From 3f313cab814458fafd046ff434c7e4cde1e825e5 Mon Sep 17 00:00:00 2001 From: christian Date: Wed, 6 Nov 2019 18:29:44 +0100 Subject: [PATCH] add some lecture notes --- ws2019/ana/lectures/analysis3.pdf | Bin 0 -> 113833 bytes ws2019/ana/lectures/analysis3.tex | 140 ++++++++++++++++ ws2019/ana/lectures/analysis4.pdf | Bin 0 -> 132822 bytes ws2019/ana/lectures/analysis4.tex | 176 ++++++++++++++++++++ ws2019/ana/lectures/analysis5.pdf | Bin 0 -> 137138 bytes ws2019/ana/lectures/analysis5.tex | 268 ++++++++++++++++++++++++++++++ ws2019/ana/lectures/linear4.pdf | Bin 0 -> 131858 bytes ws2019/ana/lectures/linear4.tex | 233 ++++++++++++++++++++++++++ ws2019/ana/lectures/linear5.pdf | Bin 0 -> 118241 bytes ws2019/ana/lectures/linear5.tex | 259 +++++++++++++++++++++++++++++ ws2019/la/lectures/linear4.pdf | Bin 0 -> 131858 bytes ws2019/la/lectures/linear4.tex | 233 ++++++++++++++++++++++++++ ws2019/la/lectures/linear5.pdf | Bin 0 -> 118241 bytes ws2019/la/lectures/linear5.tex | 259 +++++++++++++++++++++++++++++ ws2019/la/uebungen/la2.pdf | Bin 0 -> 95928 bytes ws2019/la/uebungen/la2.tex | 215 ++++++++++++++++++++++++ ws2019/theo/uebungen/theo.pdf | Bin 0 -> 105973 bytes ws2019/theo/uebungen/theo.tex | 197 ++++++++++++++++++++++ 18 files changed, 1980 insertions(+) create mode 100644 ws2019/ana/lectures/analysis3.pdf create mode 100644 ws2019/ana/lectures/analysis3.tex create mode 100644 ws2019/ana/lectures/analysis4.pdf create mode 100644 ws2019/ana/lectures/analysis4.tex create mode 100644 ws2019/ana/lectures/analysis5.pdf create mode 100644 ws2019/ana/lectures/analysis5.tex create mode 100644 ws2019/ana/lectures/linear4.pdf create mode 100644 ws2019/ana/lectures/linear4.tex create mode 100644 ws2019/ana/lectures/linear5.pdf create mode 100644 ws2019/ana/lectures/linear5.tex create mode 100644 ws2019/la/lectures/linear4.pdf create mode 100644 ws2019/la/lectures/linear4.tex create mode 100644 ws2019/la/lectures/linear5.pdf create mode 100644 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Es gilt: + \[ + \sum_{k=1}^{n} k = \frac{n(n+1)}{2} + .\] + + \begin{proof} + Induktionsanfang für $n=1$: + \[ + \sum_{k=1}^{1} k = 1 = \frac{1(1+2)}{2} = 1 + .\] + Induktionsschritt + \[ + \sum_{k=1}^{n+1} k = \sum_{k=1}^{n} k + n + 1 = \frac{n(n+1)}{2} + n + 1 + = \frac{n(n+1)+2n+2}{2} = \frac{(n+1)(n+2)}{2} + .\] + \end{proof} +\end{bsp} + +\begin{definition} + Seien $m, n \in \N, m \le n$\\ + $a_{m}, a_{m+1}, \ldots, a_n \in \R$. Dann + $a_m + a_{m+1} + \ldots + a_n = \sum_{k=m}^{n} a_{k}$. + Falls $m>n$, dann $\sum_{k=m}^{n} a_{k} := 0$ + +\end{definition} + +\begin{bsp} + Definiere rekursiv für $x \in \R$: + $x^0 := 1$ und $x^{n+1} := x \cdot x^n, n \in \N_0$ + Betrachte + \[ + \sum_{k=0}^n x^k = 1 + x + x^2 + x^3 + x^4 + \ldots + x^n, x \in \R + .\] + Dann heißt + \[ + \sum_{k=0}^n x^{k} = \frac{1 - x^{n+1}}{1 - x} + \] geometrische Summenformel. + + \begin{proof} + Induktionsanfang für $n = 1$: + \[ + 1+x = \frac{(1+x)(1-x)}{1-x} = \frac{1-x^2}{1-x} + .\] + + Induktionsschritt: $n \to n + 1$ + + \begin{align*} + \sum_{n=0}^{n+1} x^k &= \sum_{k=0}^{n} x^k + x^{n+1}\\ + &= \frac{1-x^{n+1}}{1-x} + x^{n+1} + &= \frac{1-x^{n+1}}{1-x} + \frac{(1-x)(x^{n+1})}{1-x} + &= \frac{1 - x^{n+2}}{1-x} + .\end{align*} + \end{proof} + + \begin{proof}[Alternativbeweis mit Teleskop-Summe] + \begin{align*} + 1-x^{n+1} &= 1 - x + x - x^2 + x^2 - \ldots - x^n + x^n - x^n+1 \\ + &= \sum_{k=0}^{n} x^{k} - \sum_{k=1}^{n+1} x^{k} \\ + &= \sum_{k=0}^{n} x^{k} - \sum_{k=0}^{n} x^{k+1} \\ + &= \sum_{k=0}^{n} x^{k} - x \\ + .\end{align*} + \end{proof} + +\end{bsp} + +Als Anwendung der geometrischen Summenformel ergeben sich nützliche Formeln, z.B. +$ \forall a, b \in \R$ und $n \in \N$ gilt: + +\begin{align*} + a^n - b^n = (a - b) \sum_{k=0}^{n-1} a^{n-1-k} b^k +\end{align*} + +\begin{proof} + Für $a=0$ und $a = b$ stimmt die Formel offenbar.\\ + Betrachte geometrische Reihe mit $x := \frac{b}{a} \neq 1$ + \[ + 1 - \left(\frac{b}{a}\right)^n = 1 - x^n = (1 - x) \sum_{k=0}^{n-1} x^k + = (1 - \frac{b}{a}) \sum_{k=0}^{n-1} \left(\frac{b}{a}\right)^k + \] + \[ + a^n - b^n = (a-b) \sum_{k=0}^{n-1} b^k a^{-k} a^{n-1} = (a - b) \sum_{k=0}^{n-1} a^{n-1-k} b^k + \] +\end{proof} + +\subsection{Elemente der Kombinatorik} + +Für $n \in \N$ ist die Fakultät $n!$ rekursiv definiert durch: +\[ + 1! := 1 \text{ und } \forall n \in \N: (n + 1)! = n!(n+1) +.\] Per Definition $0! := 1$ + +\begin{satz}[Permutationen] + Die Anzahl aller Anordnungen (oder Permutationen) von $n \in \N$ Elementen ist $n!$. +\end{satz} + +\begin{proof} + Induktionsanfang: + + $n=1$: Eine Anordnung 1 \\ + $n=2$: Zwei Anordnungen 12, 21 + + Induktionsschritt $n \to n+1$: Anzahl von Anordnungen der Elemente ${1, \ldots, n+1}$, + die das Element $(n+1)$ auf Platz 1 hat bei beliebiger Anordnung der + anderen Elemente nach Induktionsannahme ist $n!$. Für jedes der $n+1$ + Plätze ergeben sich wieder $n!$ Anordnungen, d.h. insgesamt: + $n!(n+1) = (n+1)!$ +\end{proof} + +\begin{definition}[Binomialkoeffizient] + Für $n, k \in \N$ definieren wir:\\ + \begin{align*} + n \ge k \ge 1:& \binom{n}{k} := \frac{n(n-1) \ldots (n -k + 1)}{k!} \\ + k = 0:& \binom{n}{0} := 1 + \end{align*} + $\binom{n}{k}$ ist die Anzahl der k-Elementigen Teilmengen einer n-Elementigen Menge, z.B.: Lotto $\binom{49}{6} = 13.983.816$. + \begin{align*} + \binom{n}{k} &= \frac{n(n-1) \ldots (n - k +1)}{k!}\\ + &= \frac{n(n-1) \ldots (n-k+1)(n-k)!}{k!(n-k)!}\\ + &= \frac{n!}{k!(n-k)!} = \binom{n}{n-k}\\ + .\end{align*} + Es folgt $\binom{n}{0} = 1$, $\binom{n}{n} = 1$, + $\binom{n}{1} = \binom{n}{n-1} = n, \forall n \in \N.$ +\end{definition} + +\begin{figure}[ht] + \centering + \incfig{figur1} + \caption{figur1} + \label{fig:figur1} +\end{figure} + +\end{document} diff --git a/ws2019/ana/lectures/analysis4.pdf b/ws2019/ana/lectures/analysis4.pdf new file mode 100644 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+ + Ordnungsrelation für $a, b \in K$ + \begin{align*} + a < b \iff b - a \in K^{+} \\ + b > a:\iff a < b \\ + a \le b: \iff a < b \wedge a = b \\ + b \ge a: \iff a \le b \\ + .\end{align*} + + Für je zwei $a \in K, b \in K$ gilt genau eine der Relationen + $ab$. +\end{definition} + +Es gelten Regeln: +\begin{itemize} + \item $a < b, b < c \implies a < c$ Transitivität + \item $a < b \implies a + c < b +c, c \in K$ + \item $a < b \implies a \cdot c < b \cdot c, c \in K^{+}$ + \item $a \ge b, b \ge a \iff a = b$ + \item $a < b, a > 0, b > 0 \iff \frac{1}{a} > \frac{1}{b}$ +\end{itemize} + +\begin{bsp}[Positivität auf $\Q$] + \[ + \Q^{+} := \left\{a \in Q \mid a = \frac{r}{s} , r, s \in \N\right\} + .\] +\end{bsp} + +\begin{definition}[Absolutbetrag] + Sei $(K, +, \cdot, >$ ein angeordneter Körper + Dann + \[ + |a| := \begin{cases} + a & \text{für } a > 0 \\ + 0 & \text{für } a = 0 \\ + -a & \text{für } a < 0 \\ + \end{cases} + .