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******************************************************************************** + +%% ******************************************************************************** +%% DOCUMENT STRUCTURE: +%% ~~~~~~~~~~~~~~~~~~~ +%% +%% - root.tex; +%% | +%% ---- parameters.tex; +%% | +%% ---- ../loesungen/src/index.tex; +%% | +%% ---- ########; +%% | +%% ---- ../loesungen/src/setup-type.tex; +%% | +%% ---- ../loesungen/src/setup-packages.tex; +%% | +%% ---- ../loesungen/src/setup-parameters.tex; +%% | +%% ---- ../loesungen/src/setup-macros.tex; +%% | +%% ---- ../loesungen/src/setup-environments.tex; +%% | +%% ---- ../loesungen/src/setup-layout.tex; +%% | +%% ---- ../loesungen/src/setup-localmacros.tex; +%% | +%% ---- front/index.tex; +%% | +%% ---- front/title.tex; +%% | +%% ---- front/foreword.tex; +%% | +%% ---- front/contents.tex; +%% | +%% ---- body/index.tex; +%% | +%% ---- body/linear-extensions.tex; +%% | +%% ---- back/index.tex; +%% | +%% ---- ./../loesungen/back/quelle.bib; +%% +%% DOCUMENT-RANDOM-SEED: 5637845 +%% ******************************************************************************** + +%% ******************************************************************************** +%% FILE: root.tex +%% ******************************************************************************** + +%% ******************************************************************************** +%% FILE: parameters.tex +%% ******************************************************************************** + +%% ******************************************************************************** +%% FILE: ../loesungen/src/index.tex +%% ******************************************************************************** + +\makeatletter + +%% ******************************************************************************** +%% FILE: ../loesungen/src/setup-type.tex +%% ******************************************************************************** + +\documentclass[ + 12pt, + a4paper, + oneside, + openright, + center, + chapterbib, + crosshair, + fleqn, + headcount, + headline, + indent, + indentfirst=false, + portrait, + phonetic, + oldernstyle, + onecolumn, + sfbold, + upper, +]{scrbook} + +%% ******************************************************************************** +%% FILE: ../loesungen/src/setup-packages.tex +%% ******************************************************************************** + +\PassOptionsToPackage{T2A,OT1}{fontenc} % T1,OT1,T2A,OT2 +\PassOptionsToPackage{utf8}{inputenc} % utf8 +\PassOptionsToPackage{british,english,ngerman,russian}{babel} +\PassOptionsToPackage{ + english, + ngerman, + russian, + capitalise, +}{cleveref} +\PassOptionsToPackage{ + bookmarks=true, + bookmarksopen=false, + bookmarksopenlevel=0, + bookmarkstype=toc, + colorlinks=false, + raiselinks=true, + hyperfigures=true, +}{hyperref} +\PassOptionsToPackage{ + reset, + left=1in, + right=1in, + top=20mm, + bottom=20mm, + heightrounded, +}{geometry} +\PassOptionsToPackage{ + framemethod=TikZ, +}{mdframed} +\PassOptionsToPackage{normalem}{ulem} +\PassOptionsToPackage{ + amsmath, + thmmarks, +}{ntheorem} +\PassOptionsToPackage{table}{xcolor} +\PassOptionsToPackage{ + all, + color, + curve, + frame, + import, + knot, + line, + movie, + rotate, + textures, + tile, + tips, + web, + xdvi, +}{xy} + +\usepackage{amsfonts} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{ntheorem} % <— muss nach den ams* Packages vorkommen!! +\usepackage{array} +\usepackage{babel} +\usepackage{bbding} +\usepackage{bbm} +\usepackage{calc} +\usepackage{sectsty} +\usepackage{titlesec} +\usepackage{fancyhdr} +\usepackage{footmisc} +\usepackage{geometry} +\usepackage{graphicx} +\usepackage{ifpdf} +\usepackage{ifthen} +\usepackage{ifnextok} +\usepackage{longtable} +\usepackage{multicol} +\usepackage{multirow} +\usepackage{nameref} +\usepackage{nowtoaux} +\usepackage{paralist} +\usepackage{enumerate} %% nach [paralist] +\usepackage{pgf} +\usepackage{pgfplots} +\usepackage{proof} +\usepackage{refcount} +\usepackage{relsize} +\usepackage{savesym} +\usepackage{stmaryrd} +\usepackage{subfigure} +\usepackage{yfonts} %% <— Altgotische