From dbced3cda6f2c99a1ccc42b14499895ea90298db Mon Sep 17 00:00:00 2001 From: raj_mathe Date: Wed, 10 Mar 2021 15:23:54 +0100 Subject: [PATCH] =?UTF-8?q?master=20>=20master:=20Musterl=C3=B6sung=20Kl1-?= =?UTF-8?q?A6?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .gitignore | 2 + docs/musterloesung_WiSe_2020_2021_Kl1_A6.pdf | Bin 0 -> 185937 bytes docs/musterloesung_WiSe_2020_2021_Kl1_A6.tex | 1872 ++++++++++++++++++ 3 files changed, 1874 insertions(+) create mode 100644 docs/musterloesung_WiSe_2020_2021_Kl1_A6.pdf create mode 100644 docs/musterloesung_WiSe_2020_2021_Kl1_A6.tex diff --git a/.gitignore b/.gitignore index 2944b3c..b270aae 100644 --- a/.gitignore +++ b/.gitignore @@ -17,6 +17,8 @@ !/docs/loesungen.pdf !/docs/zusatz.tex !/docs/zusatz.pdf +!/docs/musterloesung_WiSe_2020_2021_Kl1_A6.tex +!/docs/musterloesung_WiSe_2020_2021_Kl1_A6.pdf !/code !/code/**/*.md diff --git a/docs/musterloesung_WiSe_2020_2021_Kl1_A6.pdf 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z4|KXuW?8BN&`}1O3fTpTB^kx3i3(*Ixa> zGJ{XvD;S$VkN8mlN#>LQovH7Z0}Kn7)a3k>RFK)lCBU%E<*K+<8|0gP$U$K5^YAkl zWwJKifqP- zktsrNMUlplo>dG7B|Z;!WpXKm9N* z#ovl)^{*Jqgp)~v|4i1oT;e>ds@c9hyw>UbW5)6~;q*e2SDXe_ z1r;)zr|7uvP*|gMhSOh3hr?f4CzykI`;%KTz0y3dBX2KU_-(hBZ{_eb&{4MjN`HtCat5dTXUF=`8Z%jUQ6WHVU!q`WYz{Oc2&5w|AIJk` zDiF^_A=<{x)ZE$B#T=N#EnE#vO&u)^U5#8VEL>g8T}@n^EiLU72rB{Ff7c6iC28(sXbn56)Ul|Ei*@OgMPeUV9d$oFl3vg~X~)J1 z7Pg(ti*wZc_}=L4lCW#wUdSeF*SL8z=eJ$4bvymeeO7rs=T2JM?}KZsSTijcmui9C@-HD%dnTe~^fCvXz-;TKdcI zkpb`bhZZU+x%Rc&8T$?dTCn*)RJ3VN&1iFrd^h?3H5UB^qLVfyEzaL^z-G=GfuED- zG?`iOepg_ATfk$k!`yY*)HkBL4F>mT; zUGI2v`;_aZH`j} MnSymbolA5 + <6-7> MnSymbolA6 + <7-8> MnSymbolA7 + <8-9> MnSymbolA8 + <9-10> MnSymbolA9 + <10-12> MnSymbolA10 + <12-> MnSymbolA12 +}{} +\DeclareFontShape{U}{MnSymbolA}{b}{n}{ + <-6> MnSymbolA-Bold5 + <6-7> MnSymbolA-Bold6 + <7-8> MnSymbolA-Bold7 + <8-9> MnSymbolA-Bold8 + <9-10> MnSymbolA-Bold9 + <10-12> MnSymbolA-Bold10 + <12-> MnSymbolA-Bold12 +}{} +\DeclareSymbolFont{MnSyA}{U}{MnSymbolA}{m}{n} +\DeclareMathSymbol{\lcirclearrowright}{\mathrel}{MnSyA}{252} +\DeclareMathSymbol{\lcirclearrowdown}{\mathrel}{MnSyA}{255} +\DeclareMathSymbol{\rcirclearrowleft}{\mathrel}{MnSyA}{250} +\DeclareMathSymbol{\rcirclearrowdown}{\mathrel}{MnSyA}{251} + +\DeclareFontFamily{U}{MnSymbolC}{} +\DeclareSymbolFont{MnSyC}{U}{MnSymbolC}{m}{n} +\DeclareFontShape{U}{MnSymbolC}{m}{n}{ + <-6> MnSymbolC5 + <6-7> MnSymbolC6 + <7-8> MnSymbolC7 + <8-9> MnSymbolC8 + <9-10> MnSymbolC9 + <10-12> MnSymbolC10 + <12-> MnSymbolC12% +}{} +\DeclareMathSymbol{\powerset}{\mathord}{MnSyC}{180} + +%% ******************************************************************************** +%% FILE: src/setup-parameters.tex +%% ******************************************************************************** + +\def\boolwahr{true} +\def\boolfalsch{false} +\def\boolleer{} + +\let\documenttwosided\boolfalsch +\let\boolinappendix\boolfalsch +\let\boolinmdframed\boolfalsch +\let\eqtagset\boolfalsch +\let\eqtaglabel\boolleer +\let\eqtagsymb\boolleer + +\newcount\bufferctr +\newcount\bufferreplace +\newcounter{columnanzahl} + +\newlength\rtab +\newlength\gesamtlinkerRand +\newlength\gesamtrechterRand +\newlength\ownspaceabovethm +\newlength\ownspacebelowthm +\setlength{\rtab}{0.025\textwidth} +\setlength{\ownspaceabovethm}{0.5\baselineskip} +\setlength{\ownspacebelowthm}{0.5\baselineskip} +\setlength{\gesamtlinkerRand}{0pt} +\setlength{\gesamtrechterRand}{0pt} + +\def\secnumberingpt{$\cdot$} +\def\secnumberingseppt{.} +\def\subsecnumberingseppt{} +\def\thmnumberingpt{$\cdot$} +\def\thmnumberingseppt{} +\def\thmForceSepPt{.} + +\definecolor{leer}{gray}{1} +\definecolor{hellgrau}{gray}{0.85} +\definecolor{dunkelgrau}{gray}{0.5} +\definecolor{maroon}{rgb}{0.6901961,0.1882353,0.3764706} +\definecolor{dunkelgruen}{rgb}{0.015625,0.363281,0.109375} +\definecolor{dunkelrot}{rgb}{0.5450980392,0,0} +\definecolor{dunkelblau}{rgb}{0,0,0.5450980392} +\definecolor{blau}{rgb}{0,0,1} +\definecolor{newresult}{rgb}{0.6,0.6,0.6} +\definecolor{improvedresult}{rgb}{0.9,0.9,0.9} +\definecolor{hervorheben}{rgb}{0,0.9,0.7} +\definecolor{starkesblau}{rgb}{0.1019607843,0.3176470588,0.8156862745} +\definecolor{achtung}{rgb}{1,0.5,0.5} +\definecolor{frage}{rgb}{0.5,1,0.5} +\definecolor{schreibweise}{rgb}{0,0.7,0.9} +\definecolor{axiom}{rgb}{0,0.3,0.3} + +%% ******************************************************************************** +%% FILE: src/setup-macros.tex +%% ******************************************************************************** + +%% **************************************************************** +%% TEX: +%% **************************************************************** + +\def\let@name#1#2{\expandafter\let\csname #1\expandafter\endcsname\csname #2\endcsname\relax} +\DeclareRobustCommand\crfamily{\fontfamily{ccr}\selectfont} +\DeclareTextFontCommand{\textcr}{\crfamily} + +\def\nichtzeigen#1{\phantom{#1}} + +%% **************************************************************** +%% SPACING: +%% **************************************************************** + +\def\ifthenelseleer#1#2#3{\ifthenelse{\equal{#1}{}}{#2}{#1#3}} +\def\bedingtesspaceexpand#1#2#3{\ifthenelseleer{\csname #1\endcsname}{#3}{#2#3}} +\def\voritemise{\leavevmode\nvraum{1}} +\def\hraum{\null\hfill\null} +\def\vraum{\null\vfill\null} +\def\nvraum{\@ifnextchar\bgroup{\nvraum@c}{\nvraum@bes}} + \def\nvraum@c#1{\vspace*{-#1\baselineskip}} + \def\nvraum@bes{\vspace*{-\baselineskip}} +\def\erlaubeplatz{\relax\ifmmode\else\@\xspace\fi} +\def\entferneplatz{\relax\ifmmode\else\expandafter\@gobble\fi} +\def\forceindent{\hspace*{20pt}} %% * nötig, damit am Anfang/Ende einer Zeile nicht ignoriert wird + +%% **************************************************************** +%% TAGS / BEZEICHNUNGEN / LABELLING: +%% **************************************************************** + +\def\send@toaux#1{\@bsphack\protected@write\@auxout{}{\string#1}\@esphack} + +%% \rlabel{LABEL}[CTR]{CREF-SHORT}{CREF-LONG}{DISPLAYTEXT} +\def\rlabel#1[#2]#3#4#5{#5\rlabel@aux{#1}[#2]{#3}{#4}{#5}} + \def\rlabel@aux#1[#2]#3#4#5{% + \send@toaux{\newlabel{#1}{{\@currentlabel}{\thepage}{{\unexpanded{#5}}}{#2.