itshape Fakultät für Mathematik und Informatik/Institut für Philosophie}\hraum\\
+    \hraum{\small \itshape Fakultät für Mathematik und Informatik}\hraum\\
     \hraum{\small \itshape Universität Leipzig.}\hraum\\
     \hraum{\small Wintersemester 2020/2021 }\hraum
 \end{titlepage}
@@ -1398,10 +1402,9 @@
 \chapter*{Vorwort}
 
 Dieses Dokument enthält Lösungsansätze zu den Übungsserien, Selbstkontrollenaufgaben, und Quizzes.
-Diese werden natürlich \emph{nach} Abgabefristen hochgeladen
-und dienen \emph{nicht} als Musterlösungen!
+Diese werden natürlich \emph{nach} Abgabefristen hochgeladen.
 Der Zweck dieser Lösungen ist es vielmehr, Ansätze zu präsentieren,
-mit denen man seine \emph{eigenen} Versuche vergleichen kann.
+mit denen man seine eigenen Versuche vergleichen kann.
 
 %% ********************************************************************************
 %% FILE: front/contents.tex
@@ -3076,11 +3079,6 @@ Seien $X$, $Y$ nicht leere Mengen und ${f:X\to Y}$ eine Funktion.
 \chapter[Woche 4]{Woche 4}
     \label{ueb:4}
 
-\textbf{ACHTUNG.}
-Diese Lösungen dienen \emph{nicht} als Musterlösungen sondern eher als Referenz.
-Hier wird eingehender gearbeitet, als generell verlangt wird.
-Das Hauptziel hier ist, eine Variant anzubieten, gegen die man seine Versuche vergleichen kann.
-
 %% AUFGABE 4-1
 \let\altsectionname\sectionname
 \def\sectionname{Aufgabe}
@@ -3110,13 +3108,13 @@ Das Hauptziel hier ist, eine Variant anzubieten, gegen die man seine Versuche ve
 
             \begin{kompaktenum}[\rtab][\rtab]
                 \item[\uwave{{\itshape Reflexivität:}}]
-                    Sei $(a,b)\in X$ beliebig.
+                    Sei $(a,b)\in X$ beliebig.\\
                     \textbf{Zu zeigen:} $(a,b)\sim(a,b)$.\\
                     Offensichtlich gilt $ab=ab$.\\
                     Per Konstruktion gilt also $(a,b)\sim(a,b)$.
 
                 \item[\uwave{{\itshape Symmetrie:}}]
-                    Seien $(a,b),(a',b')\in X$ beliebig.
+                    Seien $(a,b),(a',b')\in X$ beliebig.\\
                     \textbf{Zu zeigen:} ${(a,b)\sim(a',b')\Rightarrow(a',b')\sim(a,b)}$.\\
                     Es gilt
 
@@ -3185,7 +3183,7 @@ Das Hauptziel hier ist, eine Variant anzubieten, gegen die man seine Versuche ve
         für $(a,b),(a',b')\in X$ definiert wird.
 
         \begin{claim*}
-            $(X,\leq)$ ist ist eine Halbordnung aber \fbox{nicht total}.
+            $(X,\leq)$ ist eine partielle Ordnung aber \fbox{nicht total}.
         \end{claim*}
 
         \begin{proof}
@@ -3193,7 +3191,7 @@ Das Hauptziel hier ist, eine Variant anzubieten, gegen die man seine Versuche ve
 
             \begin{kompaktenum}[\rtab][\rtab]
                 \item[\uwave{{\itshape Reflexivität:}}]
-                    Sei $(a,b)\in X$ beliebig.
+                    Sei $(a,b)\in X$ beliebig.\\
                     \textbf{Zu zeigen:} $(a,b)\leq(a,b)$.\\
                     Offensichtlich gilt $a\leq a$ und $b\leq b$.\\
                     Per Konstruktion gilt also $(a,b)\leq(a,b)$.
@@ -3254,7 +3252,7 @@ Das Hauptziel hier ist, eine Variant anzubieten, gegen die man seine Versuche ve
                         \end{mathe}
             \end{kompaktenum}
 