\] + + Abbildung $|\cdot|$: $K \to K$ mit Eigenschaften: + \begin{itemize} + \item $|a| = 0 \iff a = 0$ (Definiertheit) + \item $|ab| = |a| |b|$ (Multiplikativität) + \item $|a+b| \le |a| + |b|$ (Dreiecksungleichung) + \end{itemize} +\end{definition} + +\begin{proof}[Beweis der Dreiecksungleichung] + Beobachtung: $\pm a \le |a| \implies a + b \le |a| + |b| + \implies -(a+b) \le |a| + |b|$ +\end{proof} + +Es folgt aus den Eigenschaften: +\begin{itemize} + \item $|a-b| = 0 \implies a = b$ + \item $|-a| = |a|$ + \item $|\frac{a}{b}| = \frac{|a|}{|b|}, b \neq 0$ + \item $| |a| - |b| | \le |a - b| $ + (folgt aus: $|a| = |a-b+b| \le |a-b| + |b|$ und + $|b| = |b - a + a| \le |b-a| + |a|$) +\end{itemize} + +\begin{satz}[Dezimalbruchdarstellung] + Jede rationale Zahl $a$ besitzt eine endliche + oder periodische Dezimalbruchdarstellung der Form: + \[ + a = \pm (a_0 + 0,d_1\ldots d_s): \iff + a = \pm \left(a_0 + \sum_{k=1}^{s} ds \cdot 10^{-k}\right) + .\] bzw. + \[ + a = \pm (a_0 + 0,d_1\ldots d_s \overline{d_{s+1} \ldots d_{s+t}}) + .\] + $a_0 \in N_0, d_1\ldots d_s \in \{0, 1, \ldots ,9\}$ Ziffern + + Umgekehrt stellt jede Dezimalbruchzerlegung dieser Art eine rationale + Zahl dar.\\ + Hier: bei periodischen Dezimalbrüchen ist die Periode $\overline{9}$ + nicht zugelassen: + \[ + a_0,d_1\ldots d_{k-1} d_k \overline{9} + := a_0 + 0,d_1\ldots d_k (d_k+1), d_k < 9 + .\] +\end{satz} + +\begin{proof} + Siehe Lehrbuch +\end{proof} + +\section{Die Reellen Zahlen} + +\subsection{Von den rationalen Zahlen zu den reellen Zahlen} + +\begin{lemma}[Irrationalität der Quadratwurzel] + Die quadratische Gleichung $x^2 = 2$ besitzt keine rationale + Lösung. +\end{lemma} + +\begin{proof}[Beweis durch Widerspruch] + Angenommen: Es existiert eine rationale Lösung + \[ + x := \sqrt{2} = \frac{r}{s} + .\] mit Zahlen $r \in \Z$ und $s \in \N$. + + O.B.d.A. (Ohne Beschränkung der Allgemeinheit) nehmen wir + an, dass $r$ und $s$ teilerfremd sind. + + Dann gilt: $r \neq 0$ und $r^2 = 2s^2$ und $\frac{1}{2}r^2 = s^2$. + Also muss $r^2$ und auch $r$ gerade sein, denn $(2n+1)^2 = 4n^2+ 4n +1$ + ungerade (Kontraposition). + Damit sind auch $\frac{1}{2}r^2$ gerade und $s^2$ gerade. + + Aber wegen Teilerfremdheit können $r^2$ und $s^2$ nicht beide + durch zwei teilbar sein. $\implies$ Widerspruch zur Annahme +\end{proof} + +\begin{bem} +Allgemeiner: ,,quadratische'' Gleichung +\[ +a+bx +c x^2 = 0 +.\] ist nicht für beliebig gewählte $a, b, c \in \Q$ durch +ein $x \in \Q$ lösbar. +\end{bem} + +\begin{bem}[Beweisarten] + Direkter Beweeis: + \[ + E \implies E_1 \implies E_2 \implies \ldots \implies E_k \implies V + .\] Indirekter Beweis: Zeigen $E$ und $\neg V$ immer + falsch. Da $E$ immer wahr ist, muss $\neg V$ falsch sein. + Da $\neg V$ falsch ist, dann ist $V$ wahr. +\end{bem} + +\textbf{Ziel}: Konstruiere rationale Zahlen, welche die +Gleichung $x^2 = 2$ mit zunehmender Genauigkeit erfüllen, +z.B. rekursiv durch Einschließung mit Hilfe von Dezimalbrüchen. + +Wir nutzen die Eigenschaft: $a, b > 0$ und $a^2 < b^2 \implies a < b$, folgt +aus: +\[ + b^2 - a^2 = (b-a)(b+a), (b+a > 0) +.\] + +Start: $a_1 := 1,4$, $b_1 := 1,5$ mit +$a_1 < b_1, a_1^2 = 1,96 < 2 < 2,25 = b_1^2$ + +2 Fälle: + +Fall a) Es liege für ein $n \in \N$ eine Einschließung vor: +\[ + a_n = 1,d_1d_2, \ldots d_{n-1}d_n < b_n = 1,d_1d_2\ldots d_{n-1}(d_{n+1}) +.\] +\[ + a_n^2 < 2 < b_n^2 +.\] $d_k \in \{0,1, \ldots, 9\}, k=1 \ldots n-1, dn \le 8 $ + +Die nächste Einschließung ist +\[ + a_{n+1} := 1,d_1\ldots d_n, d_{n+1}, d_{n+1} \in \{0,1,\ldots_,9\} +.\] $a_{n+1}$ möglichst groß aber $a_{n+1}^2 < 2$. + +und +\[ +b_{n+1} := \begin{cases} + 1,d_1, \ldots d_n (d_{n+1} + 1) & \text{für } d_{n+1} \le 8 \\ + 1,d_1, \ldots (d_n + 1) 0 & für d_{n+1} = 9 \\ +\end{cases} +.\] Nach Konstruktion: +\[ + a_n < a_{n+1} < b_{n+1} < b_n +.\] +\[ +a_{n+1}^2 < 2 < b_{n+1}^2 +.\] + +Fall b) Für ein $n \in \N$ liegt eine Einschließung vor +\[ + a_1 = 1, d_1\ldots d_{n-1} d_{n} < b_{n} = 1, d_1 d_2 \ldots d_{n-1} (d_n + 1) 0 \ldots 0 +.\] +\[ + a_n^2 < 2 < b_n^2 \text{ mit } d_k \in \{0, 1, \ldots ,9\}, k=1, n = 1 +.\] +\[ +d_n \le 8, d_{n+1} = \ldots = d_n = 9 +.\] + +Die nächste Einschließung +\[ + a_{n+1} := 1,d_1 \ldots d_{n}, d_{n+1}, d_{n+1} \in {0, 1, \ldots, 9} +.\] $a_{n+1}$ möglichst groß, aber $a_{n+1}^2 < 2$. + +\[ +b_{n+1} = \begin{cases} + 1,d_1\ldots d_n (d_{n+1} + 1) & \text{für } d_{n+1} \le 8 \\ + 1,d_1\ldots d_{m-1}(d_m + 1) 0 \ldots 0 & \text{für } d_{n+1} = 9 +\end{cases} +.\] Der Fall b) kann nur endlich oft hintereinander auftreten, dann wäre +$a_n = b_n$ ab einem gewissen n und folglich $a_n ^2 = 2$ + +Nach Konstruktion: +\[ +a_n < a_{n+1} < b_{n+1} < b_n +.\] +\[ +a_{n+1}^2 < 2 < b_{n+1}^2 +.\] Wir erhalten 2 Folgen $(a_n)_{n \in \N}$ und $(b_n)_{n \in N}$ mit +den Eigenschaften +\[ +1,4 = a_1 \le \ldots \le a_n \le a_{n+1} < b_{n+1} \le b_n \le \ldots b_1 = 1,5 +.\] Konkret: $a_1 = 1,4$, $a_2 = 1,41$, $a_3 = 1,414$ +$b_1 = 1,5$, $b_2 = 1,42$, $b_3 = 1,415$ + +Abstand $b_n - a_n \le 10^{-n}$, $n \in \N$ wird immer kleiner +$\implies$ wir sollen die Zahl $\sqrt{2}$ eventuell erfassen! + +\begin{definition}[Zahlenfolge] + Eine Menge $(a_n)_{n \in \N}$ nummerierter rationaler Zahlen wird + ,,Folge'' genannt. +\end{definition} + +\begin{bsp} + $a_n = 1 + \frac{1}{n}$ + + $a_1 = 2$, $a_2 = \frac{3}{2}$, $a_3 = \frac{5}{4}$ +\end{bsp} + +Offenbar, $1 + \frac{1}{n} \to 1, n \to \infty$ +bzw. +\[ + \lim_{n \to \infty} (1+ \frac{1}{n}) = 1 +.\] d.h. Folge konvergiert gegen 1 + +\begin{definition}[Konvergenz] + Eine Folge $(a_n)_{n \in \N}$ heißt konvergent, gegen einen ,,Limes'' + a, wenn gilt: + \[ + |a_n - a| \to 0, n \to \infty + .\] Falls $|a_n|$, $n \to \infty$ heißt $(a_n)_{n \in \N}$ + strikt divergent. + + Präziser (Cauchy) + + Eine Folge $(a_n)_{n \in N}$ ist ,,konvergent'' gegen einen + Grenzwert a, wenn: + \[ + \forall \epsilon > 0: \exists n := n(\epsilon) = n_{\epsilon} + .\] sodass + \[ + |a_n - a| < \epsilon \text{ für } n \ge n_{\epsilon} + .\] +\end{definition} + +\end{document} diff --git a/ws2019/ana/lectures/linear4.pdf b/ws2019/ana/lectures/linear4.pdf new file mode 100644 index 0000000000000000000000000000000000000000..885b40f16fedf14c0ef09d028a7a013e40aead84 GIT binary patch literal 131858 zcmb5WV{~QRx`i9twr$(CZQEAGPAax-+ZEfkjjB}a-1_#}=eFDKPHU(3{I!10@r)O} zzhlfrsvsgp%SguxMLN4Uv|ZLBPz+&Q9>J2Z~*de3vTP!}w`U z--tp&FnGu_bz)6juvMKb#6_BDW6qzD_-vJI_g)d#;!Qlo_QA~QnccrJ6hpEaAGnHA 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Die Abb $g \circ f: M \to K, m \mapsto g(f(m))$ + heißt die Komposition von f und g. Die Komposition kann man auch als + Mengenabbildung auffassen: + \[ + \circ: Abb(M, N) \times Abb(N, K) \to Abb(M, K) + \] + \[ + (f, g) \mapsto g \circ f + .\] +\end{definition} + +\begin{lemma}[] + Seien $I$ und $M$ Mengen und es sei: $(M_{i})_{i \in I}$ die + Familie von (immergleichen) Mengen $M_{i} = M$ indiziert + über $i \in I$. Dann existiert eine natürliche Bijektion + \[ + \Phi: Abb(I, M) \to^{\thicksim} \prod_{}^{} (M_{i})_{i \in I} (= M^I) + .\] +\end{lemma} +\begin{proof} + rechts: Tupel $(m_{i})_{i \in I}, m_{i} \in M_{i} = M$ + links: Abbildung $f: I \to M$. + Eine solche Abbildung ist dadurch gegeben, dass man jedem $i \in I$ + ein $m_{i} = f(i) \in M$ zuordnet. Wir definieren $\Phi$ durch die + Zuordnung: + \[ + \Phi: f \in Abb(I, M) \mapsto (f(i))_{i \in I} \in \prod_{}^{} (M_{i})_{i \in I} + .