\csname the#2\endcsname}{}}}\relax% + } + +%% \tag@rawscheme{CREF-SHORT}{CREF-LONG}[CTR]{LEFT-BRKT}{RIGHT-BRKT} [LABEL]{DISPLAYTEXT} +\def\tag@rawscheme#1#2[#3]#4#5{\@ifnextchar[{\tag@rawscheme@{#1}{#2}[#3]{#4}{#5}}{\tag@rawscheme@{#1}{#2}[#3]{#4}{#5}[*]}} + \def\tag@rawscheme@#1#2[#3]#4#5[#6]{\@ifnextchar\bgroup{\tag@rawscheme@@{#1}{#2}[#3]{#4}{#5}[#6]}{\tag@rawscheme@@{#1}{#2}[#3]{#4}{#5}[#6]{}}} + \def\tag@rawscheme@@#1#2[#3]#4#5[#6]#7{% + \ifthenelse{\equal{#6}{*}}{% + \ifthenelse{\equal{#7}{\boolleer}}{\refstepcounter{#3}#4\csname the#3\endcsname#5}{#4#7#5}% + }{% + \refstepcounter{#3}#4% + \ifthenelse{\equal{#7}{\boolleer}}{\rlabel{#6}[#3]{#1}{#2}{\csname the#3\endcsname}}{\rlabel{#6}[#3]{#1}{#2}{#7}}% + #5% + }% + } +%% \tag@scheme{CREF-SHORT}{CREF-LONG}[CTR] [LABEL]{DISPLAYTEXT} +\def\tag@scheme#1#2[#3]{\tag@rawscheme{#1}{#2}[#3]{\upshape(}{\upshape)}} + +%% \eqtag[LABEL]{DISPLAYTEXT} +\def\eqtag@post#1{\makebox[0pt][r]{#1}} +\def\eqtag@pre{\tag@scheme{Eq}{Equation}[Xe]} +\def\eqtag{\@ifnextchar[{\eqtag@}{\eqtag@[*]}} + \def\eqtag@[#1]{\@ifnextchar\bgroup{\eqtag@@[#1]}{\eqtag@@[#1]{}}} + \def\eqtag@@[#1]#2{\eqtag@post{\eqtag@pre[#1]{#2}}} + +\def\eqcref#1{\text{(\ref{#1})}} +\def\ptcref#1{\ref{#1}} +\def\punktlabel#1{\label{it:#1:\beweislabel}} +\def\punktcref#1{\eqcref{it:#1:\beweislabel}} +\def\crefit#1#2{\cref{#1}~\eqcref{it:#2:#1}} +\def\Crefit#1#2{\Cref{#1}~\eqcref{it:#2:#1}} + +%% UNDER/OVERSET BEFEHLE +\def\opfromto[#1]_#2^#3{\underset{#2}{\overset{#3}{#1}}} +\def\textoverset#1#2{\overset{\text{#1}}{#2}} +\def\textunderset#1#2{\underset{#2}{\text{#1}}} +\def\crefoverset#1#2{\textoverset{\cref{#1}}{#2}} +\def\Crefoverset#1#2{\textoverset{\Cref{#1}}{#2}} +\def\crefunderset#1#2{\textunderset{#2}{\cref{#1}}} +\def\Crefunderset#1#2{\textunderset{#2}{\Cref{#1}}} +\def\eqcrefoverset#1#2{\textoverset{\eqcref{#1}}{#2}} +\def\eqcrefunderset#1#2{\textunderset{#2}{\eqcref{#1}}} +\def\mathclap#1{#1} +\def\oberunterset#1{\@ifnextchar^{\oberunterset@oben{#1}}{\oberunterset@unten{#1}}} + \def\oberunterset@oben#1^#2_#3{\underset{\mathclap{#3}}{\overset{\mathclap{#2}}{#1}}} + \def\oberunterset@unten#1_#2^#3{\underset{\mathclap{#2}}{\overset{\mathclap{#3}}{#1}}} + \def\breitunderbrace#1_#2{\underbrace{#1}_{\mathclap{#2}}} + \def\breitoverbrace#1^#2{\overbrace{#1}^{\mathclap{#2}}} + \def\breitunderbracket#1_#2{\underbracket{#1}_{\mathclap{#2}}} + \def\breitoverbracket#1^#2{\overbracket{#1}^{\mathclap{#2}}} + +\def\generatenestedsecnumbering#1#2#3{% + \expandafter\gdef\csname thelong#3\endcsname{% + \expandafter\csname the#2\endcsname% + \secnumberingpt% + \expandafter\csname #1\endcsname{#3}% + }% + \expandafter\gdef\csname theshort#3\endcsname{% + \expandafter\csname #1\endcsname{#3}% + }% +} +\def\generatenestedthmnumbering#1#2#3{% + \expandafter\gdef\csname the#3\endcsname{% + \expandafter\csname the#2\endcsname% + \thmnumberingpt% + \expandafter\csname #1\endcsname{#3}% + }% + \expandafter\gdef\csname theshort#3\endcsname{% + \expandafter\csname #1\endcsname{#3}% + }% +} + +%% **************************************************************** +%% ALLG. MACROS: +%% **************************************************************** + +\def\+#1{\addtocounter{#1}{1}} +\def\setcounternach#1#2{\setcounter{#1}{#2}\addtocounter{#1}{-1}} +\def\textsubscript#1{${}_{\textup{#1}}$} +\def\rome#1{\overline{\underline{#1}}} +\def\textTODO{\text{[{\large\textcolor{red}{More work needed!}}]}} +\def\hlineEIGENpt{\hdashline[0.5pt/5pt]} +\def\clineEIGENpt#1{\cdashline{#1}[0.5pt/5pt]} + +\def\forcepunkt#1{#1\IfEndWith{#1}{.}{}{.}} +\def\lateinabkuerzung#1#2{% + \expandafter\gdef\csname #1\endcsname{\emph{#2}\@ifnextchar.{\entferneplatz}{\erlaubeplatz}} +} +\def\deutscheabkuerzung#1#2{% + \expandafter\gdef\csname #1\endcsname{{#2}\@ifnextchar.{\entferneplatz}{\erlaubeplatz}} +} + +%% **************************************************************** +%% MATHE +%% **************************************************************** + +\def\matrix#1{\left(\begin{array}{#1}} + \def\endmatrix{\end{array}\right)} +\def\smatrix{\left(\begin{smallmatrix}} + \def\endsmatrix{\end{smallmatrix}\right)} + +\def\multiargrekursiverbefehl#1#2#3#4#5#6#7#8{% + \expandafter\gdef\csname#1\endcsname #2##1#4{\csname #1@anfang\endcsname##1#3\egroup} + \expandafter\def\csname #1@anfang\endcsname##1#3{#5##1\@ifnextchar\egroup{\csname #1@ende\endcsname}{#7\csname #1@mitte\endcsname}} + \expandafter\def\csname #1@mitte\endcsname##1#3{#6##1\@ifnextchar\egroup{\csname #1@ende\endcsname}{#7\csname #1@mitte\endcsname}} + \expandafter\def\csname #1@ende\endcsname##1{#8} +} +\multiargrekursiverbefehl{svektor}{[}{;}{]}{\begin{smatrix}}{}{\\}{\\\end{smatrix}} +\multiargrekursiverbefehl{vektor}{[}{;}{]}{\begin{matrix}{c}}{}{\\}{\\\end{matrix}} +\multiargrekursiverbefehl{vektorzeile}{}{,}{;}{}{&}{}{} +\multiargrekursiverbefehl{matlabmatrix}{[}{;}{]}{\begin{smatrix}\vektorzeile}{\vektorzeile}{;\\}{;\end{smatrix}} + +\def\cases[#1]#2{\left\{\begin{array}[#1]{#2}} + \def\endcases{\end{array}\right.} + +\def\BeweisRichtung[#1]{\@ifnextchar\bgroup{\@BeweisRichtung@c[#1]}{\@BeweisRichtung@bes[#1]}} + \def\@BeweisRichtung@bes[#1]{{\bfseries(#1).~}} + \def\@BeweisRichtung@c[#1]#2#3{{\bfseries(#2#1#3).~}} +\def\erzeugeBeweisRichtungBefehle#1#2{ + \expandafter\gdef\csname #1text\endcsname##1##2{\BeweisRichtung[#2]{##1}{##2}} + \expandafter\gdef\csname #1\endcsname{% + \@ifnextchar\bgroup{\csname #1@\endcsname}{\csname #1text\endcsname{}{}}% + } + \expandafter\gdef\csname #1@\endcsname##1##2{% + \csname #1text\endcsname{\punktcref{##1}}{\punktcref{##2}}% + } +} +\erzeugeBeweisRichtungBefehle{hinRichtung}{$\Longrightarrow$} +\erzeugeBeweisRichtungBefehle{herRichtung}{$\Longleftarrow$} +\erzeugeBeweisRichtungBefehle{hinherRichtung}{$\Longleftrightarrow$} + +\def\cal#1{\mathcal{#1}} +\def\brkt#1{\langle{}#1{}\rangle} +\def\mathfrak#1{\mbox{\usefont{U}{euf}{m}{n}#1}} +\def\kurs#1{\textit{#1}} +\def\rectangleblack{\text{\RectangleBold}} +\def\rectanglewhite{\text{\Rectangle}} +\def\squareblack{\blacksquare} +\def\squarewhite{\Box} + +%% ******************************************************************************** +%% FILE: src/setup-environments.tex +%% ******************************************************************************** + +%% ********************************************************************** +%% CLEVEREF: ************************************************************ + + \def\crefname@full#1#2#3{\crefname{#1}{#2}{#3}\Crefname{#1}{#2}{#3}} + \crefname@full{chapter}{Kapitel}{Kapitel} + \crefname@full{section}{Abschnitt}{Abschnitte} + \crefname@full{figure}{Fig.}{Fig.} + \crefname@full{subfigure}{Fig.}{Fig.} + + \crefname@full{proof}{Beweis}{Beweise} + \crefname@full{thm}{Theorem}{Theoreme} + \crefname@full{satz}{Satz}{Sätze} + \crefname@full{claim}{Behauptung}{Behauptungen} + \crefname@full{lemm}{Lemma}{Lemmata} + \crefname@full{cor}{Korollar}{Korollarien} + \crefname@full{folg}{Folgerung}{Folgerungen} + \crefname@full{prop}{Proposition}{Propositionen} + \crefname@full{defn}{Definition}{Definitionen} + \crefname@full{conv}{Konvention}{Konventionen} + \crefname@full{fact}{Fakt}{Fakten} + \crefname@full{rem}{Bemerkung}{Bemerkungen} + \crefname@full{qstn}{Frage}{Fragen} + \crefname@full{e.g.