-            Darum erfüllt $(X,\leq)$ die Axiome einer Halbordnung.\\
+            Darum erfüllt $(X,\leq)$ die einer partiellen Ordnung.\\
             Zum Schluss, beachte, dass $(0,1)$ und $(1,0)$ bzgl. $\leq$ unvergleichbar sind.
             Darum ist $(X,\leq)$ nicht total.
         \end{proof}
@@ -3405,8 +3403,10 @@ für $a,b\in\intgr$.
                         Seien $q\in\intgr$ und $r\in\{0,1,\ldots,n-1\}$ mit $qn+r=k$.
                         Da $k,r\in\{0,1,\ldots,n-1\}$,
                         gilt $qn=k-r\in\intgr\cap(-n,n)$.
-                        Also muss $q=0$ gelten.
-                        Also $r=k$.
+                        Das heißt, $k-r$ ist eine durch $n$ teilbare ganze Zahl
+                        in $(-n,n)$.
+                        Die einzige solche Zahl ist $0$.
+                        Also $k-r=0$.
                         Also $\modfn(k,n)=r=k$.
                     }
                     Also gilt $k\in\rho(\intgr/\lsim)$.
@@ -3429,7 +3429,7 @@ für $a,b\in\intgr$.
                     &= &\{a\in\intgr \mid \modfn(a,n)=\modfn(k,n)\}\\
                     &= &\{a\in\intgr \mid \modfn(a,n)=k\}\\
                     &= &\{a\in\intgr \mid \exists{q\in\intgr:~}a=qn+k\}\\
-                    &= &\{qn+r \mid q\in\intgr\}\\
+                    &= &\{qn+k \mid q\in\intgr\}\\
                     &= &\intgr\cdot n + k.\\
             \end{mathe}
 
@@ -6046,11 +6046,11 @@ Seien $n\in\ntrlpos$ und $\mathbf{v}_{i}\in V$ für $i\in\{1,2,\ldots,n\}$.
     Wir haben etwas ausführlicher gezeigt, dass die Menge $X'$ mindestens $n-1$ Goldfische enthält.
     Wenn wir $\tilde{x}:=x_{0}$ wählen entspricht dies der Größe des Schnitts $X_{0}\cap X_{1}$.
     Das \uline{Diagramm} mag andeuten, dass dieser Schnitt nicht leer ist, aber das Diagramm täuscht.
-    Im Induktionsschritt setzen wir nur voraus, dass $n\geq 1$.
-    Darum ist $n-1>0$ nur garantiert, wenn stattdessen $n'\geq 2$ vorausgesetzt wird.
+    Wir brauchen $n\geq 2$, damit der Schnitt nicht leer ist.
+    Aber im Induktionsschritt wird nur $n\geq 1$ vorausgesetzt.
 
-    Das heißt das Induktionsargument ist faul,
-    weil der Schritt $1\rightsquigarrow 2$ implizit übersprungen wird.
+    Kurzgesagt, das heißt das Induktionsargument ist faul,
+    \fbox{weil der Schritt $1\rightsquigarrow 2$ implizit übersprungen wird}.
 
 %% ********************************************************************************
 %% FILE: body/ska/ska5.tex
@@ -7612,6 +7612,222 @@ Wir betrachten die Komposition ${g\circ f:X\to Z}$
         \end{proof}
 \end{enumerate}
 