\] + Da die Abb. $f$ durch ihre Werte $f(i) \in M, i \in I$, gegeben ist, + ist $\Phi$ injektiv. + + Ist umgekehrt $(m_{i}) \in \prod M_{i}$ gegeben, so ist die Abbildung + $f: I \to M, i \mapsto m_{i} \in M$ ein Urbild unter $\Phi$. Daher + ist $\Phi$ surjektiv. +\end{proof} + +\section{Gruppen, Ringe, Körper} + +\subsection{Gruppen} + +\begin{definition}[Verknüpfung] + Eine (binäre) Verknüpfung auf einer Menge $M$ ist eine Abbildung: + \[ + *: M \times M \to M + .\] +\end{definition} + +\begin{definition}[Gruppe] + Eine Gruppe $(G, *, e)$ ist eine Menge $G$ mit einer Verknüfung $*$ und + einem (ausgezeichneten) Element $e \in G$, so dass: + \begin{enumerate} + \item $g*(h*k) = (g*h*)*k$ $\forall g,h,k \in G$ (Assoziativität) + \item $e * g = g$ $\forall g \in G$ ((Links)neutrales Element) + \item $\forall g \in G$: $\exists h \in G$: $h * g = e$ (Linksinverses) + \end{enumerate} + Eine Gruppe heißt kommutativ oder abelsch wenn zusätzlich gilt: + \begin{enumerate} + \item $g * h = h * g$ $\forall g, h \in G$ + \end{enumerate} +\end{definition} + +\begin{bsp}[] + \begin{enumerate} + \item $(\Z, +, 0)$ ist abelsche Gruppe + \item $(\Q, +, 0)$, $(\R, +, 0)$, $(\C, +, 0)$ sind abelsche Gruppen. + \item $(\Q \setminus{\{0\}}, \cdot, 1)$ ist eine abelsche Gruppe + \item $(\R_{>0}, \cdot, 1)$ ist abelsche Gruppe + \end{enumerate} +\end{bsp} + +\begin{bem}[] + Menge der Restklassen: + \[ + \{\overline{0}, \overline{1}, \ldots, \overline{n-1}\} = \Z / n\Z + .\] + $(\Z / n\Z, +, \overline{0})$ ist eine abelsche Gruppe. + Wie ist die Summe von Restklassen definiert? + + Seien $A, B \in \Z / n\Z$. Vorschrift für ,,+''. + \begin{enumerate} + \item Wähle ,,Vertreter'' $a,b \in \Z$ von $A,B$, d.h. $a \in A$, + $b \in B$. + \item bilde $a+b$ in $\Z$ + \item $A+B =^{def} \overline{a+b}$, d.h. die Restklasse zu der + $a+b$ gehört. + \end{enumerate} + Damit diese Definition widerspruchsfrei ist (Sprich: ,,+'' + ist \textit{wohldefiniert}) muss man nachweisen, dass das Ergebnis nicht + von der Auswahl im ersten Schritt abhängt. +\end{bem} + +\begin{bsp}[] + Die symmetrische Gruppe $O_{n}$ + \[ + O_{n} := \text{die Menge aller bijektiven Abb.}\\ + \pi: \{1, \ldots, n\} \to \{1, \ldots, n\} + .\] + (sogennante Permutationen) + + $* = \circ$ Komposition von Abbildungen\\ + $e = id_{\{1, \ldots, n\}}$\\ + Wir schreiben Permutationen in der Form: + \[ + \pi = + \begin{pmatrix} + 1 & 2 & 3 & \ldots & n \\ + \pi(1) & \pi(2) & \pi(3) & \ldots & \pi(n) + \end{pmatrix} + .\] + Elementare Kombinationen: $n$ Möglichkeiten für $\pi(1)$, $(n-1)$ + Möglichkeiten, für $\pi(2)\ldots$, 1 Möglichkeit für $\pi(n)$. + \[ + \#O_{n} = n! \text{ (Fakultät)} + .\] + Verifikation der Gruppenaxiome + \begin{enumerate} + \item $g*(h*k) = g \circ (h \circ k) = (g \circ h) \circ k = (g * h) * k$ + \item $e * g = id * g = g$ + \item Sei g eine Permutation und $h = g^{-1}$ die Umkehrabbildung. + Dann gilt $h*g = g^{-1} \circ g = id = e$. + \end{enumerate} + Für $n \ge 3$ ist $\sigma_{n}$ nicht kommutativ. + \[ + \begin{pmatrix} + 1 & 2 & 3 & 4 & \ldots \\ + 2 & 1 & 3 & 4 & \ldots + \end{pmatrix} + \circ + \begin{pmatrix} + 1 & 2 & 3 & 4 & \ldots \\ + 1 & 3 & 2 & 4 & \ldots + \end{pmatrix} + = + \begin{pmatrix} + 1 & 2 & 3 & 4 & \ldots \\ + 2 & 3 & 1 & 4 & \ldots + \end{pmatrix} + .\] + \[ + \begin{pmatrix} + 1 & 2 & 3 & 4 & \ldots \\ + 1 & 3 & 2 & 4 & \ldots + \end{pmatrix} + \circ + \begin{pmatrix} + 1 & 2 & 3 & 4 & \ldots \\ + 2 & 1 & 3 & 4 & \ldots + \end{pmatrix} + = + \begin{pmatrix} + 1 & 2 & 3 & 4 & \ldots \\ + 3 & 1 & 2 & 4 & \ldots + \end{pmatrix} + .\] + +\end{bsp} + +\begin{satz}[] + Sei $G = (G, *, e)$ eine Gruppe. Dann gilt für alle $g, h, k \in G$. + \begin{enumerate} + \item $g * h = g * k \implies h = k$ (Linkskürzung) + \item $g * h = k * h \implies g = k$ (Rechtskürzung) + \item $g * e = g$ (das (links)neutrale Element ist auch rechtsneutral) + \item aus $g*h=g$ oder $h*g=g$ für ein einziges $g\in G$, so folgt $h=e$ + \item $\forall g \in G$: existiert ein eindeutig bestimmtes Element + $g^{-1} \in G$ mit $g^{-1} * g = e = g * g^{-1}$. + \item Aus $h * g = e$ oder $g*h=e$ folgt $h=g^{-1}$. + \item $\left(g^{-1}\right)^{-1} = g$ $\forall g \in G$ + \end{enumerate} +\end{satz} + +\begin{proof}[Beweis 1] + Sei $g * h = g * k$. + + Nach (G3) $\exists s \in G$, sodass $s * g = e$. + Daher gilt $s*(g*h) = (s*g)*h = e*h = k$.\\ + Analog: $s*(g*k) = (s*g)*k = e*k = k$ + Daraus folgt: $h = k$. +\end{proof} + +\begin{proof}[Beweis 3] + Nach (G3) existiert $h\in G$ und $h*g=e$. + + Es folgt $h*(g*e)=(h*g)*e=e*e=e=h*g$\\ + Nach (1) folgt $g*e=g$ +\end{proof} + +\begin{proof}[Beweis 5, Existenz] + Sei $h\in G$ mit $h*g = e$ (ex. nach G3) + + $h*(g*h) = (h*g)*h = e * h = h = h * e$\\ + Durch Linkskürzung erhalten wir $g * h = e$. +\end{proof} + +\begin{proof}[Beweis 2] + Sei $g*k=h*k$. Sei $s \in G$ so dass $k*s=e$ (existenz nach 5). + + $\implies (g*k)*s = g*(k*s) = g*e = g$, analog\\ + $\implies (h*k) * s = h*(k*s) = h*e = h$\\ + Daraus folgt $g = h$. +\end{proof} + +\begin{proof}[Beweis 4] + $g * h = g = g * e \implies h = e$, analog\\ + $h*g = g = e*g \implies h=e$ +\end{proof} + +\begin{proof}[Beweis 5 Eindeutigkeit und 6] + Seien $h, h' \in G$ mit $h*g=e=h'*g$ Nach Rechtskürzung folgt + $h = h'$. Daher ist $g^{-1} eindeutig$. Sei $h \in G$ mit + $g * h = e$. Wegen $g^{-1}*g = e$ folgt mit Linkskürzung, dass $h = g^{-1}$. +\end{proof} + +\begin{proof}[Beweis 7] + aus $g * g^{-1} = e$ folgt $\left(g^{-1}\right)^{-1} = g$ +\end{proof} + +\begin{bem}[] + $g, h \in G$, so gilt $(g * h)^{-1} = h^{-1} * g^{-1}$\\ + Grund: $(h^{-1} * g^{-1}) * (g *h) = h^{-1} * (g*g^{-1}) * h = h^{-1}* e * h = h * h^{-1} = e.$ +\end{bem} +\end{document} diff --git a/ws2019/ana/lectures/linear5.pdf b/ws2019/ana/lectures/linear5.pdf new file mode 100644 index 0000000000000000000000000000000000000000..d23ba7e6a36cf9859544184e74aa9aaec1667941 GIT binary patch literal 118241 zcmb5WRaBhavbK%8(`ayacY?dSyGw8g?he77;O;?#ySux)yE}j0z1P~`KgME=o!v)$ z&m z5Jg^flZug+>K(h^5{VT&^60`j!U6~qOOzkaw_o}7JAt7T>)KR?L;^bn@`{RHPAWZu z`E;Z=;;h=ax|Y87ERDq1{)o@=HsxX-6sM~Fkrts41Xqws28Ucw^W)_7Ei{50%;>8- zC$19D^V`|V4Le^Hc|`Tp?Oa;&!PfJFl<3|qZ3J$H+<=evtAm5L{q+SZh;2945QhlJ zu3&Cb=A-Ubl!