}{Beipsiel}{Beipsiele} + +%% **************************************************************** +%% THEOREME: +%% **************************************************************** + + \def\qedEIGEN#1{\@ifnextchar[{\qedEIGEN@c{#1}}{\qedEIGEN@bes{#1}}}%] + \def\qedEIGEN@bes#1{% + \bgroup% + \parfillskip=0pt% % so \par doesnt push \square to left + \widowpenalty=10000% % so we dont break the page before \square + \displaywidowpenalty=10000% % ditto + \finalhyphendemerits=0% % TeXbook exercise 14.32 + \leavevmode% % \nobreak means lines not pages + \unskip% % remove previous space or glue + \nobreak% % don’t break lines + \hfil% % ragged right if we spill over + \penalty50% % discouragement to do so + \hskip.2em% % ensure some space + \null% % anchor following \hfill + \hfill% % push \square to right + #1% % the end-of-proof mark + \par% + \egroup% + } + \def\qedEIGEN@c#1[#2]{% + \bgroup% + \parfillskip=0pt% % so \par doesnt push \square to left + \widowpenalty=10000% % so we dont break the page before \square + \displaywidowpenalty=10000% % ditto + \finalhyphendemerits=0% % TeXbook exercise 14.32 + \leavevmode% % \nobreak means lines not pages + \unskip% % remove previous space or glue + \nobreak% % don’t break lines + \hfil% % ragged right if we spill over + \penalty50% % discouragement to do so + \hskip.2em% % ensure some space + \null% % anchor following \hfill + \hfill% % push \square to right + {#1~{\smaller\bfseries\upshape (#2)}}% + \par% + \egroup% + } + \def\qedVARIANT#1#2{ + \expandafter\def\csname ennde#1Sign\endcsname{#2} + \expandafter\def\csname ennde#1\endcsname{\@ifnextchar[{\qedEIGEN@c{#2}}{\qedEIGEN@bes{#2}}} %] + } + \qedVARIANT{OfProof}{$\squareblack$} + \qedVARIANT{OfWork}{\rectangleblack} + \qedVARIANT{OfSomething}{$\dashv$} + \qedVARIANT{OnNeutral}{$\lozenge$} % \lozenge \bigcirc \blacklozenge + \def\qedsymbol{\enndeOfProofSign} + \def\proofSymbol{\enndeOfProofSign} + + \def\ra@pretheoremwork{ + \setlength{\theorempreskipamount}{\ownspaceabovethm} + } + \def\rathmtransfer#1#2{ + \expandafter\def\csname #2\endcsname{\csname #1\endcsname} + \expandafter\def\csname end#2\endcsname{\csname end#1\endcsname} + } + + \def\ranewthm#1#2#3[#4]{ + %% FOR \BEGIN{THM} + \theoremstyle{\current@theoremstyle} + \theoremseparator{\current@theoremseparator} + \theoremprework{\ra@pretheoremwork} + \@ifundefined{#1@basic}{\newtheorem{#1@basic}[#4]{#2}}{\renewtheorem{#1@basic}[#4]{#2}} + %% FOR \BEGIN{THM}[...] + \theoremstyle{\current@theoremstyle} + \theoremseparator{\thmForceSepPt} + \theoremprework{\ra@pretheoremwork} + \@ifundefined{#1@withName}{\newtheorem{#1@withName}[#4]{#2}}{\renewtheorem{#1@withName}[#4]{#2}} + %% FOR \BEGIN{THM*} + \theoremstyle{nonumberplain} + \theoremseparator{\thmForceSepPt} + \theoremprework{\ra@pretheoremwork} + \@ifundefined{#1@star@basic}{\newtheorem{#1@star@basic}[Xdisplaynone]{#2}}{\renewtheorem{#1@star@basic}[Xdisplaynone]{#2}} + %% FOR \BEGIN{THM*}[...] + \theoremstyle{nonumberplain} + \theoremseparator{\thmForceSepPt} + \theoremprework{\ra@pretheoremwork} + \@ifundefined{#1@star@withName}{\newtheorem{#1@star@withName}[Xdisplaynone]{#2}}{\renewtheorem{#1@star@withName}[Xdisplaynone]{#2}} + %% GENERATE ENVIRONMENTS: + \umbauenenv{#1}{#3}[#4] + \umbauenenv{#1@star}{#3}[Xdisplaynone] + %% TRANSFER *-DEFINITION + \rathmtransfer{#1@star}{#1*} + } + + \def\umbauenenv#1#2[#3]{% + %% \BEGIN{THM}... + \expandafter\def\csname #1\endcsname{\relax% + \@ifnextchar[{\csname #1@\endcsname}{\csname #1@\endcsname[*]}% + } + %% \BEGIN{THM}[ANFANG]... + \expandafter\def\csname #1@\endcsname[##1]{\relax% + \@ifnextchar[{\csname #1@@\endcsname[##1]}{\csname #1@@\endcsname[##1][*]}% + } + %% \BEGIN{THM}[ANFANG][SCHLUSS] + \expandafter\def\csname #1@@\endcsname[##1][##2]{% + \ifx*##1% + \def\enndeOfBlock{\csname end#1@basic\endcsname} + \csname #1@basic\endcsname% + \else% + \def\enndeOfBlock{\csname end#1@withName\endcsname} + \csname #1@withName\endcsname[##1]% + \fi% + \def\makelabel####1{% + \gdef\beweislabel{####1}% + \label{\beweislabel}% + }% + \ifx*##2% + \def\enndeSymbol{\qedEIGEN{#2}} + \else% + \def\enndeSymbol{\qedEIGEN{#2}[##2]} + \fi + } + %% \END{THM} + \expandafter\gdef\csname end#1\endcsname{\enndeSymbol\enndeOfBlock} + } + + %% NEWTHEOREM EINSTELLUNGSOPTIONEN: + %% F\"UR \theoremstyle + %% plain Emulates original LATEX defin, except uses param \theorem...skipamount. + %% break Header followed by line break. + %% change Header, Number and Text are interchanged, without a line break. + %% changebreak =change, but with a line break after Header. + %% margin Number in left margin, without a line break. + %% marginbreak =margin, but with a line break after the header. + %% nonumberplain =plain, without number. + %% nonumberbreak =break, without number. + %% empty No number, no name. Only the optional argument is typeset. + %% \theoremclass \theoremnumbering + %% \theorempreskip \theorempostkip \theoremindent + %% \theoremprework \theorempostwork + + \def\current@theoremstyle{plain} + \def\current@theoremseparator{\thmnumberingseppt} + \theoremstyle{\current@theoremstyle} + \theoremseparator{\current@theoremseparator} + \theoremsymbol{} + + \newtheorem{X}{X}[chapter] % for most theorems + \newtheorem{Xe}{Xe}[chapter] % for equations + \newtheorem*{Xdisplaynone}{Xdisplaynone}[chapter] % a dummy counter, that will never be displayed. + \newtheorem{Xsp}{Xsp}[chapter] % for special theorems + \generatenestedthmnumbering{arabic}{chapter}{X} + \generatenestedthmnumbering{arabic}{chapter}{Xe} + \generatenestedthmnumbering{Roman}{chapter}{Xsp} + \let\theXsp\theshortXsp + + \theoremheaderfont{\upshape\bfseries} + \theorembodyfont{\slshape} + + \ranewthm{thm}{Theorem}{\enndeOnNeutralSign}[X] + \ranewthm{satz}{Satz}{\enndeOnNeutralSign}[X] + \ranewthm{claim}{Behauptung}{\enndeOnNeutralSign}[X] + \ranewthm{lemm}{Lemma}{\enndeOnNeutralSign}[X] + \ranewthm{cor}{Korollar}{\enndeOnNeutralSign}[X] + \ranewthm{folg}{Folgerung}{\enndeOnNeutralSign}[X] + \ranewthm{prop}{Proposition}{\enndeOnNeutralSign}[X] + + \theorembodyfont{\upshape} + + \ranewthm{defn}{Definition}{\enndeOnNeutralSign}[X] + \ranewthm{conv}{Konvention}{\enndeOnNeutralSign}[X] + \ranewthm{obs}{Beobachtung}{\enndeOnNeutralSign}[X] + \ranewthm{e.g.}{Beipsiel}{\enndeOnNeutralSign}[X] + \ranewthm{fact}{Fakt}{\enndeOnNeutralSign}[X] + \ranewthm{rem}{Bemerkung}{\enndeOnNeutralSign}[X] + \ranewthm{qstn}{Frage}{\enndeOnNeutralSign}[X] + \ranewthm{exer}{Aufgabe}{\enndeOnNeutralSign}[X] + \ranewthm{soln}{Lösung}{\enndeOnNeutralSign}[X] + + \theoremheaderfont{\itshape\bfseries} + \theorembodyfont{\upshape} + + \ranewthm{proof@tmp}{Beweis}{\enndeOfProofSign}[Xdisplaynone] + \rathmtransfer{proof@tmp*}{proof} + + \def\behauptungbeleg@claim{% + \iflanguage{british}{Claim}{% + \iflanguage{english}{Claim}{% + \iflanguage{ngerman}{Behauptung}{% + \iflanguage{russian}{Утверждение}{% + Claim% + }}}}% + } + \def\behauptungbeleg@pf@kurz{% + \iflanguage{british}{Pf}{% + \iflanguage{english}{Pf}{% + \iflanguage{ngerman}{Bew}{% + \iflanguage{russian}{Доказательство}{% + Pf% + }}}}% + } + \def\behauptungbeleg{\@ifnextchar\bgroup{\behauptungbeleg@c}{\behauptungbeleg@bes}} + \def\behauptungbeleg@c#1{\item[{\bfseries \behauptungbeleg@claim\erlaubeplatz #1.}]} + \def\behauptungbeleg@bes{\item[{\bfseries \behauptungbeleg@claim.}]} + \def\belegbehauptung{\item[{\bfseries\itshape\behauptungbeleg@pf@kurz.