Fonts +\usepackage{tikz} +\usepackage{xy} +\usepackage{undertilde} +\usepackage{ulem} %% <– f\"ur besseren \underline-Befehl (\ul) +\usepackage{xcolor} +\usepackage{xspace} +\usepackage{xstring} +\usepackage{hyperref} +\usepackage{cleveref} % must vor hyperref geladen werden. + +\pgfplotsset{compat=newest} + +\usetikzlibrary{ + angles, + arrows, + automata, + calc, + decorations, + decorations.pathmorphing, + decorations.pathreplacing, + math, + positioning, + patterns, + quotes, + snakes, +} + +%% \var ≈ alter Befehl +%% \xvar ≈ wie das neue Package \var interpretieren soll. +\savesymbol{Diamond} +\savesymbol{emptyset} +\savesymbol{ggg} +\savesymbol{int} +\savesymbol{lll} +\savesymbol{RectangleBold} +\savesymbol{langle} +\savesymbol{rangle} +\savesymbol{hookrightarrow} +\savesymbol{hookleftarrow} +\savesymbol{Asterisk} +\usepackage{mathabx} +\usepackage{wasysym} +\let\varemptyset=\emptyset +\restoresymbol{x}{Diamond} +\restoresymbol{x}{emptyset} +\restoresymbol{x}{ggg} +\restoresymbol{x}{int} +\restoresymbol{x}{lll} +\restoresymbol{x}{RectangleBold} +\restoresymbol{x}{langle} +\restoresymbol{x}{rangle} +\restoresymbol{x}{hookrightarrow} +\restoresymbol{x}{hookleftarrow} +\restoresymbol{x}{Asterisk} + +\ifpdf + \usepackage{pdfcolmk} +\fi + +\usepackage{mdframed} + +%% Force-Import aus MnSymbol +\DeclareFontFamily{U}{MnSymbolA}{} +\DeclareFontShape{U}{MnSymbolA}{m}{n}{ + <-6> 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: ../loesungen/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: ../loesungen/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} + +%% **************************************************************** +%% 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: ../loesungen/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{% + \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% + } + \def\qedEIGEN@c#1[#2]{% + \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% + } + \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}[#4]{#2}}{\renewtheorem{#1@star@basic}[#4]{#2}} + %% FOR \BEGIN{THM*}[...] + \theoremstyle{nonumberplain} + \theoremseparator{\thmForceSepPt} + \theoremprework{\ra@pretheoremwork} + \@ifundefined{#1@star@withName}{\newtheorem{#1@star@withName}[#4]{#2}}{\renewtheorem{#1@star@withName}[#4]{#2}} + %% GENERATE ENVIRONMENTS: + \umbauenenv{#1}{#3}[#4] + \umbauenenv{#1@star}{#3}[#4] + %% 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: ../loesungen/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: ../loesungen/src/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}} + +\makeatother + +\begin{document} + \startdocumentlayoutoptions + + %% FRONTMATTER: + \thispagestyle{plain} + +%% ******************************************************************************** +%% FILE: front/index.tex +%% ******************************************************************************** + +%% ******************************************************************************** +%% FILE: front/title.tex +%% ******************************************************************************** + +\begin{titlepage} + \null + + \vraum + + \noindent\rule{\linewidth}{2pt} + + {\hraum\LARGE Lineare Algebra I\hraum}\\ + {\hraum\LARGE $\oast$\,\rule[0.175\baselineskip]{0.65\linewidth}{1pt}\,$\oast$ \hraum}\\ + {\hraum\Large Zusatzaufgaben aus der Übungsgruppe\hraum} + + \noindent\rule{\linewidth}{2pt} + + \vraum + + \noindent + \hraum{\footnotesize Raj Dahya}\hraum\\ + \hraum{\small \itshape Fakultät für Mathematik und Informatik}\hraum\\ + \hraum{\small \itshape Universität Leipzig.