$W{lLWd@ z`DF!7h2%Us+}AB3&?l1IoNJ|6tI`q$4dz2CorH49#IYUVVBH@qnp?BQzr|5XBJX#3 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--git a/ws2019/ana/lectures/linear5.tex b/ws2019/ana/lectures/linear5.tex new file mode 100644 index 0000000..0016c25 --- /dev/null +++ b/ws2019/ana/lectures/linear5.tex @@ -0,0 +1,259 @@ +\documentclass{lecture} + +\begin{document} +\section{Gruppen, Ringe, Körper} + +\subsection{Ringe} + +\begin{definition}[Ring] + Ein Ring $R = (R, +, \cdot, O_{R})$ ist eine Menge $R$ und zwei + Verknüpfungen $+, \cdot: R \times R \to R$ und einem + Element $O_{R} \in R$ so dass: + \begin{enumerate} + \item $(R, +, O_{R})$ ist eine abelsche Gruppe + \item $a \cdot (b \cdot c) = (a \cdot b) \cdot c$ $\forall a,b,c \in R$ + \item $a \cdot (b + c) = a \cdot b + a \cdot c$, $(a + b) \cdot c = ac + bc$ + \end{enumerate} + + Ein unitärer Ring (,,Ring mit 1'') ist ein Tupel + $(R, +, \cdot, 0_{R}, 1_{R})$, so dass $(R, +, \cdot, 0_{R})$ ein + Ring ist und $1_R \in R$, so dass gilt: + \begin{enumerate} + \item $1_{R} \cdot a = a = a \cdot 1_{R}$ + \end{enumerate} + + Ein Ring heißt kommutativ, wenn + \begin{enumerate} + \item $a \cdot b = b \cdot a$ $\forall a, b \in R$ + \end{enumerate} +\end{definition} + +\begin{bem}[Notation] + Das inverse Element von $a \in R$ bezüglich $+$ bezeichnet man mit + $-a$. Ein Inverses bezüglich $\cdot$ existiert i.A. nicht. + + Die Eins in einem unitären Ring ist eindeutig bestimmt. +\end{bem} + +\begin{bsp} + $(\Z, +, \cdot, 0, 1)$ ist ein kommutativer Ring mit 1 +\end{bsp} + +\begin{bsp}[$\Z / n\Z$] + ist ein kommutativer Ring mit 1. Multiplikationsvorschrift ist die + folgende: Für $A, B \in \Z / n\Z$ + \begin{enumerate} + \item Wähle Vertreter $a, b \in \Z$ von $A$ und $B$. + \item bilde $a \cdot b$ in $\Z$ + \item $A \cdot B :=$ Restklasse von $a \cdot b$ + \end{enumerate} + Nachzuweisen: Unabhängigkeit der Definition von der Auswahl der Vertreter + im ersten Schritt. +\end{bsp} + +\begin{bsp}[Die Menge der geraden ganzen Zahlen] + ist ein kommutativer Ring ohne 1. +\end{bsp} + +\begin{lemma} + $R = (R, +, \cdot, 0_{R})$ Ring. Dann gilt + \begin{enumerate} + \item $0_{R} \cdot a = 0_R = a \cdot 0_R$ + \item $a \cdot (-b) = - ab = (-a) \cdot b$ + \end{enumerate} + Ist R unitär, so gilt: + \begin{enumerate} + \item $-b = (-1_{R}) \cdot b$ + \end{enumerate} +\end{lemma} + +\begin{proof}[Beweis 1] + \[ + 0_{R} \cdot a + 0_{R} = 0_{R} = 0_{R} a = (0_R + 0_R)a = 0_R a + 0_R a + .\] + Mit kürzen folgt: $0_R = 0_R a$ + Analog für $a \cdot 0_R = 0_R$ +\end{proof} + +\begin{proof}[Beweis 2] + \[ + 0_R = a 0_R = a \left((-b) + b\right) = a(-b) + a b + \] also $a(-b) = -ab$. +\end{proof} + +\begin{proof}[Beweis 3] + Ist R unitär, so setzt man in 2 $a = 1_{R}$ ein und erhält 3. +\end{proof} + +\begin{bsp} + $R = \{0\} $ mit den einzig möglichen Verknüpfungen $+$ und $\cdot$ + heißt der \textit{Nullring}. + Nullring ist ein kommutativer Ring mit 1 ($0_R = 0 = 1_R$). + Dies ist der einzige Ring mit $1_R = 0_R$. + + Begründung: Gelte $0_R = 1_R$, so folgt für jedes $r \in R$: + \[ + r = r \cdot 1_R = r \cdot 0_R = 0_R + .\] Egal welches Element herausgenommen wird, es ist immer $0_R$, d.h. + $R$ muss ein Nullring sein. +\end{bsp} + +\begin{lemma} + Sei $R = (R, +, \cdot, 0_R, 1_R)$ ein unitärer Ring und + $R^{\times } \subset R$ die Menge der Elemente die ein Links- und + ein Rechtsinverses bezüglich $\cdot$ haben, das heißt + \[ + R^{\times } = \{r \in R \mid \exists s, t \in R : s r = 1_R = r t\} + .\] + Dann ist $(R^{x}, \cdot, 1_R)$ eine Gruppe. + Man nennt $R^{x}$ die Einheitengruppe von $R$. +\end{lemma} + +\begin{proof} + Seien $r, \overline{r} \in R^{\times }$ und $s, \overline{s}, t, \overline{t}$ + mit + \[ + s r = 1 = r t + .\] und + \[ + \overline{s} \overline{r} = 1 = \overline{r} \overline{t} + .\] Dann + \[ + (s' s)(r r') = s'(s r) r' = s'1r' = s'r' = 1 + .\] + \[ + (r r') (t' t) = r (r' t') t = r 1 t = r t = 1 + .\] $\implies r \cdot r' \in R^{\times }$ + + Wir überprüfen die Gruppenaxiome: G1 folgt aus R2 + + $1 \in R$ ist neutral $\to$ G2 + + Bleibt zu zeigen: $\forall r \in R^{\times }: \exists r' \in R: r r' = 1$ + Nach Definition: $\exists s \in R: s r = 1$. Zu zeigen: $s \in R^{\times }$. + Offenbar hat das Rechtsinverse $r$. Aber $r$ ist auch linksinvers zu $s$: + + Wähle $t \in R$ mit $r t = 1$. Dann gilt $s = s (r t) = (s r) t = t$ + $\implies$ rs = rt = 1. +\end{proof} + +\begin{bem} + $0_R \in R^{\times } \implies \exists r \in R: 0_R r = 1_R \implies 0_R = 1_R \implies$ $R$ ist der Nullring. +\end{bem} + +\begin{definition}[Körper] + Ein Körper $K$ ist ein kommutativer Ring mit 1: $(K, +, 0_K, 1_K)$ + mit $K^{\times } = K \ \{0_K\}$ +\end{definition} + +\begin{bsp} + \begin{enumerate} + \item $\Q, \R, \C$ sind Körper + \item $\Z$ ist kein Körper ($\Z^{\times } = \{+1, -1\} $) + \end{enumerate} +\end{bsp} + +\begin{lemma}[] + In einem Körper $K$ gilt, dass + \[ + a b = 0_K \implies a = 0 \wedge b = 0 + .\] +\end{lemma} + +\begin{proof} + Angenommen $a \neq 0_K$: Dann existiert $a^{-1} \in K$ mit + $a^{-1} a = 1_K$. Es folgt + \[ + b = 1_K b = a^{-1} a b = a^{-1} 0_K = 0_K + .\] +\end{proof} + +\begin{lemma} + Ist $p$ eine Primzahl, so ist $\Z / p\Z$ ein Körper. +\end{lemma} + +\begin{proof}[] + $\Z / p \Z$ ist kommutativer Ring mit 1 (siehe oben). + + Zu zeigen: $\forall A \in \Z / p\Z, A \neq \overline{0}$ ist + die $\overline{1}$ im Bild der Abbildung: + \[ + A\cdot : \Z / p \Z \to \Z / p \Z, B \mapsto A \cdot B + .\] + Wir zeigen sogar, dass $A \cdot$ surjektiv ist. + Da $\Z / p \Z$ endlich ist, genügt es z.z., dass $A \cdot$ injektiv ist. + + Angenommen es gäbe Restklassen $B, C \in \Z / p \Z$ mit + \[ + A \cdot B = A \cdot C + .\] Seien $a, b, c \in \Z$ Vertreter. + Wegen $A \neq \overline{0}$ gilt $a$ nicht durch $p$ teilbar. + Wegen $AB = AC$ gilt $ab =- ac$ mod $p \implies p$ teilt $a (b -c)$. + + Weil $p$ eine Primzahl ist und $p$ teilt nicht $a$ folgt + $p / (b -c)$, also $b =- c$ mod $p \implies B = C$ +\end{proof} + +\begin{bem} + Ist $n \in N$ keine Primzahl, so ist $\Z / n \Z$ kein Körper. +\end{bem} + +\begin{proof} + Für $n=1$ ist $\Z / n\Z$ der Nullring (kein Körper). + + Nun sei $n > 1$ keine Primzahl + $\implies \exists a, b \in \N, 1 < a, b < n$ mit $a b = n$ Für die + Restklassen bedeutet dies: $\overline{a} \neq \overline{0}$, + $\overline{b} \neq \overline{0}$ aber + $\overline{a} \cdot \overline{b} = \overline{a \cdot b} = \overline{n} = \overline{0}$. Wir erhalten Widerspruch zu Lemma 1.15. Also + ist $\Z / n \Z$ kein Körper. +\end{proof} + +\begin{definition}[Charakteristik] + Sei $K$ Köper. Die kleinste natürliche Zahl $n$ mit n mal + $1_K + \ldots + 1_K = 0_K$ (in $K$) heißt die + Charakteristik von K. + + Notation: char(K). + Gibt es eine solche Zahl $n$ nicht, dann setzt man char(K) = 0. +\end{definition} + +\begin{bem} + + \begin{enumerate} + \item char(K) $=0$ oder char(K) $\ge 2$ (wegen $0_K \neq 1_K$). + \item \Q, \R, \C haben Charakteristik Null. + \item \Z / p \Z hat die Charakteristik p. + \end{enumerate} +\end{bem} + +\begin{satz} + char(K) ist entweder 0 oder Primzahl. +\end{satz} + +\begin{proof} + Sei char(K) $\neq 0$, also char(K) $= n \ge 2$. + + Wäre $n$ keine Primzahl, so existieren $a, b \in \N, 1 < a ,b < n$ + mit $ab = n$. Dann gilt: + \[ + (1_K + \ldots + 1_K) \cdot (1_K + \ldots + 1_K) + = (1_K + \ldots + 1_K) = 0 + .\] Aus dem Satz vom Nullprodukt folgt $(1_K + \ldots + 1_K)$ = $0_K$ + oder $(1_K + \ldots + 1_K) = 0_K$ + + Das Widerspricht der Minimalität von n. +\end{proof} + +\subsection{Homomorphismen} + +\begin{definition} + Seien $(G, *_G, e_G)$ und $(H, *_H, e_H)$ Gruppen. + Eine Abbildung $f: G \to H$ heißt Gruppenhomomorphismus wenn für alle + $g, g' \in G$ gilt: + \[ + f(g *_G g') = f(g) *_H f(g') + .\] +\end{definition} + +\end{document} diff --git a/ws2019/la/lectures/linear4.pdf b/ws2019/la/lectures/linear4.pdf new file mode 100644 index 0000000000000000000000000000000000000000..885b40f16fedf14c0ef09d028a7a013e40aead84 GIT binary patch literal 131858 zcmb5WV{~QRx`i9twr$(CZQEAGPAax-+ZEfkjjB}a-1_#}=eFDKPHU(3{I!10@r)O} zzhlfrsvsgp%SguxMLN4Uv|ZLBPz+&Q9>J2Z~*de3vTP!