}]} + +%% **************************************************************** +%% ALTE UMGEBUNGEN: +%% **************************************************************** + + \newcolumntype{\RECHTS}[1]{>{\raggedleft}p{#1}} + \newcolumntype{\LINKS}[1]{>{\raggedright}p{#1}} + \newcolumntype{m}{>{$}l<{$}} + \newcolumntype{C}{>{$}c<{$}} + \newcolumntype{L}{>{$}l<{$}} + \newcolumntype{R}{>{$}r<{$}} + \newcolumntype{0}{@{\hspace{0pt}}} + \newcolumntype{\LINKSRAND}{@{\hspace{\@totalleftmargin}}} + \newcolumntype{h}{@{\extracolsep{\fill}}} + \newcolumntype{i}{>{\itshape}} + \newcolumntype{t}{@{\hspace{\tabcolsep}}} + \newcolumntype{q}{@{\hspace{1em}}} + \newcolumntype{n}{@{\hspace{-\tabcolsep}}} + \newcolumntype{M}[2]{% + >{\begin{minipage}{#2}\begin{math}}% + {#1}% + <{\end{math}\end{minipage}}% + } + \newcolumntype{T}[2]{% + >{\begin{minipage}{#2}}% + {#1}% + <{\end{minipage}}% + } + \setlength{\LTpre}{\baselineskip} + \setlength{\LTpost}{0pt} + \def\center{\centering} + \def\endcenter{} + + \def\punkteumgebung@genbefehl#1#2#3{ + \punkteumgebung@genbefehl@{#1}{#2}{#3}{}{} + \punkteumgebung@genbefehl@{multi#1}{#2}{#3}{ + \setlength{\columnsep}{10pt}% + \setlength{\columnseprule}{0pt}% + \begin{multicols}{\thecolumnanzahl}% + }{\end{multicols}\nvraum{1}} + } + \def\punkteumgebung@genbefehl@#1#2#3#4#5{ + \expandafter\gdef\csname #1\endcsname{ + \@ifnextchar\bgroup{\csname #1@c\endcsname}{\csname #1@bes\endcsname} + }%] + \expandafter\def\csname #1@c\endcsname##1{ + \@ifnextchar[{\csname #1@c@\endcsname{##1}}{\csname #1@c@\endcsname{##1}[\z@]} + }%] + \expandafter\def\csname #1@c@\endcsname##1[##2]{ + \@ifnextchar[{\csname #1@c@@\endcsname{##1}[##2]}{\csname #1@c@@\endcsname{##1}[##2][\z@]} + }%] + \expandafter\def\csname #1@c@@\endcsname##1[##2][##3]{ + \let\alterlinkerRand\gesamtlinkerRand + \let\alterrechterRand\gesamtrechterRand + \addtolength{\gesamtlinkerRand}{##2} + \addtolength{\gesamtrechterRand}{##3} + \advance\linewidth -##2% + \advance\linewidth -##3% + \advance\@totalleftmargin ##2% + \parshape\@ne \@totalleftmargin\linewidth% + #4 + \begin{#2}[\upshape ##1]% + \setlength{\parskip}{0.5\baselineskip}\relax% + \setlength{\topsep}{\z@}\relax% + \setlength{\partopsep}{\z@}\relax% + \setlength{\parsep}{\parskip}\relax% + \setlength{\itemsep}{#3}\relax% + \setlength{\listparindent}{\z@}\relax% + \setlength{\itemindent}{\z@}\relax% + } + \expandafter\def\csname #1@bes\endcsname{ + \@ifnextchar[{\csname #1@bes@\endcsname}{\csname #1@bes@\endcsname[\z@]} + }%] + \expandafter\def\csname #1@bes@\endcsname[##1]{ + \@ifnextchar[{\csname #1@bes@@\endcsname[##1]}{\csname #1@bes@@\endcsname[##1][\z@]} + }%] + \expandafter\def\csname #1@bes@@\endcsname[##1][##2]{ + \let\alterlinkerRand\gesamtlinkerRand + \let\alterrechterRand\gesamtrechterRand + \addtolength{\gesamtlinkerRand}{##1} + \addtolength{\gesamtrechterRand}{##2} + \advance\linewidth -##1% + \advance\linewidth -##2% + \advance\@totalleftmargin ##1% + \parshape\@ne \@totalleftmargin\linewidth% + #4 + \begin{#2}% + \setlength{\parskip}{0.5\baselineskip}\relax% + \setlength{\topsep}{\z@}\relax% + \setlength{\partopsep}{\z@}\relax% + \setlength{\parsep}{\parskip}\relax% + \setlength{\itemsep}{#3}\relax% + \setlength{\listparindent}{\z@}\relax% + \setlength{\itemindent}{\z@}\relax% + } + \expandafter\gdef\csname end#1\endcsname{% + \end{#2}#5 + \setlength{\gesamtlinkerRand}{\alterlinkerRand} + \setlength{\gesamtlinkerRand}{\alterrechterRand} + } + } + + \def\ritempunkt{{\Large\textbullet}} % \textbullet, $\sqbullet$, $\blacktriangleright$ + \setdefaultitem{\ritempunkt}{\ritempunkt}{\ritempunkt}{\ritempunkt} + \punkteumgebung@genbefehl{itemise}{compactitem}{\parskip}{}{} + \punkteumgebung@genbefehl{kompaktitem}{compactitem}{\z@}{}{} + \punkteumgebung@genbefehl{enumerate}{compactenum}{\parskip}{}{} + \punkteumgebung@genbefehl{kompaktenum}{compactenum}{\z@}{}{} + + \let\ALTthebibliography\thebibliography + \renewenvironment{thebibliography}[1]{% + \begin{ALTthebibliography}{#1} + \addcontentsline{toc}{part}{\bibname} + }{% + \end{ALTthebibliography} + } + +%% **************************************************************** +%% NEUE UMGEBUNGEN: +%% **************************************************************** + + \def\matrix#1{\left(\begin{array}[mc]{#1}} + \def\endmatrix{\end{array}\right)} + \def\smatrix{\left(\begin{smallmatrix}} + \def\endsmatrix{\end{smallmatrix}\right)} + \def\vector{\begin{matrix}{c}} + \def\endvector{\end{matrix}} + \def\svector{\begin{smatrix}} + \def\endsvector{\end{smatrix}} + + \def\multiargrekursiverbefehl#1#2#3#4#5#6#7#8{% + \expandafter\gdef\csname#1\endcsname #2##1#4{\csname #1@anfang\endcsname##1#3\egroup} + \expandafter\def\csname #1@anfang\endcsname##1#3{#5##1\@ifnextchar\egroup{\csname #1@ende\endcsname}{#7\csname #1@mitte\endcsname}} + \expandafter\def\csname #1@mitte\endcsname##1#3{#6##1\@ifnextchar\egroup{\csname #1@ende\endcsname}{#7\csname #1@mitte\endcsname}} + \expandafter\def\csname #1@ende\endcsname##1{#8} + } + \multiargrekursiverbefehl{svektor}{[}{;}{]}{\begin{smatrix}}{}{\\}{\\\end{smatrix}} + \multiargrekursiverbefehl{vektor}{[}{;}{]}{\begin{matrix}{c}}{}{\\}{\\\end{matrix}} + \multiargrekursiverbefehl{vektorzeile}{}{,}{;}{}{&}{}{} + \multiargrekursiverbefehl{matlabmatrix}{[}{;}{]}{\begin{smatrix}\vektorzeile}{\vektorzeile}{;\\}{;\end{smatrix}} + + \def\underbracenodisplay#1{% + \mathop{\vtop{\m@th\ialign{##\crcr + $\hfil\displaystyle{#1}\hfil$\crcr + \noalign{\kern3\p@\nointerlineskip}% + \upbracefill\crcr\noalign{\kern3\p@}}}}\limits% + } + + \def\mathe[#1]#2{% + \ifthenelse{\equal{\boolinmdframed}{\boolwahr}}{}{\begin{escapeeinzug}} + \noindent% + \let\eqtagset\boolfalsch + \let\eqtaglabel\boolleer + \let\eqtagsymb\boolleer + \let\alteqtag\eqtag + \def\eqtag{\@ifnextchar[{\eqtag@loc@}{\eqtag@loc@[*]}}% + \def\eqtag@loc@[##1]{\@ifnextchar\bgroup{\eqtag@loc@@[##1]}{\eqtag@loc@@[##1]{}}}% + \def\eqtag@loc@@[##1]##2{% + \gdef\eqtagset{\boolwahr} + \gdef\eqtaglabel{##1} + \gdef\eqtagsymb{##2} + }% + \def\verticalalign{}% + \IfBeginWith{#1}{t}{\def\verticalalign{t}}{}% + \IfBeginWith{#1}{m}{\def\verticalalign{c}}{}% + \IfBeginWith{#1}{b}{\def\verticalalign{b}}{}% + \def\horizontalalign{\null\hfill\null}% + \IfEndWith{#1}{l}{}{\null\hfill\null}% + \IfEndWith{#1}{r}{\def\horizontalalign{}}{}% + \begin{math} + \begin{array}[\verticalalign]{0#2}% + } + \def\endmathe{% + \end{array} + \end{math}\horizontalalign% + \let\eqtag\alteqtag + \ifthenelse{\equal{\eqtagset}{\boolwahr}}{\eqtag[\eqtaglabel]{\eqtagsymb}}{} + \ifthenelse{\equal{\boolinmdframed}{\boolwahr}}{}{\end{escapeeinzug}}% + } + + \def\longmathe[#1]#2{\relax + \let\altarraystretch\arraystretch + \renewcommand\arraystretch{1.2}\relax + \begin{longtable}[#1]{\LINKSRAND #2} + } + \def\endlongmathe{ + \end{longtable} + \renewcommand\arraystretch{\altarraystretch} + } + + \def\einzug{\@ifnextchar[{\indents@}{\indents@[\z@]}}%] + \def\indents@[#1]{\@ifnextchar[{\indents@@[#1]}{\indents@@[#1][\z@]}}%] + \def\indents@@[#1][#2]{% + \begin{list}{}{\relax + \setlength{\topsep}{\z@}\relax + \setlength{\partopsep}{\z@}\relax + \setlength{\parsep}{\parskip}\relax + \setlength{\listparindent}{\z@}\relax + \setlength{\itemindent}{\z@}\relax + \setlength{\leftmargin}{#1}\relax + \setlength{\rightmargin}{#2}\relax + \let\alterlinkerRand\gesamtlinkerRand + \let\alterrechterRand\gesamtrechterRand + \addtolength{\gesamtlinkerRand}{#1} + \addtolength{\gesamtrechterRand}{#2} + }\relax + \item[]\relax + } + \def\endeinzug{% + \setlength{\gesamtlinkerRand}{\alterlinkerRand} + \setlength{\gesamtlinkerRand}{\alterrechterRand} + \end{list}% + } + + \def\escapeeinzug{\begin{einzug}[-\gesamtlinkerRand][-\gesamtrechterRand]} + \def\endescapeeinzug{\end{einzug}} + + \def\programmiercode{ + \modulolinenumbers[1] + \begin{einzug}[\rtab][\rtab]% + \begin{linenumbers}% + \fontfamily{cmtt}\fontseries{m}\fontshape{u}\selectfont% + \setlength{\parskip}{1\baselineskip}% + \setlength{\parindent}{0pt}% + } + \def\endprogrammiercode{ + \end{linenumbers} + \end{einzug} + } + + \def\schattiertebox@genbefehl#1#2#3{ + \expandafter\gdef\csname #1\endcsname{% + \@ifnextchar[{\csname #1@args\endcsname}{\csname #1@args\endcsname[#3]}%]% + } + \expandafter\def\csname #1@args\endcsname[##1]{% + \@ifnextchar[{\csname #1@args@l\endcsname[##1]}{\csname #1@args@n\endcsname[##1]}%]% + } + \expandafter\def\csname #1@args@l\endcsname[##1][##2]{% + \@ifnextchar[{\csname #1@args@l@r\endcsname[##1][##2]}{\csname #1@args@l@n\endcsname[##1][##2]}%]% + } + \expandafter\def\csname #1@args@n\endcsname[##1]{% + \let\boolinmdframed\boolwahr + \begin{mdframed}[#2leftmargin=0,rightmargin=0,outermargin=0,innermargin=0,##1] + } + \expandafter\def\csname #1@args@l@n\endcsname[##1][##2]{% + \let\boolinmdframed\boolwahr + \begin{mdframed}[#2leftmargin=##2/2,rightmargin=##2/2,outermargin=##2/2,innermargin=##2/2,##1] + } + \expandafter\def\csname #1@args@l@r\endcsname[##1][##2][##3]{% + \let\boolinmdframed\boolwahr + \begin{mdframed}[#2leftmargin=##2,rightmargin=##3,outermargin=##2,innermargin=##3,##1] + } + \expandafter\gdef\csname end#1\endcsname{% + \end{mdframed} + \let\boolinmdframed\boolfalsch + } + } + \schattiertebox@genbefehl{schattiertebox}{ + splittopskip=0,% + splitbottomskip=0,% + frametitleaboveskip=0,% + frametitlebelowskip=0,% + skipabove=1\baselineskip,% + skipbelow=1\baselineskip,% + linewidth=2pt,% + linecolor=black,% + roundcorner=4pt,% + }{ + backgroundcolor=leer,% + nobreak=true,% + } + + \schattiertebox@genbefehl{schattierteboxdunn}{ + splittopskip=0,% + splitbottomskip=0,% + frametitleaboveskip=0,% + frametitlebelowskip=0,% + skipabove=1\baselineskip,% + skipbelow=1\baselineskip,% + linewidth=1pt,% + linecolor=black,% + roundcorner=2pt,% + }{ + backgroundcolor=leer,% + nobreak=true,% + } + + \def\algorithm{\schattiertebox[backgroundcolor=hellgrau,nobreak=false]} + \def\endalgorithm{\endschattiertebox} + + \def\tikzsetzenode#1{% + \tikz[remember picture,baseline,overlay]{\node #1;}% + } + \def\tikzsetzepfeil#1{% + \begin{tikzpicture}[remember picture,overlay,>=latex]% + \draw #1;% + \end{tikzpicture}% + } + \def\tikzsetzeoverlay#1{% + \begin{tikzpicture}[remember picture,overlay,>=latex]% + #1% + \end{tikzpicture}% + } + \def\tikzsetzekreise[#1]#2#3{% + \tikzsetzepfeil{% + [rounded corners,#1]% + ([shift={(-\tabcolsep,0.75\baselineskip)}]#2)% + rectangle% + ([shift={(\tabcolsep,-0.5\baselineskip)}]#3) + }% + } + + \tikzset{ + >=stealth, + auto, + thick, + main node/.style={ + circle,draw,font=\sffamily\Large\bfseries,minimum size=0pt + }, + } + +%% ******************************************************************************** +%% FILE: src/setup-layout.tex +%% ******************************************************************************** + +\pagestyle{fancyplain} + +\@ifundefined{setcitestyle}{% + %% do nothing +}{% + \setcitestyle{numeric-comp,open={[},close={]}} +} +\def\crefpairconjunction{ und } +\def\crefmiddleconjunction{, } +\def\creflastconjunction{, und } + +\raggedbottom %% <- pushes footers up +\sloppy +\def\headrulewidth{0pt} +\def\footrulewidth{0pt} +\setlength{\columnsep}{20pt} +\setlength{\columnseprule}{1pt} +\setlength{\headheight}{11pt} +\setlength{\partopsep}{0pt} +\setlength{\topsep}{\baselineskip} +\setlength{\topskip}{0.5\baselineskip} +\setlength{\footskip}{-1\baselineskip} +\setlength{\maxdepth}{0pt} +\renewcommand{\baselinestretch}{1} +\renewcommand{\arraystretch}{1} +\setcounter{LTchunksize}{\infty} +\setlength{\abovedisplayskip}{0pt} +\setlength{\parskip}{1\baselineskip} +\def\firstparagraph{\noindent} +\def\continueparagraph{\noindent} + +\hypersetup{ + hidelinks=true, +} + + \@addtoreset{chapter}{part} %% nötig für Hyperref. + + \def\partfont{\documentfont\fontseries{bx}\Huge\selectfont} + \def\chapterfont{\documentfont\fontseries{bx}\huge\selectfont} + \def\sectionfont{\documentfont\fontseries{bx}\Large\selectfont} + \def\subsectionfont{\documentfont\fontseries{bx}\large\selectfont} + + \def\thepart{\Roman{part}} + \generatenestedsecnumbering{arabic}{part}{chapter} + \generatenestedsecnumbering{arabic}{chapter}{section} + \generatenestedsecnumbering{arabic}{section}{subsection} + \generatenestedsecnumbering{arabic}{subsection}{subsubsection} + \def\theunitnamepart{\thepart} + \def\theunitnamechapter{\theshortchapter} + \def\theunitnamesection{\thelongsection} + \def\theunitnamesubsection{\thelongsubsection} + \def\theunitnamesubsubsection{\thelongsubsubsection} + + \def\partname{Teil\erlaubeplatz} + \def\chaptername{Kapitel\erlaubeplatz} + \def\sectionname{\S\erlaubeplatz} + \def\subsectionname{} + \def\subsubsectionname{} + + \let\appendix@orig\appendix + \def\appendix{% + \appendix@orig% + \let\boolinappendix\boolwahr + \addcontentsline{toc}{part}{\appendixname}% + \addtocontents{toc}{\protect\setcounter{tocdepth}{0}} + \def\sectionname{Appendix}% + \def\theunitnamesection{\Alph{section}}% + } + \def\notappendix{% + \let\boolinappendix\boolfalse + \addtocontents{toc}{\protect\setcounter{tocdepth}{1 }} + \def\sectionname{}% + \def\theunitnamesection{\arabic{section}}% + } + +%% \titlespacing{} +%% {linker einzug}{platz oberhalb}{platz unterhalb}[rechter einzug] + + \titlespacing{\section}{0pt}{\baselineskip}{\baselineskip} + \titlespacing{\subsection}{0pt}{\baselineskip}{\baselineskip} + \titlespacing{\subsubsection}{0pt}{\baselineskip}{\baselineskip} + \titlespacing{\paragraph}{0pt}{0pt}{1em} + + \titleformat{\part}[display] + {\normalfont\headingfont\bfseries\Huge\centering} + {% + \ifthenelse{\equal{\partname}{}}{% + \theunitnamepart% + }{% + \MakeUppercase{\partname}~\theunitnamepart% + }% + }{0pt}{% + }[\thispagestyle{empty}] + \titleformat{\chapter}[frame] + {\normalfont\headingfont\bfseries\Large} + {% + \bedingtesspaceexpand{chaptername}{~}{\theunitnamechapter}% + }{0.5em}{% + }[\thispagestyle{empty}]%\titlerule%[2pt]% + \titleformat{\section}[hang] + {\normalfont\headingfont\bfseries\flushleft\large} + {% + \bedingtesspaceexpand{sectionname}{~}{\theunitnamesection}% + }{0.5em} + {% + } + [% + \nvraum{0.25}% + ] + \titleformat{\subsection}[hang] + {\normalfont\headingfont\bfseries\flushleft\large} + {% + \bedingtesspaceexpand{subsectionname}{~}{\theunitnamesubsection}% + }{0.5em} + {% + } + [% + \nvraum{0.25}% + ] + \titleformat{\subsubsection}[hang] + {\normalfont\headingfont\bfseries\flushleft\large} + {% + \bedingtesspaceexpand{subsubsectionname}{~}{\theunitnamesubsubsection}% + }{0.5em} + {% + } + [% + \nvraum{0.25}% + ] + +\def\rafootnotectr{20} +\def\incrftnotectr#1{% + \addtocounter{#1}{1}% + \ifnum\value{#1}>\rafootnotectr\relax + \setcounter{#1}{0}% + \fi% +} +\def\footnoteref[#1]{\protected@xdef\@thefnmark{\ref{#1}}\@footnotemark} +\let\altfootnotetext\footnotetext + \def\footnotetext[#1]#2{\incrftnotectr{footnote}\altfootnotetext[\value{footnote}]{\label{#1}#2}} +\let\altfootnotemark\footnotemark + %% Undesirable solution, as the text is not hyperlinked. + \def\footnotemark[#1]{\text{\textsuperscript{\getrefnumber{#1}}}} + +\DefineFNsymbols*{custom}{abcdefghijklmnopqrstuvwxyz} +\setfnsymbol{custom} +\def\footnotelayout{\documentfont\scriptsize} +\def\thefootnote{\fnsymbol{footnote}} + +\def\kopfzeileleer{ + \lhead[]{} + \chead[]{} + \rhead[]{} + \lfoot[]{} + \cfoot[]{} + \rfoot[]{} +} +\def\kopfzeiledefault{ + \lhead[]{} + \lhead[]{} + \chead[]{} + \rhead[]{} + \lfoot[]{} + \cfoot{\footnotesize\thepage} + \rfoot[]{} +} + +\DeclareRobustCommand\crfamily{\fontfamily{pcr}\selectfont} +\def\headingfont{\fontfamily{cmss}\selectfont} +\def\documentfancyfont{% + \gdef\headingfont{\crfamily}% + \fontfamily{ccr}\fontseries{m}\selectfont% +} +\def\documentfont{% + \gdef\headingfont{\fontfamily{cmss}\selectfont}% + \fontfamily{cmss}\fontseries{m}\selectfont% + \renewcommand{\sfdefault}{phv}% + \renewcommand{\ttdefault}{pcr}% + \renewcommand{\rmdefault}{cmr}% <— funktionieren nicht mit {ptm} + \renewcommand{\bfdefault}{bx}% + \renewcommand{\itdefault}{it}% + \renewcommand{\sldefault}{sl}% + \renewcommand{\scdefault}{sc}% + \renewcommand{\updefault}{n}% +} + +\allowdisplaybreaks +\let\altcleardoublepage\cleardoublepage +\let\cleardoublepage\clearpage + +\def\startdocumentlayoutoptions{ + \selectlanguage{ngerman} + \setlength{\parskip}{0.5\baselineskip} + \setlength{\parindent}{0pt} + \kopfzeiledefault + \documentfont + \normalsize +} + +\def\highlightTerm#1{\emph{#1}} + +%% ******************************************************************************** +%% FILE: srclocal/setup-locallayout.tex +%% ******************************************************************************** + +\def\theunitnamesection{\theshortsection} + +%% ******************************************************************************** +%% FILE: srclocal/setup-localmacros.tex +%% ******************************************************************************** + +%% **************************************************************** +%% MATHE: +%% **************************************************************** + +\def\cal#1{\mathcal{#1}} +\def\reell{\mathbb{R}} +\def\kmplx{\mathbb{C}} +\def\Torus{\mathbb{T}} +\def\rtnl{\mathbb{Q}} +\def\intgr{\mathbb{Z}} + +\def\ntrl{\mathbb{N}} +\def\ntrlpos{\mathbb{N}} +\def\ntrlzero{\mathbb{N}_{0}} +\def\reellNonNeg{\reell_{+}} + +\def\imageinh{\imath} +\def\ReTeil{\mathop{\mathfrak{R}\text{\upshape e}}} +\def\ImTeil{\mathop{\mathfrak{I}\text{\upshape m}}} + +\def\leer{\emptyset} +\def\restr#1{\vert_{#1}} +\def\ohne{\mathbin{\setminus}} +\def\Pot{\mathop{\mathcal{P}}} +\def\einser{\mathbf{1}} +\def\supp{\mathop{\mathrm{supp}}} + +\def\brkt#1{\langle{}#1{}\rangle} +\def\lsim{\mathop{\sim}} +\def\lneg{\mathop{\neg}} +\def\land{\mathop{\wedge}} +\def\lor{\mathop{\vee}} + +\def\eps{\varepsilon} +\let\altphi\phi +\let\altvarphi\varphi + \def\phi{\altvarphi} + \def\varphi{\altphi} + +\def\vectorspacespan{\mathop{\text{\upshape Lin}}} +\def\dim{\mathop{\text{\upshape dim}}} +\def\rank{\mathop{\text{\upshape Rang}}} +\def\onematrix{\text{\upshape\bfseries I}} +\def\zeromatrix{\text{\upshape\bfseries 0}} +\def\zerovector{\text{\upshape\bfseries 0}} + +\def\graph{\mathop{\text{\upshape Gph}}} +\def\domain{\mathop{\text{\upshape dom}}} +\def\range{\mathop{\text{\upshape Bild}}} +\def\ker{\mathop{\text{\upshape Kern}}} +\def\functionspace{\mathop{\text{\upshape Abb}}} +\def\id{\text{\upshape id}} +\def\modfn{\mathop{\text{\upshape mod}}} +\def\divides{\mathbin{\mid}} +\def\ndivides{\mathbin{\nmid}} +\def\ggT{\mathop{\text{\upshape ggT}}} +\def\choose#1#2{\begin{smatrix}#1\\#2\\\end{smatrix}} + +\def\punktschema{ + \footnotesize + \hraum + \begin{tabular}[mc]{|p{0.1\textwidth}|p{0.7\textwidth}|} + \hline + \multicolumn{2}{|l|}{{\bfseries NOTENSCHEMA}}\\ + Punkte &Beschreibung\\ + \hline + \hline +} + \def\endpunktschema{ + \hline + \end{tabular} + \hraum + } + +\def\headingTeilaufgabe#1{ + \uwave{{\bfseries\large Aufgabe #1}} +} + +\makeatother + +\begin{document} + \startdocumentlayoutoptions + + %% HAUPTTEXT: + +%% ******************************************************************************** +%% FILE: body/index.tex +%% ******************************************************************************** + +%% ******************************************************************************** +%% FILE: body/A6.tex +%% ******************************************************************************** + +%% AUFGABE 6 +\let\altsectionname\sectionname +\def\sectionname{Aufgabe} +\setcounternach{section}{6} +\section[]{(Klausur${}_{1}$, WiSe 2020/2021)} + \label{ueb:1:ex:1} +\let\sectionname\altsectionname + +Es sSei $V\neq\{0\}$ ein Vektorraum über einem Körper, $K$. +Eine lineare Abbildung ${\phi:V\to V}$ heißt dann \emph{stark kontrahierend}, +wenn $\exists{n\in\ntrlpos:~}\phi^{n}=0(\cdot)$. + +%% ******************************************************************************** +%% FILE: body/A6a.tex +%% ******************************************************************************** + +%% AUFGABE 6a +\headingTeilaufgabe{6a} + +\begin{claim*} + Sei ${\phi:V\to V}$ linear. + Dann gilt $\phi$ \emph{stark kontrahierend} $\Rightarrow$ $\phi$ nicht invertierbar. +\end{claim*} + +Es gibt hierfür mehrere Ansätze. +In jedem der u.\,s. Möglichkeiten fixieren wir ein $n\in\ntrlpos$, so dass $\phi^{n}=\zerovector$, +und wir nehmen an, $\phi$ sei \emph{stark kontrahierend}. + +Als möglicherweise einfachtsten Ansätze kann man auf der Ebene von Abbildungen argumentieren. + +\setcounternach{enumi}{1} +\begin{enumerate}{\bfseries {Ansatz} I.} + \item + \textbf{Zu zeigen}, $\phi$ sei nicht invertierbar. + Angenommen, dies sei nicht der Fall. + Aus \textbf{Lemma 6.3.2} im Skript (siehe insbes. den Beweis dort) + ist die Menge der invertierbaren lineare Abbildungen + unter „Multiplikation“ (d.\,h. Komposition) abgeschlossen. + Darum sind + $\phi$, $\phi^{2}=\phi\circ\phi$, $\phi^{3}=\phi\circ\phi^{2}$, \ldots\, + alle invertierbar. + Insbesondere muss $\phi^{n}(=0(\cdot))$ \fbox{invertierbar} sein. + Da $V\neq\{\zerovector\}$ ist die $0(\cdot)$-Abbildung \fbox{nicht invertierbar}. + Widerspuch! + Also ist $\phi$ nicht invertierbar. + \enndeOfProof + \item + Es reicht aus \textbf{zu zeigen}, dass $\phi$ nicht surjektiv ist. + Angenommen, dies sei nicht der Fall. + Aus \textbf{Satz 2.3.6(1)} im Skript (siehe insbes. den Beweis) + sind surjektive Abbildungen unter Komposition abgeschlossen. + Darum sind + $\phi$, $\phi^{2}=\phi\circ\phi$, $\phi^{3}=\phi\circ\phi^{2}$, \ldots\, + alle surjektiv. + Insbesondere muss $\phi^{n}$ surjektiv sein. + Da $\phi^{n}$ linear ist, muss also \fbox{$\range(\phi^{n})=V$} gelten. + Da $\phi^{n}=0(\cdot)$, gilt aber $\range(\phi^{n})=\{\zerovector\}$. + Darum muss $V=\{\zerovector\}$ gelten, was der Voraussetzung auf $V$ widerspricht. + Darum haben wir einen Widerspruch erreicht. + Also ist $\phi$ nicht surjektiv, w.z.z.w. + \enndeOfProof + \item + Es reicht aus \textbf{zu zeigen}, dass $\phi$ nicht injektiv ist. + Angenommen, dies sei nicht der Fall. + Aus \textbf{Satz 2.3.6(1)} im Skript (siehe insbes. den Beweis) + sind injektive Abbildungen unter Komposition abgeschlossen. + Darum sind + $\phi$, $\phi^{2}=\phi\circ\phi$, $\phi^{3}=\phi\circ\phi^{2}$, \ldots\, + alle injektiv. + Insbesondere muss $\phi^{n}$ injektiv sein. + Da $\phi^{n}$ linear ist, muss also \fbox{$\ker(\phi^{n})=\{\zerovector\}$} gelten. + Da $\phi^{n}=0(\cdot)$, gilt aber $\ker(\phi^{n})=V$ (weil $\phi^{n}(u)=\zerovector$ für alle $u\in V$). + Darum muss $\{\zerovector\}=V$ gelten, was der Voraussetzung auf $V$ widerspricht. + Darum haben wir einen Widerspruch erreicht. + Also ist $\phi$ nicht injektiv, w.z.z.w. + \enndeOfProof +\end{enumerate} + +Als direkter Ansatz kann man auf der Ebene von Vektoren argumentieren +und konstruktiv vorgehen. + +\begin{enumerate}{\bfseries {Ansatz} I.