}\hraum\\ + \hraum{\small Wintersemester 2020/2021 }\hraum +\end{titlepage} + +%% ******************************************************************************** +%% FILE: front/foreword.tex +%% ******************************************************************************** + +\chapter*{Vorwort} + +Dieses Dokument enthält zusätzliche Aufgaben und Themen, +die in den Übungsgruppen erörtert wurden. + +%% ******************************************************************************** +%% FILE: front/contents.tex +%% ******************************************************************************** + +\kopfzeiledefault +\footnotesize +\setcounter{tocdepth}{1} +\def\contentsname{Inhaltsverzeichnis} + + \tableofcontents + + %% HAUPTTEXT: + +%% ******************************************************************************** +%% FILE: body/index.tex +%% ******************************************************************************** + +\def\chaptername{} + +%% ******************************************************************************** +%% FILE: body/linear-extensions.tex +%% ******************************************************************************** + +\chapter[Lineare Ausdehnung]{Lineare Ausdehnung} + \label{ch:lin-ext} + +In der Übungsgruppe in Woche 12 (am 3.2.2021) diskutierten wir +verzwickte Situationen und Fragentypen, die zum Thema linearer Ausdehnung vorkommen können. +Wir hatten das größtenteils theoretisch ausgelegt. +Hier wollen wir ein paar Aufgaben komplett durchrechnen. + +\textbf{Beachte!} Hier geht es niemals darum, +eine lineare Ausdehnung \emph{exmplizit darzustellen}, +sondern vielmehr +(1) \cite[Satz~6.1.13]{sinn2020} als zentrales Resultat anzuwenden, +(2) eine Basis aus den Inputvektoren zu generieren + (ggf. durch Entfernung von „linear abhängigen“ Vektoren, ggf. durch Basiserweiterung, ggf. durch beides!) +(3) die Input und Outputvektoren in der partielldefinierten Funktion +zu untersuchen, und \uline{rein aufgrund dessen} ein Urteil zu treffen, +ob + (3a) eine lineare Ausdehnung überhaupt möglich ist, + (3b) eine injektive/nicht injektive lineare Ausdehnung möglich ist, + (3c) eine surjektive/nicht surjektive lineare Ausdehnung möglich ist, + (3d) eine Isomorphismus (=Bijektion)/nicht-Isomorphismus als lineare Ausdehnung möglich ist. + +Nun, im Falle von Funktionen ${\phi:U\to V}$, wobei $U,V$ Vektorräume mit $\dim(U)=\dim(V)$, +sind wegen \cite[Korollar~6.1.11]{sinn2020} die Nebenfragen (3a)–(3c) alle äquivalent. +Im Falle $\dim(U)\neq\dim(V)$ machen wir von folgender Beobachtung Gebrauch: + +\begin{obs*} + Seien $U$, $V$ (endlich dimensionale) Vektorräume über einem Körper $K$ + und sei ${\phi:U\to V}$ linear. + Da $\range(\phi)\subseteq V$ gilt offensichtlich $\dim(\range(\phi))\leq\dim(V)$. + Und wenn wir eine Basis ${\{u_{1},u_{2}\ldots,u_{n}\}\subseteq U}$ + für $U$ fixieren, mit $n=\dim(U)$, + so gilt wegen Linearität + ${\range(\phi)=\vectorspacespan\{\phi(u_{1}),\phi(u_{2})\ldots,\phi(u_{n})\}}$. + Das heißt, $\{\phi(u_{1}),\phi(u_{2})\ldots,\phi(u_{n})\}$ + ist ein Erzeugendensystem für $\range(\phi)$. + Folglich gilt $\dim(\range(\phi))\leq n=\dim(U)$. + Da per Definition $\rank(\phi)=\dim(\range(\phi))$, + haben wir gezeigt, + dass + ${\rank(\phi)\leq\dim(V)}$ + und ${\rank(\phi)\leq\dim(U)}$ + \uline{stets gelten}. + Kürzer formuliert: + + \begin{mathe}[mc]{rcl} + \eqtag[eq:lin-abb-leq:ch:lin-ext] + \rank(\phi) &\leq &\min\{\dim(U),\dim(V)\}\\ + \end{mathe} + + gilt immer für alle lineare Abbildungen ${\phi:U\to V}$ + und alle Vektorräume $U, V$. +\end{obs*} + +Aus dieser Beobachtung können wir über (3b–3d) folgende Urteile generell treffen, wenn $\dim(U)\neq\dim(V)$: + + \begin{kompaktitem} + \item + Falls $\dim(U)>\dim(V)$ kann es bei offensichtlich höchstens nicht-injektive lineare Ausdehnungen geben, + weil für ${\phi:U\to V}$ linear gilt $\rank(\phi)\leq\dim(V)<\dim(U)$, + sodass laut \cite[Korollar~6.3.15(1)]{sinn2020} $\phi$ niemals injektiv sein kann. + \item[] + Darum lautet die Antwort zu (3b/3d) \emph{Gibt es injektive/bijektive...?} immer nein. + Die Fragen (3b/3d) \emph{Gibt es nicht-injektive/nicht-bijektive...?} sind dann äquivalent zu (3a). + \item + Falls $\dim(U)<\dim(V)$ kann es bei (3c) offensichtlich höchstens nicht-surjektive lineare Ausdehnungen geben, + weil für ${\phi:U\to V}$ linear gilt $\rank(\phi)\leq\dim(U)<\dim(V)$, + sodass laut \cite[Korollar~6.3.15(2)]{sinn2020} $\phi$ niemals surjektiv sein kann. + \item[] + Darum lautet die Antwort zu (3c/3d) \emph{Gibt es surjektive/bijektive...?