}w`U z--tp&FnGu_bz)6juvMKb#6_BDW6qzD_-vJI_g)d#;!Qlo_QA~QnccrJ6hpEaAGnHA zQrEcg+clUn4xDum2nOHi}nAQ06bmB)+aa6)DCu~KV13bZ-j|KTP 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Die Abb $g \circ f: M \to K, m \mapsto g(f(m))$ + heißt die Komposition von f und g. Die Komposition kann man auch als + Mengenabbildung auffassen: + \[ + \circ: Abb(M, N) \times Abb(N, K) \to Abb(M, K) + \] + \[ + (f, g) \mapsto g \circ f + .\] +\end{definition} + +\begin{lemma}[] + Seien $I$ und $M$ Mengen und es sei: $(M_{i})_{i \in I}$ die + Familie von (immergleichen) Mengen $M_{i} = M$ indiziert + über $i \in I$. Dann existiert eine natürliche Bijektion + \[ + \Phi: Abb(I, M) \to^{\thicksim} \prod_{}^{} (M_{i})_{i \in I} (= M^I) + .\] +\end{lemma} +\begin{proof} + rechts: Tupel $(m_{i})_{i \in I}, m_{i} \in M_{i} = M$ + links: Abbildung $f: I \to M$. + Eine solche Abbildung ist dadurch gegeben, dass man jedem $i \in I$ + ein $m_{i} = f(i) \in M$ zuordnet. Wir definieren $\Phi$ durch die + Zuordnung: + \[ + \Phi: f \in Abb(I, M) \mapsto (f(i))_{i \in I} \in \prod_{}^{} (M_{i})_{i \in I} + .\] + Da die Abb. $f$ durch ihre Werte $f(i) \in M, i \in I$, gegeben ist, + ist $\Phi$ injektiv. + + Ist umgekehrt $(m_{i}) \in \prod M_{i}$ gegeben, so ist die Abbildung + $f: I \to M, i \mapsto m_{i} \in M$ ein Urbild unter $\Phi$. Daher + ist $\Phi$ surjektiv. +\end{proof} + +\section{Gruppen, Ringe, Körper} + +\subsection{Gruppen} + +\begin{definition}[Verknüpfung] + Eine (binäre) Verknüpfung auf einer Menge $M$ ist eine Abbildung: + \[ + *: M \times M \to M + .\] +\end{definition} + +\begin{definition}[Gruppe] + Eine Gruppe $(G, *, e)$ ist eine Menge $G$ mit einer Verknüfung $*$ und + einem (ausgezeichneten) Element $e \in G$, so dass: + \begin{enumerate} + \item $g*(h*k) = (g*h*)*k$ $\forall g,h,k \in G$ (Assoziativität) + \item $e * g = g$ $\forall g \in G$ ((Links)neutrales Element) + \item $\forall g \in G$: $\exists h \in G$: $h * g = e$ (Linksinverses) + \end{enumerate} + Eine Gruppe heißt kommutativ oder abelsch wenn zusätzlich gilt: + \begin{enumerate} + \item $g * h = h * g$ $\forall g, h \in G$ + \end{enumerate} +\end{definition} + +\begin{bsp}[] + \begin{enumerate} + \item $(\Z, +, 0)$ ist abelsche Gruppe + \item $(\Q, +, 0)$, $(\R, +, 0)$, $(\C, +, 0)$ sind abelsche Gruppen. + \item $(\Q \setminus{\{0\}}, \cdot, 1)$ ist eine abelsche Gruppe + \item $(\R_{>0}, \cdot, 1)$ ist abelsche Gruppe + \end{enumerate} +\end{bsp} + +\begin{bem}[] + Menge der Restklassen: + \[ + \{\overline{0}, \overline{1}, \ldots, \overline{n-1}\} = \Z / n\Z + .\] + $(\Z / n\Z, +, \overline{0})$ ist eine abelsche Gruppe. + Wie ist die Summe von Restklassen definiert? + + Seien $A, B \in \Z / n\Z$. Vorschrift für ,,+''. + \begin{enumerate} + \item Wähle ,,Vertreter'' $a,b \in \Z$ von $A,B$, d.h. $a \in A$, + $b \in B$. + \item bilde $a+b$ in $\Z$ + \item $A+B =^{def} \overline{a+b}$, d.h. die Restklasse zu der + $a+b$ gehört. + \end{enumerate} + Damit diese Definition widerspruchsfrei ist (Sprich: ,,+'' + ist \textit{wohldefiniert}) muss man nachweisen, dass das Ergebnis nicht + von der Auswahl im ersten Schritt abhängt. +\end{bem} + +\begin{bsp}[] + Die symmetrische Gruppe $O_{n}$ + \[ + O_{n} := \text{die Menge aller bijektiven Abb.}\\ + \pi: \{1, \ldots, n\} \to \{1, \ldots, n\} + .\] + (sogennante Permutationen) + + $* = \circ$ Komposition von Abbildungen\\ + $e = id_{\{1, \ldots, n\}}$\\ + Wir schreiben Permutationen in der Form: + \[ + \pi = + \begin{pmatrix} + 1 & 2 & 3 & \ldots & n \\ + \pi(1) & \pi(2) & \pi(3) & \ldots & \pi(n) + \end{pmatrix} + .\] + Elementare Kombinationen: $n$ Möglichkeiten für $\pi(1)$, $(n-1)$ + Möglichkeiten, für $\pi(2)\ldots$, 1 Möglichkeit für $\pi(n)$. + \[ + \#O_{n} = n! \text{ (Fakultät)} + .\] + Verifikation der Gruppenaxiome + \begin{enumerate} + \item $g*(h*k) = g \circ (h \circ k) = (g \circ h) \circ k = (g * h) * k$ + \item $e * g = id * g = g$ + \item Sei g eine Permutation und $h = g^{-1}$ die Umkehrabbildung. + Dann gilt $h*g = g^{-1} \circ g = id = e$. + \end{enumerate} + Für $n \ge 3$ ist $\sigma_{n}$ nicht kommutativ. + \[ + \begin{pmatrix} + 1 & 2 & 3 & 4 & \ldots \\ + 2 & 1 & 3 & 4 & \ldots + \end{pmatrix} + \circ + \begin{pmatrix} + 1 & 2 & 3 & 4 & \ldots \\ + 1 & 3 & 2 & 4 & \ldots + \end{pmatrix} + = + \begin{pmatrix} + 1 & 2 & 3 & 4 & \ldots \\ + 2 & 3 & 1 & 4 & \ldots + \end{pmatrix} + .\] + \[ + \begin{pmatrix} + 1 & 2 & 3 & 4 & \ldots \\ + 1 & 3 & 2 & 4 & \ldots + \end{pmatrix} + \circ + \begin{pmatrix} + 1 & 2 & 3 & 4 & \ldots \\ + 2 & 1 & 3 & 4 & \ldots + \end{pmatrix} + = + \begin{pmatrix} + 1 & 2 & 3 & 4 & \ldots \\ + 3 & 1 & 2 & 4 & \ldots + \end{pmatrix} + .\] + +\end{bsp} + +\begin{satz}[] + Sei $G = (G, *, e)$ eine Gruppe. Dann gilt für alle $g, h, k \in G$. + \begin{enumerate} + \item $g * h = g * k \implies h = k$ (Linkskürzung) + \item $g * h = k * h \implies g = k$ (Rechtskürzung) + \item $g * e = g$ (das (links)neutrale Element ist auch rechtsneutral) + \item aus $g*h=g$ oder $h*g=g$ für ein einziges $g\in G$, so folgt $h=e$ + \item $\forall g \in G$: existiert ein eindeutig bestimmtes Element + $g^{-1} \in G$ mit $g^{-1} * g = e = g * g^{-1}$. + \item Aus $h * g = e$ oder $g*h=e$ folgt $h=g^{-1}$. + \item $\left(g^{-1}\right)^{-1} = g$ $\forall g \in G$ + \end{enumerate} +\end{satz} + +\begin{proof}[Beweis 1] + Sei $g * h = g * k$. + + Nach (G3) $\exists s \in G$, sodass $s * g = e$. + Daher gilt $s*(g*h) = (s*g)*h = e*h = k$.\\ + Analog: $s*(g*k) = (s*g)*k = e*k = k$ + Daraus folgt: $h = k$. +\end{proof} + +\begin{proof}[Beweis 3] + Nach (G3) existiert $h\in G$ und $h*g=e$. + + Es folgt $h*(g*e)=(h*g)*e=e*e=e=h*g$\\ + Nach (1) folgt $g*e=g$ +\end{proof} + +\begin{proof}[Beweis 5, Existenz] + Sei $h\in G$ mit $h*g = e$ (ex. nach G3) + + $h*(g*h) = (h*g)*h = e * h = h = h * e$\\ + Durch Linkskürzung erhalten wir $g * h = e$. +\end{proof} + +\begin{proof}[Beweis 2] + Sei $g*k=h*k$. Sei $s \in G$ so dass $k*s=e$ (existenz nach 5). + + $\implies (g*k)*s = g*(k*s) = g*e = g$, analog\\ + $\implies (h*k) * s = h*(k*s) = h*e = h$\\ + Daraus folgt $g = h$. +\end{proof} + +\begin{proof}[Beweis 4] + $g * h = g = g * e \implies h = e$, analog\\ + $h*g = g = e*g \implies h=e$ +\end{proof} + +\begin{proof}[Beweis 5 Eindeutigkeit und 6] + Seien $h, h' \in G$ mit $h*g=e=h'*g$ Nach Rechtskürzung folgt + $h = h'$. Daher ist $g^{-1} eindeutig$. Sei $h \in G$ mit + $g * h = e$. Wegen $g^{-1}*g = e$ folgt mit Linkskürzung, dass $h = g^{-1}$. +\end{proof} + +\begin{proof}[Beweis 7] + aus $g * g^{-1} = e$ folgt $\left(g^{-1}\right)^{-1} = g$ +\end{proof} + +\begin{bem}[] + $g, h \in G$, so gilt $(g * h)^{-1} = h^{-1} * g^{-1}$\\ + Grund: $(h^{-1} * g^{-1}) * (g *h) = h^{-1} * (g*g^{-1}) * h = h^{-1}* e * h = h * h^{-1} = e.$ +\end{bem} +\end{document} diff --git a/ws2019/la/lectures/linear5.pdf b/ws2019/la/lectures/linear5.pdf new file mode 100644 index 0000000000000000000000000000000000000000..d23ba7e6a36cf9859544184e74aa9aaec1667941 GIT binary patch literal 118241 zcmb5WRaBhavbK%8(`ayacY?dSyGw8g?he77;O;?