} + \setcounternach{enumi}{4} + \item + Es reicht aus \textbf{zu zeigen}, dass $\ker(\phi)\neq\{\zerovector\}$, d.\,h. dass $\phi$ nicht injektiv ist. + Wir zeigen dies direkt. + Da $V\neq\{\zerovector\}$ existiert ein $v\in V\ohne\{\zerovector\}$. + Betrachten wir die Elemente: + + \begin{mathe}[mc]{ccccc} + (\zerovector\neq)v=\phi^{0}(v), + &\phi^{1}(v), + &\phi^{2}(v), + &\ldots, + &\phi^{n}(v)(=\zerovector)\\ + \end{mathe} + + Dann existiert ein $k\in\ntrlzero$ mit $1\leq k0$. + (Anhand Elementarkenntnisse wissen wir, dass dies zu $\ker(\phi)\neq\{\zerovector\}$ äquivalent ist, + was wiederum zu der Nichtinjektivität von $\phi$ äquivalent ist, + was die Nichtinvertierbarkeit von $\phi$ impliziert.)\\ + \forceindent + O.\,E. können wir auch annehmen, dass $n$ \uline{minimal} gewählt wird, so dass $\phi^{n}=0(\cdot)$.\\ + \textbf{Fall 1.} $n=1$. + Dann gilt $\phi=\phi^{1}=0(\cdot)$, + woraus sich $\ker(\phi)=V$ und $\dim(\ker(\phi))=\dim(V)>0$ ergibt, + da $V\neq\{\zerovector\}$.\\ + \textbf{Fall 2.} $n>1$. + Dann gilt $\phi\circ\phi^{n-1}=\phi^{n}=0(\cdot)$, + woraus sich \fbox{$\range(\phi^{n-1})\subseteq\ker(\phi)$}\textsuperscript{~($\ast$)} ergibt. + Nun, wegen \uline{Minimalität} von $n$ kann $\phi^{n-1}=0(\cdot)$ \uline{nicht} gelten. + Es gilt nun + + \begin{mathe}[mc]{rcl} + \phi^{n-1}\neq 0(\cdot) + &\Longrightarrow + &\range(\phi^{n-1})\neq\{\zerovector\}\\ + &\Longrightarrow + &\dim(\range(\phi^{n-1}))>0\\ + &\Longrightarrow + &\dim(\ker(\phi))\geq\dim(\range(\phi^{n-1}))>0.\\ + \end{mathe} + + In der letzten Aussage gilt die erste Ungleichung gilt wegen ($\ast$). + + Darum gilt in allen Fällen $\dim(\ker(\phi))>0$, wzzw. + \enndeOfProof +\end{enumerate} + +\begin{punktschema} + 3 &Argument vollständig (=ausführlich) und logisch gültig.\\ + \hdashline + 2 &Der Ansatz war richtig aber z.\,B.:\\ + &es fehlte an Ausführlichkeit (enthielt jedoch genug von dem nicht trivialen Teil);\\ + &oder die Aufgabe war in (a) falsch, aber versteckteweise in (b) vorhanden (und zwar vollständig+gültig);\\ + &oder er baute z.\,T. auf einem inkorrekt präsentierten Resultat (was dann z.\,B. auf die Ausführlichkeit eine Auswirkung hatte).\\ + \hdashline + 1 &Ansatz enthielt eine richtige Idee, aber wurde nicht korrekt/ausführlich ausgeführt, + od. man schließt die (nicht triviale) Lücke zw. Aussage über $\phi^{n}$ und Aussage über $\phi$ nicht\\ + \hdashline + 0 &sonst.\\ +\end{punktschema} + +{\footnotesize +\textbf{Bemerkung.} +Da es sich hier um die Bewertung von Argumentationen handelt, +kann man in Wirklichkeit hier kein Schema festlegen. +Stattdessen musste ich über die Qualität ein Urteil treffen. +In erster Linie kriegt man volle Punkte, wenn man +vollständig (idealerweise auch ausführlich) + gültig + überzeugend +argumentierte. +Ab dann musste ich anhand unterschiedlicher Defizite empirische Graduierungen implementieren. +Wenn etwas unvollständig oder ungültig war, bekam der Versuch einen Abzug. +Wenn etwas zu unordentlich oder inkohärent war, wurde meistens auf $0$ Pkt gegeben, +aber diese wurde verschont, wenn die Argumentation eine richtige Idee enthielt. +Es gab einen Fall, wo leider ein Denkfehler (ungültiger Schritt) vorlag, +aber der Ansatz war sonst sauber aufgeschrieben, +sodass der Versuch mindestens $1$ Pkt verdiente. +} + +\clearpage + +%% ******************************************************************************** +%% FILE: body/A6b.tex +%% ******************************************************************************** + +%% AUFGABE 6b +\headingTeilaufgabe{6b} + +\begin{claim*} + Sei ${\phi:V\to V}$ linear und \emph{stark kontrahierend}. + Sei $v\in V$ mit $v\neq\zerovector$ und $\phi(v)\neq\zerovector$. + Dann gilt $\ker(\phi)\cap\range(\phi)\neq\{\zerovector\}$. +\end{claim*} + +Beachte, dass (offensichtlich) + $\zerovector\in\ker(\phi)$ + und + $\zerovector\in\range(\phi)$ +gelten, sodass $\ker(\phi)\cap\range(\phi)\supseteq\{\zerovector\}$. +Was wir also eigentlich in dieser Behauptung zeigen, ist + $\ker(\phi)\cap\range(\phi)\supset\{\zerovector\}$ (strikte Inklusion), +d.\,h. dass ein Element $u\in V\ohne\{\zerovector\}$ existiert, +so dass \uline{dasselbe} Element $u$ sowohl in $\ker(\phi)$ +als auch in $\range(\phi)$ liegt. + +Es gibt hierfür mehrere Ansätze. +In jedem der u.\,s. Möglichkeiten fixieren wir ein $n\in\ntrlpos$, so dass $\phi^{n}=\zerovector$, +und wir nehmen an, $\phi$ sei \emph{stark kontrahierend}. + +\begin{enumerate}{\bfseries {Ansatz} I.} + \item + \textbf{Zu zeigen:} Es gibt ein Element + $u\in V\ohne\{\zerovector\}$ + mit $u\in\ker(\phi)$ und $u\in\range(\phi)$. + + Um ein solches $u$ zu konstruieren, betrachten wir die Elemente: + + \begin{mathe}[mc]{ccccc} + v=\phi^{0}(v), + &\phi^{1}(v), + &\phi^{2}(v), + &\ldots, + &\phi^{n}(v)(=\zerovector)\\ + \end{mathe} + + Sei $k\in\ntrlzero$ \uline{minimal} mit $\phi^{k}(v)=\zerovector$. + Da $\phi^{n}(v)=\zerovector$, ist dies wohldefiniert. + Da + $\phi^{0}(v)\neq\zerovector$ + und + $\phi^{1}(v)\neq\zerovector$, + gilt $k\geq 2$.\\ + \forceindent + Darum können wir den Vektor, $u:=\phi^{k-1}(v)$ betrachten. + Wegen \uline{Minimalität} von $k$ gilt $u\neq\zerovector$. + Da $k\geq 2$, gilt $u=\phi(\phi^{k-2}(v))\in\range(\phi)$. + Und per Wahl von $k$ gilt $\phi(u)=\phi(\phi^{k-1}(v))=\phi^{k}(v)=\zerovector$, + sodass $u\in\ker(u)$ gilt. + Darum haben wir ein passendes Element gefunden, + und die Behauptung ist bewiesen. + \enndeOfProof + \item + \textbf{Zu zeigen:} $\ker(\phi)\cap\range(\phi)\neq\{\zerovector\}$. + + Angenommen, dies sei nicht der Fall. + Dann + + \begin{mathe}[mc]{rcl} + \eqtag{$\star$} + \ker(\phi)\cap\range(\phi) &= &\{\zerovector\}.\\ + \end{mathe} + + Sei $k\in\ntrlzero$ \uline{minimal} mit $\phi^{k}(v)=\zerovector$. + Da $\phi^{n}(v)=\zerovector$, ist dies wohldefiniert + und per Voraussetzung auf $v$ gilt $k\geq 2$.\\ + \forceindent + Man betrachte nun $u:=\phi^{k-1}(v)$. + Wegen \uline{Minimalität} von $k$ gilt $u\neq\zerovector$. + Da $k\geq 2$, gilt $u=\phi(\phi^{k-2}(v))\in\range(\phi)$. + Und per Wahl von $k$ gilt $\phi(u)=\phi(\phi^{k-1}(v))=\phi^{k}(v)=\zerovector$, + sodass $u\in\ker(u)$ gilt. + Darum + $% + u\in\ker(\phi)\cap\range(\phi)% + \overset{(\star)}{=}\{\zerovector\}% + $. + Also $\phi^{k-1}(v)=u=\zerovector$, was ein Widerspruch zur \uline{Minimalität} von $k$ ist. + Da wir einen Widerspruch erreicht haben, + gilt doch $\ker(\phi)\cap\range(\phi)\neq\{\zerovector\}$. + \enndeOfProof +\end{enumerate} + +Beachte, dass diese Ansätze eigentlich äquivalent sind: +in dem II. Ansatz haben wir das gesuchte Element in I konstruiert. +Aber die Zielsetzungen sind anders. + +\begin{punktschema} + 3 &Argument vollständig (=ausführlich) und logisch gültig.\\ + \hdashline + 2 &Fehlte was Kleines (aber Wichtiges), wie, + explizit zu sagen/zeigen, + dass $\phi^{k-1}(v)\in\range(\phi)$ + oder dass $\phi^{k-1}(v)\in\ker(\phi)$ (aber wenn beides fehlten kriegt man natürlich weniger als $2$), + oder dass $k\geq 2$ (was nötig ist, damit man $\phi^{k-1}(v)=\phi^{k-2}(v)$ schreiben darf). + Oder man hat mit $k=n-1$ gearbeitet, obwohl das nicht unbedingt stimmt (außer man wählte $n$ minimal für $v$, aber das muss man dann sagen).