} immer nein. + Die Fragen (3b/3d) \emph{Gibt es nicht-surjektive/nicht-bijektive...?} sind dann äquivalent zu (3a). + \end{kompaktitem} + +Daher können wir die Fragentypen in den Aufgaben immer teilweise sofort beantworten +und zum Teil vereinfachen, +je nachdem, ob $\dim(U)=\dim(V)$, oder $\dim(U)<\dim(V)$, oder $\dim(U)>\dim(V)$ +gelten. + +%% AUFGABE 1 +\clearpage +\let\altsectionname\sectionname +\def\sectionname{Aufgabe} +\section[Aufgabe 1]{} + \label{sec:1} +\let\sectionname\altsectionname + + Betrachten wir die Vektorräume $U:=\reell^{4}$ und $V:=\reell^{2}$. + Betrachten wir die Vektoren $u_{1},u_{2},u_{3}\in U$ und $v_{1},v_{2},v_{3}\in V$ + gegeben durch + + \begin{mathe}[mc]{ccc} + u_{1} = \begin{svector} 0\\ 0\\ 1\\ 1\\\end{svector}, + &u_{2} = \begin{svector} 3\\ 0\\ 1\\ 1\\\end{svector}, + &u_{3} = \begin{svector} 30\\ 0\\ 0\\ 0\\\end{svector},\\ + \end{mathe} + + \begin{mathe}[mc]{ccc} + v_{1} = \begin{svector} 5\\ 8\\\end{svector}, + &v_{2} = \begin{svector} -9\\ 11\\\end{svector}, + &v_{3} = \begin{svector} -140\\ 30\\\end{svector}.\\ + \end{mathe} + + \begin{qstn} + Gibt es eine lineare Abbildung, ${\phi:U\to V}$, + so dass + + \setcounter{columnanzahl}{3} + \begin{multikompaktenum}{\bfseries i)}[\rtab][\rtab] + \item + $\phi(u_{1})=v_{1}$ + \item + $\phi(u_{2})=v_{2}$ + \item + $\phi(u_{3})=v_{3}$ + \end{multikompaktenum} + + \uline{alle} erfüllt sind? + \end{qstn} + + \begin{soln*} + Wir beachten zuerst, dass $\{u_{1},u_{2}\}$ linear unabhängig sind\footnote{ + ich lasse hier die Beweise weg, + aber man sollte die zeigen, + z.\,B. durch das Gaußverfahren. + } + und dass $u_{3}\in\vectorspacespan\{u_{1},_{2}\}$, + da ${u_{3}=10u_{2}-10u_{1}}$. + Wir beachten auch, dass + + \begin{mathe}[mc]{rcccl} + 10v_{2}-10v_{1} &= &\begin{svector} -140\\ 30\\\end{svector} &= &v_{3}\\ + \end{mathe} + + gilt. + Darum können wir die Frage auf Bedingungen {i) + ii)} reduzieren: + existiert eine lineare Abbildung, die {i) + ii)} erfüllt, + dann wird wegen Linearität Bedingung iii) automatisch mit erfüllt. + Existiert keine lineare Abbildung, die {i) + ii)} erfüllt, + dann existiert natürlich auch keine, die i)--iii) erfüllt. + + Wir \uline{erweitern} nun die lineare unabhängige Menge + $\{u_{1},u_{2}\}$ + zu einer Basis + $\{u_{1},u_{2},u'_{3},u'_{4}\}$ + von $U$. + Wähle außerdem beliebige Vektoren, $v'_{3},v'_{4}\in V$. + Da $\{u_{1},u_{2},u'_{3},u'_{4}\}$ eine Basis von $U$ + ist und $v_{1},v_{2},v'_{3},v'_{4}\in V$, + existiert laut \cite[Satz~6.1.13]{sinn2020} + eine lineare Ausdehnung, + ${\phi:U\to V}$, + so dass + + \begin{mathe}[mc]{cccc} + \phi(u_{1})=v_{1}, + &\phi(u_{2})=v_{2}, + &\phi(u'_{3})=v'_{3}, + &\phi(u'_{4})=v'_{4}, + \end{mathe} + + gelten. Insbesondere sind Bedingungen {i) + ii)} erfüllt. + Also lautet wie oben argumentiert, die Antwort auf die Originalfrage: »Ja«! + \end{soln*} + + \begin{qstn} + Gibt es eine + + \setcounter{columnanzahl}{3} + \begin{multikompaktenum} + \item[] b) injektive + \item[] b') nicht-injektive + \item[] c) surjektive + \item[] c') nicht-surjektive + \item[] d) bijektive\footnote{ + also einen »Isomorphismus« + } + \item[] d') nicht-bijektive + \end{multikompaktenum} + + lineare Abbildung, ${\phi:U\to V}$, + so dass i)--iii) erfüllt sind? + \end{qstn} + + \begin{soln*} + Da $\dim(U)>\dim(V)$, + kann es generell keine injektiven linearen Abbildungen + von $U$ nach $V$ geben. + Also lauten die Antworten auf b), d) »Nein«, + und da es mindestens eine lineare Ausdehnung existiert, + lautet die Antwort auf b') und d') »Ja«. + + Es bleiben nur noch c) und c') zu bestimmen. + Sei ${\phi:U\to V}$ eine lineare Ausdehnung von i)--iii). + Dann wegen Bedingungen {i) + ii)} und Linearität gilt + + \begin{mathe}[mc]{rcccccl} + \range(\phi) + &\supseteq + &\vectorspacespan\{\phi(u_{1}),\phi(u_{2})\} + &= &\vectorspacespan\{v_{1},v_{2}\} + &= &V.