From 3128aa2401544e6c56a5adc1c8116481a5cdc684 Mon Sep 17 00:00:00 2001 From: henrydatei Date: Thu, 14 Jun 2018 10:49:31 +0200 Subject: [PATCH] VL LAAG 14.6. --- .../LAAG/TeX_files/Das_Lemma_von_Zorn.tex | 18 ++++- 2. Semester/LAAG/TeX_files/Der_Dualraum.tex | 73 ++++++++++++++++++ 2. Semester/LAAG/TeX_files/Quadriken.tex | 11 ++- 2. Semester/LAAG/Vorlesung LAAG.pdf | Bin 403433 -> 440181 bytes 2. Semester/LAAG/Vorlesung LAAG.tex | 1 + 5 files changed, 99 insertions(+), 4 deletions(-) create mode 100644 2. Semester/LAAG/TeX_files/Der_Dualraum.tex diff --git a/2. Semester/LAAG/TeX_files/Das_Lemma_von_Zorn.tex b/2. Semester/LAAG/TeX_files/Das_Lemma_von_Zorn.tex index cb1be97..4c20ced 100644 --- a/2. Semester/LAAG/TeX_files/Das_Lemma_von_Zorn.tex +++ b/2. Semester/LAAG/TeX_files/Das_Lemma_von_Zorn.tex @@ -108,12 +108,24 @@ Sei $K$ ein Körper und $U,V,W$ seien $K$-Vektorräume. Zudem sei $X$ eine Menge Sei $(X,\le)$ eine Halbordnung, die nicht leer ist. Wenn jede Kette eine obere Schranke hat, dann hat $X$ ein maximales Element. \end{theorem} \begin{proof} - Dieses Theorem ist äquivalent zum Auswahlaxiom. \frownie{} + Dieses Theorem ist äquivalent zum Auswahlaxiom. \frownie{}. Wir wollen zumindest die Hinrichtung zeigen, d.h. aus dem Lemma von Zorn folgt das Auswahlaxiom. \end{proof} -\begin{conclusion} +\begin{conclusion}[Auswahlaxiom] Zu jeder Familie $(x_i)$, nicht leer, gibt es eine \begriff{Auswahlfunktion}, das heißt eine Abbildung: \begin{align} - f: I\to \bigcup X_i\text{ mit } f(i)\in X_i\quad\forall i\notag + f: I\to \bigcup_{i\in I} X_i\text{ mit } f(i)\in X_i\quad\forall i\notag \end{align} \end{conclusion} +\begin{proof} + Sei $\mathcal{F}$ die Menge der Paare $(J,f)$ bestehend aus einer Teilmenge $J\subseteq I$ und einer Abbildung $f:I\to \bigcup_{i\in I} X_i$ mit $f(i)\in X_i\quad\forall i\in J$. Definieren wir $(J,f)\le (J',f')\iff J\subseteq J'$ und $f'\vert_J = f$, so ist $\le$ eine Halbordnung auf $\mathcal{F}$. Da $(\emptyset,\emptyset)\in\mathcal{F}$ ist $\mathcal{F}$ nichtleer. Ist $\mathcal{G}\subseteq\mathcal{F}$ eine nichtleere Kette, so wird auf $J':=\bigcup_{(J,f)\in\mathcal{G}} J$ durch $f'(j)=f(j)$ falls $(J,f)\in\mathcal{G}$ und $j\in J$ eine wohldefinierte Abbildung $f':J\to \bigcup_{i\in J}X_i$ mit $f'(i)\in X_i\quad\forall i\in J'$ gegeben. Das Paar $(J',f')$ ist eine obere Schranke der Kette $\mathcal{G}$. Nach dem Lemma von Zorn besitzt $\mathcal{F}$ ein maximales Element $(J,f)$. Wir behaupten, dass $J=I$. Andernfalls nehmen wir ein $i'\in I\backslash J$ und ein $x'\in X_{i'}$ und definieren $J':= U\cup\{i'\}$ und $f':J'\to \bigcup_{i\in J'} X_i$, $j\mapsto\begin{cases}f(j)&j\in J\\ x'&j=i'\end{cases}$. Dann ist $(J',f')\in\mathcal{F}$ und $(J,f)<(J',f')$ im Widerspruch zur Maximalität von $(J,f)$. +\end{proof} + + +\begin{conclusion}[Basisergänzungssatz] + \proplbl{3_1_11} + Sei $V$ ein $K$-Vektorraum. Jede linear unabhängige Teilmenge $X_0\subseteq V$ ist in einer Basis von $V$ enthalten. +\end{conclusion} +\begin{proof} + Sei $\mathcal{X}=\{X\subseteq V\mid X\text{ ist linear unabhängig, } X_0\subseteq X\}$ geordnet durch Inklusion. Dann ist $X_0\in\mathcal{X}$, also $\mathcal{X}\neq\emptyset$. Ist $\mathcal{Y}$ eine nichtleere Kette in $\mathcal{X}$, so ist auch $Y=\bigcup\mathcal{Y}\subseteq V$ linear unabhängig. Sind $y_1,...,y_n\in Y$ paarweise verschieden, so gibt es $Y_1,...,Y_n\in\mathcal{Y}$ mit $y_i\in Y_i$ für $i=1,...,n$. Da $\mathcal{Y}$ total geordnet ist, besitzt $\{Y_1,...,Y_n\}$ ein größtes Element, o.E. $Y_1$. Also sind $y_1,...,y_n\in Y_1$ und somit linear unabhängig. Folglich ist $Y_1\in \mathcal{X}$ eine obere Schranke von $\mathcal{Y}$. Nach dem Lemma von Zorn besitzt $\mathcal{X}$ ein maximales Element $X$. Das heißt, $X$ ist eine maximal linear unabhängige Teilmenge von $V$, nach LAAG1 II.3.5 also eine Basis von $V$. %TODO: Verlinkung + \end{proof} \ No newline at end of file diff --git a/2. Semester/LAAG/TeX_files/Der_Dualraum.tex b/2. Semester/LAAG/TeX_files/Der_Dualraum.tex new file mode 100644 index 0000000..5d88a82 --- /dev/null +++ b/2. Semester/LAAG/TeX_files/Der_Dualraum.tex @@ -0,0 +1,73 @@ +\section{Der Dualraum} + +Sei $V$ ein $K$-Vektorraum. + +\begin{definition}[Dualraum] + Der \begriff{Dualraum} zu $V$ ist der $K$-Vektorraum + \begin{align} + V^*=\Hom_K(V,K)=\{\phi:V\to K\text{ linear}\}\notag + \end{align} + Die Elemente von $V^*$ heißen \begriff{Linearformen} auf $V$. +\end{definition} + +\begin{example} + Ist $V=K^n=\Mat_{n\times 1}(K)$, so wird $V^*=\Hom_K(V,K)$ durch $\Mat_{1\times n}(K)\cong K^n$. Wir können also die Elemente von $V$ als Spaltenvektoren und die Linearformen auf $V$ als Zeilenvektoren auffassen. +\end{example} + +\begin{lemma} + Ist $B(x_1)_{i\in I}$ eine Basis von $V$, so gibt es zu jedem $i\in I$ genau $x_i^*\in V^*$ mit $x_i^*(x_j)=\delta_{ij}\quad\forall j\in I$. +\end{lemma} +\begin{proof} + Siehe LAAG1 III.5.1, angewandt auf die Familie $(y_j)_{j\in I}$, $y_j\delta_{i.j}$ in $W=K$. %TODO: Verlinkung +\end{proof} + +\begin{proposition} + Ist $B=(x_1)_{i\in I}$ eine Basis von $V$, so ist $B^*=(x_i^*)_{i\in I}$ linear unabhängig. Ist $I$ endlich, so ist $B^*$ eine Basis von $V^*$. +\end{proposition} +\begin{proof} + Ist $\phi=\sum_{i\in I} \lambda_ix_i^*$, $\lambda_i\in K$, fast alle gleich 0, so ist $\phi(x_j)=\sum_{i\in I} \lambda_j x_i^*(x_j)=\lambda_j$ für jedes $j\in I$. Ist also $\phi=0$, so ist $\lambda_j=\phi(x_j)=0\quad\forall j\in I$, $B^*$ ist somit linear unabhängig. \\ + Ist zudem $I$ endlich und $\psi\in V^*$, so ist $\psi=\psi'=\sum_{i\in I} \psi(x_i)x_i^*$, denn $\psi'(x_j)=\sum_{i\in I} \psi(x_i)x_i^*(x_j)=\psi(x_i)\quad\forall j\in I$, und somit ist $B^*$ ein Erzeugendensystem von $V^*$. +\end{proof} + +\begin{definition}[duale Basis] + Ist $B=(x_i)_{i\in I}$ eine endliche Basis von $V$, so nennt man $B^*=(x_i^*)_{i\in I}$ die zu $B$ \begriff{duale Basis}. +\end{definition} + +\begin{conclusion} + Zu jeder Basis $B$ von $V$ gibt es einen eindeutig bestimmtem Monomorphismus + \begin{align} + f_V\to V^*\text{ mit } f(B)=B^*\notag + \end{align} + Ist $\dim_K(V)<\infty$, so ist dieser ein Isomorphismus. +\end{conclusion} + +\begin{conclusion} + Zu jedem $=0\neq x\in V$ gibt es eine Linearform $\phi\in V$ mit $\phi(x)=1$. +\end{conclusion} +\begin{proof} + Ergänze $x_1=x$ zu einer Basis $(x_i)_{i\in I}$ von $V$ (\propref{3_1_11}) und $\phi=x_1^*$. +\end{proof} + +\begin{example} + Ist $V=K^n$ mit Standardbasis $\mathcal{E}=(e_1,...,e_n)$, so können wir $V^*$ mit dem Vektorraum der Zeilenvektoren identifizieren, und dann ist + \begin{align} + e_i^* = e_i^t\notag + \end{align} +\end{example} + +\begin{definition}[Bidualraum] + Der \begriff{Bidualraum} zu $V$ ist der $K$-Vektorraum + \begin{align} + V^{**}=(V^*)^*=\Hom_K(V^*,K)\notag + \end{align} +\end{definition} + +\begin{proposition} + Die kanonische Abbildung + \begin{align} + \iota:\begin{cases} + V\to V^{**} \\ x\to \iota_x + \end{cases}\text{ wobei } \iota_x(\phi)=\phi(x)\notag + \end{align} + ist ein Monomorphismus. Ist $\dim_K(V)<\infty$, so ist $\iota$ ein Isomorphismus. +\end{proposition} \ No newline at end of file diff --git a/2. Semester/LAAG/TeX_files/Quadriken.tex b/2. Semester/LAAG/TeX_files/Quadriken.tex index 742663b..22d490a 100644 --- a/2. Semester/LAAG/TeX_files/Quadriken.tex +++ b/2. Semester/LAAG/TeX_files/Quadriken.tex @@ -98,6 +98,15 @@ Sei $n\in\natur$. \end{proof} +\begin{conclusion} + Sei $Q\subseteq \real^n$ eine Quadrik. Es gibt eine invertierbare affine Abbildung $f\in\Aff_{\real}(\real^n)$ für die $f(Q)$ eine der folgenden 3 Formen annimmt: + \begin{itemize} + \item \itemEq{f(Q)=\left\lbrace x\in\real^n\mid \sum_{i=1}^k x_i^2-\sum_{i=k+1}^r x_i^2=0\right\rbrace \quad k\ge r-k\notag} + \item \itemEq{f(Q)=\left\lbrace x\in\real^n\mid \sum_{i=1}^k x_i^2-\sum_{i=k+1}^r x_i^2=1\right\rbrace\notag} + \item \itemEq{f(Q)=\left\lbrace x\in\real^n\mid \sum_{i=1}^k x_i^2-\sum_{i=k+1}^r x_i^2-2x_{r+1}=0\right\rbrace\quad k\ge r-k,rQ*bT}u(cW6wr$(CZF9%w8{4*RJK3>q+ctNS`OZ{L&D5O#t}ptsyX#r2*To#x z%Pmd<88AmOun9FFG@c(ZNC+;>!(V)?JMhVSufrG#_0WhU+o|7^C{o0EVZ@LT)6v=S za*5(@DNPg-&mH<#mD8<>rWB$@u0TVnJz$o}7)i}S8$(i)E74Ezox(FqCO7fl?uV>Z z*{Gv4A52_a-jxjswos0Uj#y$uL;}uGOJZ|5)}DXCVA2`4cNDBBg7Ke?7;yQ-kkW}E zrdc(W@ZmiICvYk=Jsv6Z(VV7c)Rm-415v_>q&sQBU;C?LP_Qw<&2-@y5HvI*CL#wD z8yG%57)CjBdrMajRv>F78Ba;H!YV+u$DmH;W>%*JPH0L6n2)o5)B8& zAL+ojg7JFBO45c@8K>w!4b@ARmtHKFB2w@;>#=9_%8c@tQ_#$G|KZ_LBsdk_k2bI~J 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Semester/LAAG/Vorlesung LAAG.tex b/2. Semester/LAAG/Vorlesung LAAG.tex index 7b8050d..fca5dd1 100644 --- a/2. Semester/LAAG/Vorlesung LAAG.tex +++ b/2. Semester/LAAG/Vorlesung LAAG.tex @@ -641,6 +641,7 @@ \chapter{Dualität} \input{./TeX_files/Das_Lemma_von_Zorn} +\include{./TeX_files/Der_Dualraum} \chapter{Moduln}