\\ + \hdashline + 1,5 &Der Ansatz enthielt eine richtige Idee, aber wurde nicht korrekt oder ausführlich ausgeführt. + Es lag zumindest vor, dass man ein gemeinsames Element in $\ker(\phi)$ und im $\range(\phi)$ + zeigen musste.\\ + \hdashline + 1 &Der Ansatz enthielt eine richtige Idee, aber wurde nicht korrekt oder ausführlich ausgeführt. + Unterschied zu 1,5: Entweder fehlte zu viel oder war an Stellen inkohärent.\\ + \hdashline + 0 &sonst.\\ +\end{punktschema} + +{\footnotesize +\textbf{Bemerkung.} Hier lagen ähnliche Schwierigkeiten vor, ein Schema festzulegen. +Dafür wandte ich ähnliche Prinzipien an wie in der Bemerkung am Ende von A6a. +Spezifisch zu dieser Aufgabe konnte ich folgendes Beobachten: + +\begin{kompaktitem} + \item Denkfehler im Ansatz: Viele haben ein Element in $\ker(\phi)$ gesucht, + dann eins in $\range(\phi)$, aber nicht ein \uline{gemeinsames Element}. + \item Technische Kleinigkeiten (die jedoch keine Lappalien sind): + Damit man $\phi^{k-1}(v)$ und $\phi^{k-2}(v)$ überhaupt bilden darf, + muss man \uline{begründen}, dass $k\geq 1$ bzw. $k\geq 2$. + Ein sorgfältiger Umgang mit Randfällen und zu prüfen, dass etwas nicht jenseits eines Randes, + sind allgemein wichtig in allen technischen Bereichen. + \item Zu unterscheiden dazwischen, wann etwas trivial ist, und wann etwas explizit/ausführlich begründet werden soll. + \item Man argumentiert für Ergebnisse, + die schon in anderen Teilaufgaben vorhanden sind. + Das weist darauf hin, + dass man sich der Bedeutung der Resultate bzw. der Zusammenhänge nicht bewusst ist. + Auch wenn eine Prüfung größtenteils sachlich ist, + ist es generell sinnvoll, + sich zu überlegen, wie die Teile einer Aufgabe aufeinander aufbauen + und wie sie konzipiert sind. +\end{kompaktitem} +} + +\clearpage + +%% ******************************************************************************** +%% FILE: body/A6c.tex +%% ******************************************************************************** + +%% AUFGABE 6c +\headingTeilaufgabe{6c} + +Hier müssen wir für $V=\reell^{2}$ ein ${\phi:V\to V}$, +so dass $\phi\neq 0(\cdot)$ +und so dass $\phi$ \kurs{stark kontrahierend} ist. +Äquivalent können wir eine passende +Matrixdarstellung, $A\in M_{2\times 2}(\reell)$, +konstruieren. + +Hier ein paar Möglichkeiten: + +\begin{kompaktitem} + \item + $A:=\begin{matrix}{cc} + 1 &-1\\ + 1 &-1\\ + \end{matrix}$. + Dann ist $A$ offensichtlich ungleich $\zeromatrix$ (die Nullmatrix). + Und $A^{2}=\begin{matrix}{cc} + 0 &0\\ + 0 &0\\ + \end{matrix}$, + sodass $A$ (bzw. $\phi_{A}$) auch \kurs{stark kontrahierend} ist. + \item + $A:=\begin{matrix}{cc} + 0 &1\\ + 0 &0\\ + \end{matrix}$. + Dann ist $A$ offensichtlich ungleich $\zeromatrix$. + Und $A^{2}=\begin{matrix}{cc} + 0 &0\\ + 0 &0\\ + \end{matrix}$, + sodass $A$ (bzw. $\phi_{A}$) auch \kurs{stark kontrahierend} ist. +\end{kompaktitem} + +Es gibt natürlich viel mehr Möglichkeiten. + +\begin{punktschema} + 2 &Konstruktion beide Eigenschaften erfüllt + stark kontrahierend begründet.\\ + \hdashline + 1 &Konstruktion beide Eigenschaften erfüllt, aber unbegründet.\\ + \hdashline + 0 &Konstruktion erfüllt mind. eine der zwei Bedingungen nicht.\\ +\end{punktschema} + +\clearpage + +%% ******************************************************************************** +%% FILE: body/A6d.tex +%% ******************************************************************************** + +%% AUFGABE 6d +\headingTeilaufgabe{6d} + +Hier müssen wir für $V=\reell^{2}$ ein ${\phi:V\to V}$, +so dass $\phi$ nicht invertierbar +und so dass $\phi$ \uline{nicht} \kurs{stark kontrahierend} ist. +Äquivalent können wir eine passende +Matrixdarstellung, $A\in M_{2\times 2}(\reell)$, +konstruieren. + +Hier ein paar Möglichkeiten: + +\begin{kompaktitem} + \item + $A:=\begin{matrix}{cc} + 1 &1\\ + 1 &1\\ + \end{matrix}$. + Da $\rank(A)=1$ ist $A$ nicht invertierbar. + (Auch möglich: man weise darauf hin, dass die Spalten in $A$ nicht linear unabhängig sind.) + Es gilt nun $A^{2}=\begin{matrix}{cc} + 2 &2\\ + 2 &2\\ + \end{matrix}=2A$. + Darum + $A^{3} = A^{2}\cdot A = 2A\cdot A = 2\cdot A^{2} = 2\cdot 2A = 2^{2}A$, + usw. + Man sieht per Induktion, dass $A^{n}=2^{n}A(\neq 0)$ für alle $n\in\ntrlpos$. + Darum ist $A$ (bzw. $\phi_{A}$) \uline{nicht} \kurs{stark kontrahierend}. + \item + $A:=\begin{matrix}{cc} + p &1-p\\ + p &1-p\\ + \end{matrix}$ + für $p\in[0,1]$. + Da $\rank(A)=1$ ist $A$ nicht invertierbar. + (Auch möglich: man weise darauf hin, dass die Spalten in $A$ nicht linear unabhängig sind.) + Es gilt nun $A^{2}=\begin{matrix}{cc} + p &1-p\\ + p &1-p\\ + \end{matrix}=A$. + Darum + $A^{3} = A^{2}\cdot A = A\cdot A = A^{2} = A$, + usw. + Man sieht per Induktion, dass $A^{n}=A(\neq 0)$ für alle $n\in\ntrlpos$. + Darum ist $A$ (bzw. $\phi_{A}$) \uline{nicht} \kurs{stark kontrahierend}. + \item + $A:=\begin{matrix}{cc} + 1 &0\\ + 1 &0\\ + \end{matrix}$. + Da $\rank(A)=1$ ist $A$ nicht invertierbar. + (Auch möglich: berechne Zeilenstufen form und begründe dadurch.) + Es gilt nun $A^{2}=\begin{matrix}{cc} + 1 &0\\ + 1 &0\\ + \end{matrix}=A$. + Darum + $A^{3}=A^{2}\cdot A = A\cdot A = A^{2}=A$, + usw. + Man sieht per Induktion, dass $A^{n}=A(\neq 0)$ für alle $n\in\ntrlpos$. + Darum ist $A$ (bzw. $\phi_{A}$) \uline{nicht} \kurs{stark kontrahierend}. + \item + $A:=\begin{matrix}{cc} + 1 &0\\ + 0 &0\\ + \end{matrix}$. + Da $\rank(A)=1$ ist $A$ nicht invertierbar. + (Auch möglich: man weise darauf hin, dass die Spalten in $A$ nicht linear unabhängig sind.) + Es gilt nun $A^{2}=\begin{matrix}{cc} + 1 &0\\ + 0 &0\\ + \end{matrix}=A$. + Darum + $A^{3}=A^{2}\cdot A = A\cdot A = A^{2}=A$, + usw. + Man sieht per Induktion, dass $A^{n}=A(\neq 0)$ für alle $n\in\ntrlpos$. + Darum ist $A$ (bzw. $\phi_{A}$) \uline{nicht} \kurs{stark kontrahierend}. +\end{kompaktitem} + +Es gibt natürlich viel mehr Möglichkeiten. + +\begin{punktschema} + 2 &Konstruktion erfüllt beide Eigenschaften + und 2 Eigenschaften begründet.\\ + \hdashline + 1,5 &Konstruktion erfüllt beide Eigenschaften + und 1 Eigenschaft begründet.\\ + \hdashline + 1 &Konstruktion erfüllt beide Eigenschaften + und 0 Eigenschaft begründet.\\ + \hdashline + 0 &sonst.\\ +\end{punktschema} + +{\footnotesize +\textbf{Bemerkung.} +Es gab keine Teilpunkte, wenn $A$ invertierbar war. +Der Grund hierfür war, dass, +\emph{auch wenn man die anderen Aufgaben nicht beweisen konnte}, +man aus 6(a) hätte verstehen müssen, +dass + + \begin{mathe}[mc]{rcl} + \eqtag{$\star$} + \text{stark kontrahierend} &\Longrightarrow &\text{nicht invertierbar}\\ + \end{mathe} + +und damit trivialerweise + + \begin{mathe}[mc]{rcl} + \text{invertierbar} &\Longrightarrow &\text{nicht stark kontrahierend}\\ + \end{mathe} + +gelten. Der Zweck von Aufgabe 6(d) ist zu zeigen, +dass die Implikation in ($\star$) strikt ist, und dass die Umkehrung von ($\star$): + + \begin{mathe}[mc]{rcl} + \text{nicht invertierbar} &\Longrightarrow &\text{stark kontrahierend}\\ + \end{mathe} + +nicht gilt. Dies zeigt man, indem man ein $\phi$ findet, die nicht invertierbar +ist und nicht \emph{stark kontrahierend} ist. +} + +\end{document}