\\ + \end{mathe} + + Die letzte Gleichung gilt, weil $\{v_{1},v_{2}\}$ + linear unabhängig ist\footnote{ + ich lasse wieder den Beweis weg, + aber man sollte das machen + }, + und somit eine Basis von dem $2$-dimensionalen Raum, $V$, ist. + Darum ist $\range(\phi)$ surjektiv. + Da $\phi$ beliebig war haben wir tatsächlich gezeigt, + dass alle lineare Ausdehnungen von i)--iii) surjektiv sind. + Darum lautet die Antwort auf c) »Ja« und auf c') »Nein«. + \end{soln*} + +%% AUFGABE 2 +\clearpage +\let\altsectionname\sectionname +\def\sectionname{Aufgabe} +\section[Aufgabe 2]{} + \label{sec:2} +\let\sectionname\altsectionname + + Betrachten wir die Vektorräume $U:=\reell^{4}$ und $V:=\reell^{2}$. + Betrachten wir die Vektoren $u_{1},u_{2},u_{3}\in U$ und $v_{1},v_{2},v_{3}\in V$ + gegeben durch + + \begin{mathe}[mc]{cccc} + u_{1} = \begin{svector} 0\\ 0\\ 1\\ 1\\\end{svector}, + &u_{2} = \begin{svector} 3\\ 0\\ 1\\ 1\\\end{svector}, + &u_{3} = \begin{svector} 30\\ 0\\ 0\\ 0\\\end{svector}, + &u_{4} = \begin{svector} 3\\ 0\\ 2\\ 2\\\end{svector},\\ + \end{mathe} + + \begin{mathe}[mc]{cccc} + v_{1} = \begin{svector} 5\\ 8\\\end{svector}, + &v_{2} = \begin{svector} 25\\ 40\\\end{svector}, + &v_{3} = \begin{svector} -140\\ 30\\\end{svector}, + &v_{4} = \begin{svector} 30\\ 48\\\end{svector}.\\ + \end{mathe} + + \begin{qstn} + Gibt es eine lineare Abbildung, ${\phi:U\to V}$, + so dass + + \setcounter{columnanzahl}{2} + \begin{multikompaktenum}{\bfseries i)}[\rtab][\rtab] + \item + $\phi(u_{1})=v_{1}$ + \item + $\phi(u_{2})=v_{2}$ + \item + $\phi(u_{3})=v_{3}$ + \item + $\phi(u_{4})=v_{4}$ + \end{multikompaktenum} + + \uline{alle} erfüllt sind? + \end{qstn} + + \begin{soln*} + Wir beachten zuerst, dass $\{u_{1},u_{2}\}$ linear unabhängig sind + und dass $u_{3},u_{4}\in\vectorspacespan\{u_{1},_{2}\}$, + da ${u_{3}=10u_{2}-10u_{1}}$ und ${u_{4}=u_{1}+u_{2}}$. + Wir beachten auch, dass + + \begin{mathe}[mc]{rcccl} + 10v_{2}-10v_{1} &= &\begin{svector} -140\\ 30\\\end{svector} &= &v_{3},\\ + v_{1}+v_{2} &= &\begin{svector} 30\\ 48\\\end{svector} &= &v_{4}\\ + \end{mathe} + + gelten. + Darum können wir die Frage auf Bedingungen {i) + ii)} reduzieren, + weil wegen der o.\,s. Verhältnisse {iii) + iv)} + für lineare Abbildungen automatisch mit erfüllt werden. + + Wir \uline{erweitern} nun die lineare unabhängige Menge $\{u_{1},u_{2}\}$ + zu einer Basis + $\{u_{1},u_{2},u'_{3},u'_{4}\}$ + von $U$. + Wähle außerdem beliebige Vektoren, $v'_{3},v'_{4}\in V$. + Da $\{u_{1},u_{2},u'_{3},u'_{4}\}$ eine Basis von $U$ + ist und $v_{1},v_{2},v'_{3},v'_{4}\in V$, + existiert laut \cite[Satz~6.1.13]{sinn2020} + eine lineare Ausdehnung, + ${\phi:U\to V}$, + so dass + + \begin{mathe}[mc]{cccc} + \phi(u_{1})=v_{1}, + &\phi(u_{2})=v_{2}, + &\phi(u'_{3})=v'_{3}, + &\phi(u'_{4})=v'_{4}, + \end{mathe} + + gelten. Insbesondere sind Bedingungen {i) + ii)} erfüllt. + Also lautet wie oben argumentiert, die Antwort auf die Originalfrage: »Ja«! + \end{soln*} + + \begin{qstn} + Gibt es eine + + \setcounter{columnanzahl}{3} + \begin{multikompaktenum} + \item[] b) injektive + \item[] b') nicht-injektive + \item[] c) surjektive + \item[] c') nicht-surjektive + \item[] d) bijektive + \item[] d') nicht-bijektive + \end{multikompaktenum} + + lineare Abbildung, ${\phi:U\to V}$, + so dass i)--iv) erfüllt sind? + \end{qstn} + + \begin{soln*} + Da $\dim(U)>\dim(V)$, + kann es generell keine injektiven linearen Abbildungen + von $U$ nach $V$ geben. + Also lauten die Antworten auf b), d) »Nein«, + und da es mindestens eine lineare Ausdehnung existiert, + lautet die Antwort auf b') und d') »Ja«. + + Es bleiben nur noch c) und c') zu bestimmen. + Beachte, dass in der Konstruktion von $\phi$ im o.\,s. Beweis + wir $v'_{3},v'_{4}$ beliebig auswählen konnten. + + Zu c) wähle $v'_{3}:=\begin{svector} 1\\ 0\\\end{svector}$ + und $v'_{4}:=\zerovector$ + und sei ${\phi_{1}:U\to V}$ die lineare Abbildung + im o.