#ySux)yE}j0z1P~`KgME=o!v)$ z&m z5Jg^flZug+>K(h^5{VT&^60`j!U6~qOOzkaw_o}7JAt7T>)KR?L;^bn@`{RHPAWZu z`E;Z=;;h=ax|Y87ERDq1{)o@=HsxX-6sM~Fkrts41Xqws28Ucw^W)_7Ei{50%;>8- zC$19D^V`|V4Le^Hc|`Tp?Oa;&!PfJFl<3|qZ3J$H+<=evtAm5L{q+SZh;2945QhlJ zu3&Cb=A-Ubl!$W{lLWd@ z`DF!7h2%Us+}AB3&?l1IoNJ|6tI`q$4dz2CorH49#IYUVVBH@qnp?BQzr|5XBJX#3 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--git a/ws2019/la/lectures/linear5.tex b/ws2019/la/lectures/linear5.tex new file mode 100644 index 0000000..0016c25 --- /dev/null +++ b/ws2019/la/lectures/linear5.tex @@ -0,0 +1,259 @@ +\documentclass{lecture} + +\begin{document} +\section{Gruppen, Ringe, Körper} + +\subsection{Ringe} + +\begin{definition}[Ring] + Ein Ring $R = (R, +, \cdot, O_{R})$ ist eine Menge $R$ und zwei + Verknüpfungen $+, \cdot: R \times R \to R$ und einem + Element $O_{R} \in R$ so dass: + \begin{enumerate} + \item $(R, +, O_{R})$ ist eine abelsche Gruppe + \item $a \cdot (b \cdot c) = (a \cdot b) \cdot c$ $\forall a,b,c \in R$ + \item $a \cdot (b + c) = a \cdot b + a \cdot c$, $(a + b) \cdot c = ac + bc$ + \end{enumerate} + + Ein unitärer Ring (,,Ring mit 1'') ist ein Tupel + $(R, +, \cdot, 0_{R}, 1_{R})$, so dass $(R, +, \cdot, 0_{R})$ ein + Ring ist und $1_R \in R$, so dass gilt: + \begin{enumerate} + \item $1_{R} \cdot a = a = a \cdot 1_{R}$ + \end{enumerate} + + Ein Ring heißt kommutativ, wenn + \begin{enumerate} + \item $a \cdot b = b \cdot a$ $\forall a, b \in R$ + \end{enumerate} +\end{definition} + +\begin{bem}[Notation] + Das inverse Element von $a \in R$ bezüglich $+$ bezeichnet man mit + $-a$. Ein Inverses bezüglich $\cdot$ existiert i.A. nicht. + + Die Eins in einem unitären Ring ist eindeutig bestimmt. +\end{bem} + +\begin{bsp} + $(\Z, +, \cdot, 0, 1)$ ist ein kommutativer Ring mit 1 +\end{bsp} + +\begin{bsp}[$\Z / n\Z$] + ist ein kommutativer Ring mit 1. Multiplikationsvorschrift ist die + folgende: Für $A, B \in \Z / n\Z$ + \begin{enumerate} + \item Wähle Vertreter $a, b \in \Z$ von $A$ und $B$. + \item bilde $a \cdot b$ in $\Z$ + \item $A \cdot B :=$ Restklasse von $a \cdot b$ + \end{enumerate} + Nachzuweisen: Unabhängigkeit der Definition von der Auswahl der Vertreter + im ersten Schritt. +\end{bsp} + +\begin{bsp}[Die Menge der geraden ganzen Zahlen] + ist ein kommutativer Ring ohne 1. +\end{bsp} + +\begin{lemma} + $R = (R, +, \cdot, 0_{R})$ Ring. Dann gilt + \begin{enumerate} + \item $0_{R} \cdot a = 0_R = a \cdot 0_R$ + \item $a \cdot (-b) = - ab = (-a) \cdot b$ + \end{enumerate} + Ist R unitär, so gilt: + \begin{enumerate} + \item $-b = (-1_{R}) \cdot b$ + \end{enumerate} +\end{lemma} + +\begin{proof}[Beweis 1] + \[ + 0_{R} \cdot a + 0_{R} = 0_{R} = 0_{R} a = (0_R + 0_R)a = 0_R a + 0_R a + .\] + Mit kürzen folgt: $0_R = 0_R a$ + Analog für $a \cdot 0_R = 0_R$ +\end{proof} + +\begin{proof}[Beweis 2] + \[ + 0_R = a 0_R = a \left((-b) + b\right) = a(-b) + a b + \] also $a(-b) = -ab$. +\end{proof} + +\begin{proof}[Beweis 3] + Ist R unitär, so setzt man in 2 $a = 1_{R}$ ein und erhält 3. +\end{proof} + +\begin{bsp} + $R = \{0\} $ mit den einzig möglichen Verknüpfungen $+$ und $\cdot$ + heißt der \textit{Nullring}. + Nullring ist ein kommutativer Ring mit 1 ($0_R = 0 = 1_R$). + Dies ist der einzige Ring mit $1_R = 0_R$. + + Begründung: Gelte $0_R = 1_R$, so folgt für jedes $r \in R$: + \[ + r = r \cdot 1_R = r \cdot 0_R = 0_R + .\] Egal welches Element herausgenommen wird, es ist immer $0_R$, d.h. + $R$ muss ein Nullring sein. +\end{bsp} + +\begin{lemma} + Sei $R = (R, +, \cdot, 0_R, 1_R)$ ein unitärer Ring und + $R^{\times } \subset R$ die Menge der Elemente die ein Links- und + ein Rechtsinverses bezüglich $\cdot$ haben, das heißt + \[ + R^{\times } = \{r \in R \mid \exists s, t \in R : s r = 1_R = r t\} + .\] + Dann ist $(R^{x}, \cdot, 1_R)$ eine Gruppe. + Man nennt $R^{x}$ die Einheitengruppe von $R$. +\end{lemma} + +\begin{proof} + Seien $r, \overline{r} \in R^{\times }$ und $s, \overline{s}, t, \overline{t}$ + mit + \[ + s r = 1 = r t + .\] und + \[ + \overline{s} \overline{r} = 1 = \overline{r} \overline{t} + .\] Dann + \[ + (s' s)(r r') = s'(s r) r' = s'1r' = s'r' = 1 + .\] + \[ + (r r') (t' t) = r (r' t') t = r 1 t = r t = 1 + .\] $\implies r \cdot r' \in R^{\times }$ + + Wir überprüfen die Gruppenaxiome: G1 folgt aus R2 + + $1 \in R$ ist neutral $\to$ G2 + + Bleibt zu zeigen: $\forall r \in R^{\times }: \exists r' \in R: r r' = 1$ + Nach Definition: $\exists s \in R: s r = 1$. Zu zeigen: $s \in R^{\times }$. + Offenbar hat das Rechtsinverse $r$. Aber $r$ ist auch linksinvers zu $s$: + + Wähle $t \in R$ mit $r t = 1$. Dann gilt $s = s (r t) = (s r) t = t$ + $\implies$ rs = rt = 1. +\end{proof} + +\begin{bem} + $0_R \in R^{\times } \implies \exists r \in R: 0_R r = 1_R \implies 0_R = 1_R \implies$ $R$ ist der Nullring. +\end{bem} + +\begin{definition}[Körper] + Ein Körper $K$ ist ein kommutativer Ring mit 1: $(K, +, 0_K, 1_K)$ + mit $K^{\times } = K \ \{0_K\}$ +\end{definition} + +\begin{bsp} + \begin{enumerate} + \item $\Q, \R, \C$ sind Körper + \item $\Z$ ist kein Körper ($\Z^{\times } = \{+1, -1\} $) + \end{enumerate} +\end{bsp} + +\begin{lemma}[] + In einem Körper $K$ gilt, dass + \[ + a b = 0_K \implies a = 0 \wedge b = 0 + .\] +\end{lemma} + +\begin{proof} + Angenommen $a \neq 0_K$: Dann existiert $a^{-1} \in K$ mit + $a^{-1} a = 1_K$. Es folgt + \[ + b = 1_K b = a^{-1} a b = a^{-1} 0_K = 0_K + .\] +\end{proof} + +\begin{lemma} + Ist $p$ eine Primzahl, so ist $\Z / p\Z$ ein Körper. +\end{lemma} + +\begin{proof}[] + $\Z / p \Z$ ist kommutativer Ring mit 1 (siehe oben). + + Zu zeigen: $\forall A \in \Z / p\Z, A \neq \overline{0}$ ist + die $\overline{1}$ im Bild der Abbildung: + \[ + A\cdot : \Z / p \Z \to \Z / p \Z, B \mapsto A \cdot B + .\] + Wir zeigen sogar, dass $A \cdot$ surjektiv ist. + Da $\Z / p \Z$ endlich ist, genügt es z.z., dass $A \cdot$ injektiv ist. + + Angenommen es gäbe Restklassen $B, C \in \Z / p \Z$ mit + \[ + A \cdot B = A \cdot C + .\] Seien $a, b, c \in \Z$ Vertreter. + Wegen $A \neq \overline{0}$ gilt $a$ nicht durch $p$ teilbar. + Wegen $AB = AC$ gilt $ab =- ac$ mod $p \implies p$ teilt $a (b -c)$. + + Weil $p$ eine Primzahl ist und $p$ teilt nicht $a$ folgt + $p / (b -c)$, also $b =- c$ mod $p \implies B = C$ +\end{proof} + +\begin{bem} + Ist $n \in N$ keine Primzahl, so ist $\Z / n \Z$ kein Körper. +\end{bem} + +\begin{proof} + Für $n=1$ ist $\Z / n\Z$ der Nullring (kein Körper). + + Nun sei $n > 1$ keine Primzahl + $\implies \exists a, b \in \N, 1 < a, b < n$ mit $a b = n$ Für die + Restklassen bedeutet dies: $\overline{a} \neq \overline{0}$, + $\overline{b} \neq \overline{0}$ aber + $\overline{a} \cdot \overline{b} = \overline{a \cdot b} = \overline{n} = \overline{0}$. Wir erhalten Widerspruch zu Lemma 1.15. Also + ist $\Z / n \Z$ kein Körper. +\end{proof} + +\begin{definition}[Charakteristik] + Sei $K$ Köper. Die kleinste natürliche Zahl $n$ mit n mal + $1_K + \ldots + 1_K = 0_K$ (in $K$) heißt die + Charakteristik von K. + + Notation: char(K). + Gibt es eine solche Zahl $n$ nicht, dann setzt man char(K) = 0. +\end{definition} + +\begin{bem} + + \begin{enumerate} + \item char(K) $=0$ oder char(K) $\ge 2$ (wegen $0_K \neq 1_K$). + \item \Q, \R, \C haben Charakteristik Null. + \item \Z / p \Z hat die Charakteristik p. + \end{enumerate} +\end{bem} + +\begin{satz} + char(K) ist entweder 0 oder Primzahl. +\end{satz} + +\begin{proof} + Sei char(K) $\neq 0$, also char(K) $= n \ge 2$. + + Wäre $n$ keine Primzahl, so existieren $a, b \in \N, 1 < a ,b < n$ + mit $ab = n$. Dann gilt: + \[ + (1_K + \ldots + 1_K) \cdot (1_K + \ldots + 1_K) + = (1_K + \ldots + 1_K) = 0 + .