+%% ********************************************************************************
+%% FILE: body/quizzes/quiz7.tex
+%% ********************************************************************************
+
+\setcounternach{chapter}{7}
+\chapter[Woche 7]{Woche 7}
+    \label{quiz:7}
+
+\begin{enumerate}{\bfseries 1.}
+    \item
+        \begin{defn*}
+            Sei $V$ ein Vektorraum über einem Körper $K$.
+            Seien $\mathbf{v}_{1},\mathbf{v}_{2},\ldots,\mathbf{v}_{n}\in V$.
+            Die Vektoren, $\mathbf{v}_{1},\mathbf{v}_{2},\ldots,\mathbf{v}_{n}$,
+            heißen dann
+                \textbf{linear unabhängig},
+            wenn für alle Skalare, $c_{1},c_{2},\ldots,c_{n}\in K$,
+
+                \begin{mathe}[mc]{rcl}
+                    \sum_{i=1}^{n}c_{i}\cdot\mathbf{v}_{i}=\zerovector
+                    &\Longrightarrow
+                    &\forall{i\in\{1,2,\ldots,n\}:~}c_{i}=0_{K}
+                \end{mathe}
+
+            gilt.
+        \end{defn*}
+    \item
+        \begin{claim*}
+            Seien
+
+            \begin{mathe}[mc]{rclqrclqrcl}
+                \mathbf{v}_{1} &= &\begin{svector}1\\2\\2\\\end{svector}
+                &\mathbf{v}_{2} &= &\begin{svector}0\\2\\1\\\end{svector}
+                &\mathbf{v}_{3} &= &\begin{svector}2\\1\\1\\\end{svector}\\
+            \end{mathe}
+
+            Vektoren im Vektorraum $\mathbb{F}_{5}^{3}$ über dem Körper $\mathbb{F}_{5}$.
+            Diese Vektoren sind linear unabhängig.
+        \end{claim*}
+
+        \begin{proof}
+            Laut Satz \cite[Satz~5.2.4]{sinn2020} sind die Vektoren,
+                $\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3}$,
+            dann linear unabhängig, wenn $A\mathbf{x}=\zerovector$
+            nur die Triviallösung hat, wobei
+
+                \begin{mathe}[mc]{rcl}
+                    A &= &\begin{matrix}{ccc}
+1 &0 &2\\
+2 &2 &1\\
+2 &1 &1\\
+\end{matrix}.\\
+                \end{mathe}
+
+            Wir lösen also das homogene System:
+
+            \begin{algorithm}[\rtab][\rtab]
+                Zeilenoperationen
+                    ${Z_{2}\leftsquigarrow Z_{2} - 2\cdot Z_{1}}$
+                und
+                    ${Z_{3}\leftsquigarrow Z_{3} - 2\cdot Z_{1}}$
+                anwenden:\footnotemark[ft:1:aufg:2:quiz:8]
+
+                \begin{mathe}[mc]{rcl}
+                    \begin{matrix}{ccc}
+1 &0 &2\\
+0 &2 &2\\
+0 &1 &2\\
+\end{matrix}.\\
+                \end{mathe}
+
+                Zeilenoperation
+                    ${Z_{3}\leftsquigarrow 2\cdot Z_{3} - Z_{2}}$
+                anwenden:
+
+                \begin{mathe}[mc]{rcl}
+                    \begin{matrix}{ccc}
+1 &0 &2\\
+0 &2 &2\\
+0 &0 &2\\
+\end{matrix}.\\
+                \end{mathe}
+            \end{algorithm}
+
+            \footnotetext[ft:1:aufg:2:quiz:8]{
+                Beachte, dass wir im Körper $\mathbb{F}_{5}$ d.\,h. modulo 5 berechnen.
+            }
+
+            Aus der Zeilenstufenform geht hervor,
+            dass es keine freien unbekannten gibt und insbesondere,
+            dass das homogene System nur die Triviallösung besitzt.
+            Darum sind die Vektoren linear unabhängig.
+        \end{proof}
+\end{enumerate}
+
+%% ********************************************************************************
+%% FILE: body/quizzes/quiz8.tex
+%% ********************************************************************************
+
+\setcounternach{chapter}{8}
+\chapter[Woche 8]{Woche 8}
+    \label{quiz:8}
+
+\begin{claim*}
+    Sei $V$ ein Vektorraum über Körper $K$
+    und seien $U_{1},U_{2}\subseteq V$ lineare Unterräume.
+    Dann gilt
+
+        \begin{mathe}[bc]{rcl}
+            U_{1}+U_{2} = U_{2}
+            &\Longleftrightarrow
+            &U_{1}\subseteq U_{2}.\\
+        \end{mathe}
+
+    \nvraum{1}