\,s. Beweis mit diesen Vektoren in der Konstruktion. + Da $\{u_{1},u_{2},u'_{3},u'_{4}\}$ eine Basis für $U$ ist, + gilt wegen Linearität von $\phi_{1}$ + + \begin{mathe}[mc]{rcl} + \range(\phi_{1}) + &= &\phi_{1}(\vectorspacespan\{u_{1},u_{2},u'_{3},u'_{4}\})\\ + &= &\vectorspacespan\{\phi(u_{1}),\phi(u_{2}),\phi(u'_{3}),\phi(u'_{4})\}\\ + &= &\vectorspacespan\{v_{1},v_{2},v'_{3},v'_{4}\}\\ + \end{mathe} + + und damit eine Basis des 2-dimensionalen Vektorraums, $V$. + Darum gilt + $% + \rank(\phi_{1}) + \textoverset{Defn}{=}\dim(\range(\phi_{1})) + =\dim(\vectorspacespan\{v_{1},v_{2},v'_{3},v'_{4}\}) + \geq\dim(\vectorspacespan\{v_{1},v'_{3}\}) + =2 + $, + da \uline{per Wahl} $\{v_{1},v'_{3}\}$ linear unabhängig ist. + Also, $\rank(\phi_{1})\geq 2=\dim(V)$. + Folglich ist $\phi_{1}$ + laut \cite[Korollar~6.3.15(2)]{sinn2020} + surjektiv. + Die Antwort auf c) lautet also »Ja«. + + Zu c') wähle $v'_{3},v'_{4}:=\zerovector$ + und sei ${\phi_{2}:U\to V}$ die lineare Abbildung + im o.\,s. Beweis mit diesen Vektoren in der Konstruktion. + Wie oben gilt + $% + \rank(\phi_{2}) + \textoverset{Defn}{=}\dim(\range(\phi_{2})) + =\dim(\vectorspacespan\{v_{1},v_{2},v'_{3},v'_{4}\}) + =\dim(\vectorspacespan\{v_{1}\}) + \leq 1 + $, + da \uline{per Wahl} $v_{2},v_{3},v_{4}\in\vectorspacespan(\{v_{1}\})$ + und $v_{1}\neq\zerovector$. + Also, $\rank(\phi_{2})<2=\dim(V)$. + Folglich ist $\phi_{2}$ + laut \cite[Korollar~6.3.15(1)]{sinn2020} + nicht-surjektiv. + Die Antwort auf c') lautet also »Ja«. + \end{soln*} + +%% AUFGABE 3 +\clearpage +\let\altsectionname\sectionname +\def\sectionname{Aufgabe} +\section[Aufgabe 3]{} + \label{sec:3} +\let\sectionname\altsectionname + + Betrachten wir die Vektorräume $U:=\reell^{2}$ und $V:=\reell^{4}$. + Betrachten wir die Vektoren $u_{1},u_{2},u_{3}\in U$ und $v_{1},v_{2},v_{3}\in V$ + gegeben durch + + \begin{mathe}[mc]{ccc} + u_{1} = \begin{svector} 1\\ 1\\\end{svector}, + &u_{2} = \begin{svector} 0\\ 2\\\end{svector}, + &u_{3} = \begin{svector} 1\\ 3\\\end{svector}\\ + \end{mathe} + + \begin{mathe}[mc]{ccc} + v_{1} = \begin{svector} -9\\ 0\\ 0\\ 1\\\end{svector}, + &v_{2} = \begin{svector} 4\\ 0\\ 0\\ 2\\\end{svector}, + &v_{3} = \begin{svector} 5\\ 1\\ 0\\ 3\\\end{svector}.\\ + \end{mathe} + + \begin{qstn} + Gibt es eine lineare Abbildung, ${\phi:U\to V}$, + so dass + + \setcounter{columnanzahl}{3} + \begin{multikompaktenum}{\bfseries i)}[\rtab][\rtab] + \item + $\phi(u_{1})=v_{1}$ + \item + $\phi(u_{2})=v_{2}$ + \item + $\phi(u_{3})=v_{3}$ + \end{multikompaktenum} + + \uline{alle} erfüllt sind? + \end{qstn} + + \begin{soln*} + Beachte, dass $u_{3}=u_{1}+u_{2}$, aber + + \begin{mathe}[mc]{rcccl} + v_{1}+v_{2} + &= &\begin{svector} -5\\ 0\\ 0\\ 3\\\end{svector} + &\neq &v_{3}.\\ + \end{mathe} + + Angenommen, es gebe eine lineare Ausdehnung ${\phi:U\to V}$, + die i)--iii) erfüllt. + Dann muss + $v_{3}=\phi(u_{3})=\phi(u_{1}+u_{2})=\phi(u_{1})+\phi(u_{2})=v_{1}+v_{2}$ + gelten. Laut der o.\,s. Gleichung kann dies aber nicht gelten. + Darum lautet die Antwort »Nein«. + Es gibt keine lineare Ausdehnung. + \end{soln*} + +%% AUFGABE 4 +\clearpage +\let\altsectionname\sectionname +\def\sectionname{Aufgabe} +\section[Aufgabe 4]{} + \label{sec:4} +\let\sectionname\altsectionname + + Betrachten wir die Vektorräume $U:=\reell^{2}$ und $V:=\reell^{4}$. + Betrachten wir die Vektoren $u_{1},u_{2},u_{3}\in U$ und $v_{1},v_{2},v_{3}\in V$ + gegeben durch + + \begin{mathe}[mc]{ccc} + u_{1} = \begin{svector} 1\\ 1\\\end{svector}, + &u_{2} = \begin{svector} 2\\ 3\\\end{svector}, + &u_{3} = \begin{svector} 0\\ 1\\\end{svector}\\ + \end{mathe} + + \begin{mathe}[mc]{ccc} + v_{1} = \begin{svector} 8\\ 0\\ 0\\ 4\\\end{svector}, + &v_{2} = \begin{svector} 16\\ 0\\ 0\\ 9\\\end{svector}, + &v_{3} = \begin{svector} 2\\ 0\\ 0\\ 1\\\end{svector}.\\ + \end{mathe} + + \begin{qstn} + Gibt es eine lineare Abbildung, ${\phi:U\to V}$, + so dass + + \setcounter{columnanzahl}{3} + \begin{multikompaktenum}{\bfseries i)}[\rtab][\rtab] + \item + $\phi(u_{1})=v_{1}$ + \item + $\phi(u_{2})=v_{2}$ + \item + $\phi(u_{3})=v_{3}$ + \end{multikompaktenum} + + \uline{alle} erfüllt sind? + \end{qstn} + + \begin{soln*} + Wir beachten zuerst, dass $\{u_{1},u_{3}\}$ linear unabhängig ist + und dass $u_{2}\in\vectorspacespan\{u_{1}\}$, + da ${u_{2}=2u_{1}+u_{3}}$. + Wir beachten auch, dass + + \begin{mathe}[mc]{rcccl} + 2v_{1}+u_{3} &= &\begin{svector} 18\\ 0\\ 0\\ 9\\\end{svector} &= &v_{2}\\ + \end{mathe} + + gilt. + Darum können wir die Frage auf Bedingung {i) + iii)} reduzieren, + weil wegen der o.