\] Aus dem Satz vom Nullprodukt folgt $(1_K + \ldots + 1_K)$ = $0_K$ + oder $(1_K + \ldots + 1_K) = 0_K$ + + Das Widerspricht der Minimalität von n. +\end{proof} + +\subsection{Homomorphismen} + +\begin{definition} + Seien $(G, *_G, e_G)$ und $(H, *_H, e_H)$ Gruppen. + Eine Abbildung $f: G \to H$ heißt Gruppenhomomorphismus wenn für alle + $g, g' \in G$ gilt: + \[ + f(g *_G g') = f(g) *_H f(g') + .\] +\end{definition} + +\end{document} diff --git a/ws2019/la/uebungen/la2.pdf b/ws2019/la/uebungen/la2.pdf new file mode 100644 index 0000000000000000000000000000000000000000..10d1108689649c315b7179717e1bbbce9819d7e1 GIT binary patch literal 95928 zcmb@uRaBl`vabE&@Z#?7?(XjH?ykXudvJGm3liMj3GO5~!67)o+4)v~-FyG5S!1Qg z_&MUR=KWONbydv?_N#GjpJT0MNzN+1%J3==p6?SKD!I6xnaN)+jy=dCU5j zuV(^yq9(?d6@Y@qD5&=Nz^G^p_H=YM_i1mBz@pi;RsBc!wyAyb9XQ=Zw_lozC25HM 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\overline{A} \} + .\end{align*} + Das heißt es existieren mehrere inverse Elemente zu dem selben $a \in G$, + was ein Widerspruch zur Eindeutigkeit der inversen Elemente ist. +\end{proof} + +Es existiert keine Menge $G$, sodass die $P(G)$ nur aus ein-elementigen Mengen +besteht, da in jeder $P(G)$ die leere Menge liegt, die kein Inverses hat. + +\begin{aufgabe} + Es seien $A$, $B$ und $C$ Mengen und $f: A \to B$, $g: B \to C$ + Abbildungen zwischen ihnen. +\end{aufgabe} + +\textbf{a)} +\begin{proof} + Zu zeigen: $g \circ f$ injektiv $\implies$ $f$ injektiv + + Angenommen: $f$ nicht injektiv. Dann existieren + $a_1, a_2 \in A, a_1 \neq a_2$, s.d. + $f(a_1) = f(a_2)$. + + $\implies g(a_1) = g(a_2)$. Widerspruch zu $g \circ f$ injektiv. +\end{proof} + +\textbf{b)} +\begin{proof} + Zu zeigen: $g \circ f$ surjektiv $\implies$ $g$ surjektiv + + Angenommen: $g$ nicht surjektiv. Dann existiert ein $c \in C$ mit + $g^{-1}(c) = \emptyset$. Damit $(g \circ f)^{-1}(c) = \emptyset$. + Widerspruch zu $g \circ f$ surjektiv. +\end{proof} + +\textbf{c)} +\begin{proof} + Zu zeigen: Sind $f$ und $g$ bijektiv, so ist auch $g \circ f$ bijektiv + + Sei $c \in C$ beliebig, so gilt wegen $g$ bijektiv: + $g^{-1}(\{c\}) = \{b\}, b \in B$. Wegen $f$ bijektiv gilt, dann dass + $f^{-1}(\{b\}) = f^{-1}(g^{-1}(c)) = \{a\}, a \in A$. Damit ist + $f \circ g$ bijektiv. + + Zu zeigen: Es gilt $(g \circ f)^{-1} = f^{-1} \circ g^{-1}$ + + ($f^{-1} \circ g^{-1}) \circ (g \circ f)$ muss die Identitätsabbildung $id$ + ergeben, damit + $(g \circ f)^{-1}$ Umkehrabbildung von $(g \circ f)^{-1}$ ergibt. + \[ + (f^{-1} \circ g^{-1}) \circ (g \circ f) + = (f^{-1} \circ id \circ f^{-1}) + = id + .\] +\end{proof} + +\begin{aufgabe} +\end{aufgabe} + +\textbf{a)} $F_n$ ist weder injektiv noch surjektiv. + +Nicht injektiv: $0 \in \N_0$ und $1 \in \N_0$ bilden beide auf $a$ ab. + +Nicht surjektiv: Für $a = 0$ sind alle $f(n) = 0$ und damit ist $f^{-1}(1) = \emptyset$. + +\textbf{b)} $f_n^{2} = f_{n-1}f_{n+1}+(-1)^{n} \cdot a^2$ $\forall n \in \N$ + +\begin{proof}[Beweis durch vollständige Induktion] + Induktionsanfang: Für $n=1$ und für $n=2$: + \begin{align*} + &f(1)^2 = a^2 = a \cdot 2 a - a^2 = f(0)f(2) + (-1)^1 \cdot a^2 \\ + &f(2)^2 = 4a^2 = a \cdot 3 a + a^2 = f(1)f(3) + (-1)^2 \cdot a^2 \\ + .\end{align*} + + Induktionsvorraussetzung: Es existiere ein festes aber beliebiges + $n \in \N$ mit + \[ + f(n)^2 = f(n-1) f(n+1)+(-1)^{n} \cdot a^2 + .\] + Induktionsschritt $n \to n+2$: + + Zu zeigen: + \[ + f(n+2)^2 = f(n+1) f(n+3) + (-1)^{n+2} \cdot a^2 + .\] + \begin{align*} + f(n+2)^2 &= \left( f(n+1) + f(n) \right)^2\\ + &= f(n+1)^2 + 2f(n+1)f(n) + f(n)^2 \\ + &= f(n+1)^2 + 2f(n+1)f(n) + f(n-1)f(n+1) + (-1)^{n}\cdot a^2\\ + &= f(n+1) \left[ f(n+1) + 2f(n)+f(n-1)\right] + (-1)^{n+2} \cdot a^2 \\ + &= f(n+1) f(n+3) + (-1)^{n+2} \cdot a^2 + .\end{align*} + + Von den zwei Induktionsanfängen ausgehend, ergibt sich die Behauptung + für alle $n \in N$. +\end{proof} + +\begin{aufgabe} + Sei $f: X \to Y$ Abbildung. Wir definieren die Relation + \[ + R = \{(x_1, x_2) \in X \times X \mid f(x_1) = f(x_2)\} + .\] +\end{aufgabe} + +\textbf{a)} $R$ ist Äquivalenzrelation auf $X$. + +\begin{enumerate} + \item Reflexivität: Sei $a \in X$. Dann ist $a \thicksim a$, + da $f(a) = f(a)$. + \item Symmetrie: Seien $a, b \in X$ und $a \thicksim b$.\\ + $\implies f(a) = f(b) \implies f(b) = f(a) \implies b \thicksim a$ + \item Transitivität: Seien $a, b, c \in X$ und $a \thicksim b$ und + $b \thicksim c$.\\ + $\implies f(a) = f(b) \land f(b) = f(c) \implies f(a) = f(c) \implies a \thicksim c$ +\end{enumerate} + +\textbf{b)} + +Die Abbildung $\overline{f}: X / R \to im(f)$ wird definiert als +\[ + \overline{f} := A \mapsto f(a), a \in A +.\] + +Zu zeigen: $\overline{f}$ ist bijektiv. + +Injektivität: Seien $A,B \in X / R$. Entweder $A = B$, daraus folgt, dass +$\overline{f}(A) = \overline{f}(B)$. oder +$A \neq B \implies A \cap B = \emptyset$, daraus folgt, dass +$\overline{f}(A) \neq \overline{f}(B)$. + +Surjektivität: Sei $y$ in $im(f)$, so existiert ein $x \in X$ mit $f(x) = y$. +Dieses x liegt in einer Äquivalenzklasse aus $X/R$. +Damit $\overline{f}^{-1}(y) \neq \emptyset$. + +Aus Injektivität und Surjektivität folgt, dass $\overline{f}$ bijektiv ist. + +Eindeutigkeit: Die Abbildung $\overline{f}$ ist eindeutig bestimmt, durch +den Zusammenhang +\[ + \overline{f} \circ p = f +.\] Da $p$ einem $x$ immer eindeutig genau eine Äquivalenzklasse zuordnet, +nämlich, die jenige Teilmenge $A \in X / R$, deren Elemente alle auf das +selbe $f(x)$ abbilden, muss $\overline{f}$ stets genau der Äquivalenzklasse +$[x]$ ihr gemeinsames Bild $f(x)$ zuordnen, damit die Gleichheit $\overline{f} +\circ p = f$ erfüllt ist. + +\end{document} diff --git a/ws2019/theo/uebungen/theo.pdf b/ws2019/theo/uebungen/theo.pdf new 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z_|rqH_w8wW-FBjj5Dye7o{p%15K6BCPsSk=P?*CUhdD2Pu6$xO6%JpA7%6#%r^F>J zsjS^#sv-5Oi7UAqT9%wZG$Mw&-T*_W*SF?A zi%7<`yD0|$1u*RkTGb#ppUx0VRJU$puhg}cAO*fzfs9n<3Ks|m-h|Tc;X00t;Kn%Fn1CkH@QP7aD zAhnoP2t6k|oXs6zIF2uJvQb&TI*Z#dB<|_JJGo=mi7|+MMfduE=R?o8mB#NFVT#|mQGeFgV)MwXg!!)@Z+@s?atm9o!uhOrt-R~gPS zOqvWQNtUl3y{7ZYb_qa4RqqSbC0C3$u)`=>f&YCWIyxCRIJr6et~>_%UllM0ND>ke IIZ?>}2hM&OY5)KL literal 0 HcmV?d00001 diff --git a/ws2019/theo/uebungen/theo.tex b/ws2019/theo/uebungen/theo.tex new file mode 100644 index 0000000..8fe961e --- /dev/null +++ b/ws2019/theo/uebungen/theo.tex @@ -0,0 +1,197 @@ +\documentclass{lecture} + +\usepackage{siunitx} + +\begin{document} +\begin{aufgabe}[Differentialgleichungen] + Lösen Sie die folgenden Differentialgleichungen +\end{aufgabe} +\textbf{a)} $y'(x) = y^2(x)\cosh(x)$ + +Durch Umformung erhalten wir folgende Form einer homogenen DGL 1. Ordnung: +\[ + \frac{y'(x)}{y^{2}(x)} = \cosh(x) +.