+\end{claim*}
+
+    \begin{einzug}[\rtab][\rtab]
+    \begin{proof}
+        \hinRichtung Angenommen,
+
+            \begin{mathe}[bc]{rcl}
+                \eqtag[eq:1:quiz:8]
+                U_{1}+U_{2} &= &U_{2}.\\
+            \end{mathe}
+
+        \textbf{Zu zeigen:} $U_{1}\subseteq U_{2}$.
+        Da jeder lineare Unterraum den Nullvektor enthält
+        (siehe \cite[Lemma~1.4.2]{sinn2020}), gilt
+
+            \begin{mathe}[bc]{rcccccl}
+                U_{1}
+                &=
+                    &U_{1}
+                    +\underbrace{
+                        \{\zerovector\}
+                    }_{\subseteq U_{2}}
+                &\subseteq
+                    &U_{1}+U_{2}
+                &\eqcrefoverset{eq:1:quiz:8}{=}
+                    &U_{2}.\\
+            \end{mathe}
+
+        \herRichtung Angenommen, $U_{1}\subseteq U_{2}$.
+        \textbf{Zu zeigen:} $U_{1}+U_{2}=U_{2}$.\\
+        Für Mengengleichheit teilen wir dies in zwei Teile auf:
+
+        \BeweisRichtung[$\supseteq$] \textbf{Zu zeigen:} $U_{2}\subseteq U_{1}+U_{2}$.
+        Da jeder lineare Unterraum den Nullvektor enthält,
+
+            \begin{mathe}[bc]{rcccl}
+                U_{2}
+                &=
+                    &\underbrace{
+                        \{\zerovector\}
+                    }_{\subseteq U_{1}}
+                    +U_{2}
+                &\subseteq
+                    &U_{1}+U_{2}\\
+            \end{mathe}
+
+        \BeweisRichtung[$\subseteq$] \textbf{Zu zeigen:} $U_{1}+U_{2}\subseteq U_{2}$.
+        Es gilt:
+
+            \begin{mathe}[bc]{rcl}
+                U_{1}+U_{2}
+                &\subseteq &U_{2}+U_{2},
+                    \quad\text{da per Annahme $U_{1}\subseteq U_{2}$}\\
+                &\subseteq
+                    &U_{2},
+                \quad\text{da als Vektorraum $U_{2}$ unter Addition stabil ist}.\\
+            \end{mathe}
+    \end{proof}
+    \end{einzug}
+
+\begin{rem*}
+In diesem Beweis haben wir mit mengenweise Operationen gearbeitet.
+Z.\,B. aus $\{\zerovector\}\subseteq U_{1}$
+folgt $\{\zerovector\}+U_{2}\subseteq U_{1}+U_{2}$;
+und aus $U_{1}\subseteq U_{2}$
+folgt $U_{1}+U_{2}\subseteq U_{2}+U_{2}$.
+Diese Implikationen haben nichts mit linearer Algebra zu tun.
+Sie sind rein mengentheoretische Ergebnisse und lassen sich
+allgemein folgendermaßen zeigen:
+
+    \begin{mathe}[mc]{rcccccl}
+        A\star B
+        &= &\{a\star b\mid a\in A,\,b\in B\}
+        &\subseteq &\{a\star b\mid a\in A',\,b\in B\}
+        &= &A'\star B\\
+    \end{mathe}
+
+für alle Mengen, $A,A',B$ mit $A\subseteq A'$
+und alle Operationen $\star$ definiert auf $A'\times B$.
+
+Gebrauch machten wir auch von
+
+    \begin{mathe}[mc]{rcccccccl}
+        U + \{\zerovector\}
+        &= &\{u+v\mid u\in U,\,v\in\{\zerovector\}\}
+        &= &\{u+\zerovector\mid u\in U\}
+        &= &\{u\mid u\in U\}
+        &= &U\\
+    \end{mathe}
+
+für alle Teilmengen, $U\subseteq V$, und von
+
+    \begin{mathe}[mc]{rcccccccl}
+        U+U
+        &= &\{\underbrace{u+v}_{\in U}\mid u\in U,\,v\in U\}
+        &\subseteq &U\\
+    \end{mathe}
+
+für alle linearen Unterräume, $U\subseteq V$.
+\end{rem*}
+
 %% ********************************************************************************
 %% FILE: back/index.tex
 %% ********************************************************************************
@@ -7627,12 +7843,7 @@ Wir betrachten die Komposition ${g\circ f:X\to Z}$
 %% FILE: ./back/quelle.bib
 %% ********************************************************************************
 
-\begin{thebibliography}{EFT18}
-
-\bibitem[EFT18]{ebbinghaus2018}
-Heinz-Dieter Ebbinghaus, J\"org Flum, and Wolfgang Thomas.
-\newblock {\em {Einf\"uhrung in die mathematische Logik}}.
-\newblock 2018.
+\begin{thebibliography}{Wal16}
 
 \bibitem[Jec97]{jech1997}
 Thomas Jech.