\,s. Verhältnisse {ii)} + für lineare Abbildungen automatisch mit erfüllt wird. + + Wegen linearer Unabhängigkeit ist $\{u_{1},u_{3}\}$ + bereits eine Basis des $2$-dimensionalen Raums, $U$. + Deswegen brauchen wir in dieser Aufgabe keine Erweiterung zu machen. + Da $\{u_{1},u_{3}\}$ eine Basis für $U$ + und $\{v_{1},v_{3}\}\subseteq V$, + existiert laut \cite[Satz~6.1.13]{sinn2020} + eine lineare Abbildung, + ${\phi:U\to V}$, + so dass + + \begin{mathe}[mc]{cc} + \phi(u_{1})=v_{1}, &\phi(u_{3})=v_{3}, + \end{mathe} + + gelten. Insbesondere sind Bedingung {i) + iii)} erfüllt. + Also lautet wie oben argumentiert, + die Antwort auf die Originalfrage: »Ja«! + \end{soln*} + + \begin{qstn} + Gibt es eine + + \setcounter{columnanzahl}{3} + \begin{multikompaktenum} + \item[] b) injektive + \item[] b') nicht-injektive + \item[] c) surjektive + \item[] c') nicht-surjektive + \item[] d) bijektive + \item[] d') nicht-bijektive + \end{multikompaktenum} + + lineare Abbildung, ${\phi:U\to V}$, + so dass i)--iii) erfüllt sind? + \end{qstn} + + \begin{soln*} + Da $\dim(U)<\dim(V)$, + kann es generell keine surjektive linearen Abbildungen + von $U$ nach $V$ geben. + Also lauten die Antworten auf c), d) »Nein«, + und da es mindestens eine lineare Ausdehnung existiert, + lautet die Antwort auf c') und d') »Ja«. + + Es bleiben nur noch b) und b') zu bestimmen. + Sei $\phi$ eine lineare Ausdehnung, die i)---iii) erfüllt. + Dann wegen Linearität von $\phi_{1}$ + und da $\{u_{1},u_{3}\}$ eine Basis von $U$ ist, + gilt + + \begin{mathe}[mc]{rcccccl} + \range(\phi_{1}) + &= &\phi_{1}(\vectorspacespan\{u_{1},u_{3}\}) + &= &\vectorspacespan\{\phi_{1}(u_{1}),\phi_{1}(u_{3})\} + &= &\vectorspacespan\{v_{1},v_{3}\} + \end{mathe} + + und damit + $% + \rank(\phi) + \textoverset{Defn}{=}\dim(\range(\phi)) + =\dim(\vectorspacespan\{v_{1},v_{3}\}) + =\dim(\vectorspacespan\{v_{3}\}) + =1 + $, + da $v_{1}\in\vectorspacespan\{v_{3}\}$ + und $v_{3}\neq\zerovector$. + Also, $\rank(\phi)<2=\dim(U)$. + Folglich ist $\phi$ + laut \cite[Korollar~6.3.15(1)]{sinn2020} + nicht injektiv. + Da hier $\phi$ beliebig gewählt wurde, + sind alle linearen Ausdehnungen von i)--iii) + immer nicht-injektiv. + Darum lautet die Antwort auf b) »Nein« + und auf b') »Ja«. + \end{soln*} + +%% AUFGABE 5 +\clearpage +\let\altsectionname\sectionname +\def\sectionname{Aufgabe} +\section[Aufgabe 5]{} + \label{sec:5} +\let\sectionname\altsectionname + + Betrachten wir die Vektorräume $U:=\reell^{2}$ und $V:=\reell^{4}$. + Betrachten wir die Vektoren $u_{1},u_{2},u_{3}\in U$ und $v_{1},v_{2},v_{3}\in V$ + gegeben durch + + \begin{mathe}[mc]{ccc} + u_{1} = \begin{svector} 1\\ 1\\\end{svector}, + &u_{2} = \begin{svector} 2\\ 2\\\end{svector}, + &u_{3} = \begin{svector} 3\\ 3\\\end{svector}\\ + \end{mathe} + + \begin{mathe}[mc]{ccc} + v_{1} = \begin{svector} 8\\ 0\\ 0\\ 4\\\end{svector}, + &v_{2} = \begin{svector} 16\\ 0\\ 0\\ 8\\\end{svector}, + &v_{3} = \begin{svector} 24\\ 0\\ 0\\ 12\\\end{svector}.\\ + \end{mathe} + + \begin{qstn} + Gibt es eine lineare Abbildung, ${\phi:U\to V}$, + so dass + + \setcounter{columnanzahl}{3} + \begin{multikompaktenum}{\bfseries i)}[\rtab][\rtab] + \item + $\phi(u_{1})=v_{1}$ + \item + $\phi(u_{2})=v_{2}$ + \item + $\phi(u_{3})=v_{3}$ + \end{multikompaktenum} + + \uline{alle} erfüllt sind? + \end{qstn} + + \begin{soln*} + Wir beachten zuerst, dass $\{u_{1}\}$ linear unabhängig ist + und dass $u_{2},u_{3}\in\vectorspacespan\{u_{1}\}$, + da $u_{2}=2u_{1}$ und $u_{3}=3u_{1}$. + Wir beachten auch, dass + + \begin{mathe}[mc]{rcccl} + 2v_{1} &= &\begin{svector} 16\\ 0\\ 0\\ 8\\\end{svector} &= &v_{2},\\ + 3v_{1} &= &\begin{svector} 24\\ 0\\ 0\\ 12\\\end{svector} &= &v_{3}\\ + \end{mathe} + + gelten. + Darum können wir die Frage auf Bedingung {i)} reduzieren, + weil wegen der o.