\] Durch Integration erhalten wir folgenden Ausdruck +\[ + \int_{}^{} \frac{dy}{dx \cdot y^2}dx = \int_{}^{} \sinh(x)dx + \implies + -\frac{1}{y} = \sinh(x) + C +.\] und damit: +\[ + y = - \frac{1}{\sinh(x) + C} +.\] Durch Einsetzen der Anfangsbedingung $y(0) = 4$ erhalten wir: +\[ + y = -\frac{1}{\sinh(x) - \frac{1}{4}} +.\] +\textbf{b)} $y'(x) = \sin(x)\cos(x) - y(x)\sin(x)$ + +Hier liegt eine inhomogene DGL erster Ordnung vor, das heißt wir setzen +nach Umformung den inhomogenen Teil Null. +\[ + y'(x) + y(x)\sin(x) = 0 +.\] Durch Trennung der Variablen erhalten wir folgende Lösung der +homogenen Gleichung: +\[ + y = ae^{\cos(x)} +.\] Durch Variation der Konstanten $a$ durch eine Funktion $A(x)$ und +einsetzen in die DLG erhalten wir +\begin{align*} + A'(x) e^{\cos(x)} - A(x) \sin(x) e^{\cos(x)} + &= \sin(x)\cos(x)-A(x) \sin(x) e^{\cos(x)}\\ + A'(x) &= \sin(x) \cos(x) e^{-\cos(x)} +.\end{align*} +Durch Integration erhalten wir folgenden Term für $A(x)$. +\begin{align*} + A(x) &= \cos(x) e^{-\cos(x)} - \int_{}^{} -\sin(x) e^{-\cos(x)}dx \\ + &= \cos(x) e^{-\cos(x)} + e^{-\cos(x)} + C\\ + &= e^{-\cos(x)} \left( \cos(x) + 1 \right) + C +.\end{align*} und damit die allgemeine Lösung der DGL +\begin{align*} + y(x) &= \left(e^{-\cos(x)}\left( \cos(x) + 1 \right) + C\right) e^{\cos(x)}\\ + &= \cos(x) + 1 + C e^{\cos(x)} +.\end{align*} Mit der Anfangsbedingung $y(0)=0$ ergibt sich +$C = \frac{-2}{e}$, und damit: +\[ + y(x) =\cos(x)+1-2e^{\cos(x)-1} +.\] +\newpage + +\begin{aufgabe}[Start einer Rakete] +\end{aufgabe} + +\textbf{a)} Mit welcher Geschwindigkeit wird der Treibstoff aus der Sicht +eines auf der Erde ruhenden Beobachters ausgestoßen? + +Die Rakete bewegt sich mit der Geschwindigkeit $v_{R}$ nach oben, von dort +wird der Tropfen mit der Geschwindigkeit $v_{T}$ ausgestoßen. +\[ + v_{T} = v_{R} - v_{0} +.\] + +Welche Kraft erfährt die Rakete somit aus der Sicht des ruhenden Beobachters? + +Für den Impuls eines Treibstoffteilchens der Masse $\Delta m_{T}$ gilt: +\[ + p_{T} = \Delta m_{T} \cdot v_{T} = \Delta m_{T} \cdot \left( v_{R} - v_{0} \right) +.\] Für die Impulsänderung $\Delta p_{R}$ in der Zeiteinheit $\Delta t$ gilt: +\[ + \frac{\Delta p_{R}}{\Delta t} = \frac{\Delta m_{T} \cdot \left( v_{R} - v_{0} \right) }{\Delta t} +.\] Nun führen wir den Grenzübergang für $\Delta t \to 0$ durch: +\begin{align*} + \lim_{\Delta t \to 0} \left( \frac{\Delta p_{R}}{\Delta t} \right) + &= \lim_{\Delta t \to 0} \left(\frac{\Delta m_{T} \cdot \left( v_{R} - v_{0} \right) }{\Delta t}\right) \\ +.\end{align*} +Daraus erhalten wir: +\[ + \dot{p} = \dot{m} (v_{R} - v_{0}) +.\] +Die Kraft, die auf die Rakete wirkt, setzt sich zusammen aus der Kraft $F_{A}$, +die die Rakete antreibt und der Gewichtskraft $F_{G} = - m g$. + +Aus dem zweiten Newton'schen Axiom: +\[ + \dot{p} = F +\] folgt: +\[ + F = \dot{m} \left( v_{R} - v_{0} \right) - m g +.\] +\textbf{b)} Aus der Kraft $F$ folgt mit $v_{R} = v$ +\[ + \dot{m}v + m\dot{v} = \dot{m} \left( v - v_{0} \right) - m g +\] durch Umformung +\begin{align*} + \frac{\dot{m}v}{m} + \dot{v} + &= \frac{\dot{m}}{m} \left( v - v_{0} \right) - g\\ + &= \frac{\dot{m}}{m} v - \frac{\dot{m}}{m} v_{0} - g +.\end{align*} +Daraus ergibt sich: +\[ + \frac{\dot{m}}{m} v_{0} = -g -\dot{v} +.\] + +\textbf{c)} Durch Integration des obenstehenden Ausdrucks ergibt sich: +\begin{align*} + \ln(m) v_{0} &= -gt - v + m_{0}\\ + \implies v(t) &= -gt - \ln(m) v_{0} + C +.\end{align*} +Mit der Anfangsbedingung $v(0) = 0$ und der Anfangsmasse $m_{0}$ folgt: +\[ + v(0) = 0 = -v_{0} \cdot \ln(m_{0}) + C \implies C = v_{0} \ln(m_{0}) +.\] Damit: +\[ + v(t) = -gt +v_{0} \ln\left( \frac{m_{0}}{m}\right) +.\] + +\textbf{d)} Sei die Treibstoffausstoßrate $\mu$ gegeben als +$\mu = \frac{m}{t}$, so folgt: +\[ + m(t) = m_{0} - \mu t +.\] Für eine Treibstoffmasse $m_{T} < m_{0}$ ergibt sich ein Zeitpunkt +$t_{e}$ zu dem der gesamte Treibstoff aufgebraucht ist: +\[ + m(t_{e}) = m_{0} - m_{T} \implies t_{e} = \frac{m_{T}}{\mu} +.\] Mit der Rate $\mu$ ergibt sich für $v(t)$: +\[ + v(t) = -gt + v_{0} \ln\left( \frac{m_{0}}{m_{0} - \mu t} \right) +.\] Zum Zeitpunkt $t_e$ +\[ + v(t_{e}) = -gt_{e} + v_{0} \ln\left( \frac{m_{0}}{m_{0} - m_{T}} \right) +.\] Für die Höhe muss die Geschwindigkeit $v(t)$ einmal integriert werden, +mit $h_0 = 0$: +\begin{align*} + s(t) &= \int_{}^{} v(t) dt = \int_{}^{} \left( -gt + v_{0} \ln\left( \frac{m_{0}}{m_{0} - \mu t} \right) \right) dt + h_0\\ + &= -\frac{1}{2}gt^2 - v_{0} \int_{}^{} \ln\left(1-\frac{\mu}{m_{0}} t\right) \\ + &= -\frac{1}{2}gt^2 + v_{0} \frac{m_{0}}{\mu} + \left[ \left( 1 - \frac{\mu}{m_{0}} t \right) + \ln\left(1 - \frac{\mu}{m_{0}} t\right) + - \left( 1 - \frac{\mu}{m_{0}}t \right) \right] \\ + &= -\frac{1}{2}gt^2 + v_0 \frac{m_0}{\mu} \left( 1 - \frac{\mu}{m_0}t \right) \left[ \ln\left( 1 - \frac{\mu}{m_0}t \right) -1 \right] \\ + &= -\frac{1}{2}gt^2 + \left(v_0 \frac{m_0}{\mu} - v_0 t\right) + \left[ \ln\left( 1 - \frac{\mu}{m_0}t \right) \right] +.\end{align*} + +Für $s(t_{e})$ und $t_{e} = \frac{m_{T}}{\mu}$ ergibt sich: +\begin{align*} + s(t_{e}) &= -\frac{1}{2}g \left( \frac{m_{T}}{\mu} \right) ^2 + v_{0}\left( \frac{m_0}{\mu} - \frac{m_{T}}{\mu} \right) + \left[\ln\left( 1-\frac{\mu}{m_0} \frac{m_{T}}{\mu} \right) \right] \\ + &= -\frac{1}{2}g \left( \frac{m_{T}}{\mu} \right) ^2 + v_{0}\left( \frac{m_0 - m_{T}}{\mu} \right) + \left[\ln\left( 1-\frac{m_{T}}{m_0} \right) \right] \\ +.\end{align*} + +\textbf{e)} Die Geschwindigkeit, wenn der ganze Treibstoff aufgebraucht ist, ergibt sich durch +\[ + v(t_{e}) = -gt_{e} + v_{0} \ln\left( \frac{m_{0}}{m_{0} - m_{T}} \right) +.\] Wenn $m_{0} = m_{T}$ geht der obenstehnde Ausdruck gegen $\infty$. + +Eine Rakete die nur aus Treibstoff besteht, ist logischerweise nicht +physikalisch sinnvoll. Das Ergebnis, dass ein Objekt mit der Masse Null +unendliche Geschwindigkeiten erreicht, ist dahingegen denkbar. + +\textbf{f)} Zum Zeitpunkt $t_e$ hat die Rakete die Geschwindigkeit +\[ + v(t_{e}) = v_{E} +.\] Danach ist die Geschwindigkeit der Rakete durch +\[ + v(t) = v_{E} - gt +\] beschrieben. Für die Steighöhe ergibt sich daraus +\[ + s(t) = v_{E}t - \frac{1}{2}gt^2 + h_{0} +.\] Aus $v(t) = 0$ folgt $\hat{t} = \frac{v_{E}}{g}$ und damit +\begin{align*} + h(\hat{t}) &= \frac{v_{E}^2}{g} - \frac{1}{2} g\frac{v_{E}^2}{g^2} + h_0 \\ + &= \frac{1}{2} \frac{v_E^2}{g} + h_{0} +.\end{align*} + +\textbf{g)} Wasserrakete mit $v_{0} = \SI{25}{ms^{-1}}$ und $m_T = \frac{4}{5} m_0$ und $\mu = 2m_{T} \SI{}{s^{-1}}$ +\begin{align*} + v\left(\frac{1}{2}\SI{}{s}\right) &= -g \frac{1}{2}\SI{}{s} + \SI{25}{ms^{-1}} \ln\left( \frac{m_0}{m_0-\frac{4}{5}m_0} \right) \\ + &= -\SI{9.81}{ms^{-2}} \SI{0,5}{s} + \SI{25}{ms^{-1}} \ln(5) \\ + &= \SI{45.14}{ms^{-1}} \\ + s(\SI{0.5}{s}) &= -\frac{1}{2} g \left( \frac{m_{T}}{2m_{T}\SI{}{s^{-1}}} \right)^2 + + \SI{25}{ms^{-1}} \left( \frac{m_{0} - m_{T}}{2 m_T \SI{}{s^{-1}}} \right) + \ln\left( \frac{1}{5} \right) \\ + &= -\frac{1}{2}g \left( \frac{1}{2} \SI{}{s} \right)^2 + + \SI{25}{ms^{-1}} \left( \frac{1}{8} \SI{}{s} \right) + \ln\left( \frac{1}{5} \right) +.\end{align*} +\end{document}