\,s. Verhältnisse {ii) + iii)} + für lineare Abbildungen automatisch mit erfüllt werden. + + Erweitere $\{u_{1}\}$ zu einer Basis $\{u_{1},u'_{2}\}$ + des $2$-dimensionalen Raums, $U$, + und wähle einen Vektor $v'_{2}\in V$. + Da $\{u_{1},u'_{2}\}$ eine Basis für $U$ + und $\{v_{1},v'_{2}\}\subseteq V$, + existiert laut \cite[Satz~6.1.13]{sinn2020} + eine lineare Abbildung, + ${\phi:U\to V}$, + so dass + + \begin{mathe}[mc]{cc} + \phi(u_{1})=v_{1}, &\phi(u'_{2})=v'_{2}, + \end{mathe} + + gelten. Insbesondere sind Bedingung {i)} erfüllt. + Also lautet wie oben argumentiert, + die Antwort auf die Originalfrage: »Ja«! + \end{soln*} + + \begin{qstn} + Gibt es eine + + \setcounter{columnanzahl}{3} + \begin{multikompaktenum} + \item[] b) injektive + \item[] b') nicht-injektive + \item[] c) surjektive + \item[] c') nicht-surjektive + \item[] d) bijektive + \item[] d') nicht-bijektive + \end{multikompaktenum} + + lineare Abbildung, ${\phi:U\to V}$, + so dass i)--iii) erfüllt sind? + \end{qstn} + + \begin{soln*} + Da $\dim(U)<\dim(V)$, + kann es generell keine surjektive linearen Abbildungen + von $U$ nach $V$ geben. + Also lauten die Antworten auf c), d) »Nein«, + und da es mindestens eine lineare Ausdehnung existiert, + lautet die Antwort auf c') und d') »Ja«. + + Es bleiben nur noch b) und b') zu bestimmen. + Beachte, dass in der Konstruktion von $\phi$ im o.\,s. Beweis + wir $v'_{2}$ beliebig auswählen konnten. + + Zu b) wähle $v'_{2}:=\begin{svector} 0\\ 1\\ 0\\ 0\\\end{svector}$ + und sei ${\phi_{1}:U\to V}$ die lineare Abbildung + im o.\,s. Beweis mit diesen Vektoren in der Konstruktion. + Per Wahl ist $\{v_{1},v'_{2}\}$ linear unabhängig. + Da $\{u_{1},u'_{2}\}$ eine Basis für $U$ ist, + gilt wegen Linearität von $\phi_{1}$ + + \begin{mathe}[mc]{rcccccl} + \range(\phi_{1}) + &= &\phi_{1}(\vectorspacespan\{u_{1},u'_{2}\}) + &= &\vectorspacespan\{\phi_{1}(u_{1}),\phi_{1}(u'_{2})\} + &= &\vectorspacespan\{v_{1},v'_{2}\} + \end{mathe} + + und damit + $% + \rank(\phi_{1}) + \textoverset{Defn}{=}\dim(\range(\phi_{1})) + =\dim(\vectorspacespan\{v_{1},v'_{2}\}) + =2 + $, + da \uline{per Wahl} $\{v_{1},v'_{2}\}$ linear unabhängig ist. + Also, $\rank(\phi_{1})\geq 2=\dim(U)$. + Folglich ist $\phi_{1}$ + laut \cite[Korollar~6.3.15(1)]{sinn2020} + injektiv. + Die Antwort auf b) lautet also »Ja«. + + Zu b') wähle $v'_{2}:=\zerovector$ + und sei ${\phi_{2}:U\to V}$ die lineare Abbildung + im o.\,s. Beweis mit diesen Vektoren in der Konstruktion. + Wie oben gilt + $% + \rank(\phi_{2}) + \textoverset{Defn}{=}\dim(\range(\phi_{2})) + =\dim(\vectorspacespan\{v_{1},v'_{2}\}) + =\dim(\vectorspacespan\{v_{1}\}) + =1 + $, + da \uline{per Wahl} $v'_{2}\in\vectorspacespan(\{v_{1}\})$ + und $v_{1}\neq\zerovector$. + Also, $\rank(\phi_{2})<2=\dim(U)$. + Folglich ist $\phi_{2}$ + laut \cite[Korollar~6.3.15(1)]{sinn2020} + nicht-injektiv. + Die Antwort auf b') lautet also »Ja«. + \end{soln*} + +%% AUFGABE 6 +\clearpage +\let\altsectionname\sectionname +\def\sectionname{Aufgabe} +\section[Aufgabe 6]{} + \label{sec:6} +\let\sectionname\altsectionname + +{\itshape + Unter Arbeit. \textbf{TODO}: Beispiel mit $\dim(U)=\dim(V)$ und mit Antwort »Ja« auf inj + »Ja« auf nicht-inj. +} + +%% AUFGABE 7 +\let\altsectionname\sectionname +\def\sectionname{Aufgabe} +\section[Aufgabe 7]{} + \label{sec:7} +\let\sectionname\altsectionname + +{\itshape + Unter Arbeit. \textbf{TODO}: Beispiel mit $\dim(U)=\dim(V)$ und mit Antwort »Ja« auf inj + »Nein« auf nicht-inj. +} + +%% AUFGABE 8 +\let\altsectionname\sectionname +\def\sectionname{Aufgabe} +\section[Aufgabe 8]{} + \label{sec:8} +\let\sectionname\altsectionname + +{\itshape + Unter Arbeit. \textbf{TODO}: Beispiel mit $\dim(U)=\dim(V)$ und mit Antwort »Nein« auf inj + »Ja« auf nicht-inj. +} + +%% ******************************************************************************** +%% FILE: back/index.tex +%% ******************************************************************************** + +\bibliographystyle{alpha} +\def\bibname{Literaturverzeichnis} +\nocite{*} + +\bgroup +\footnotesize + +%% ******************************************************************************** +%% FILE: ./../loesungen/back/quelle.bib +%% ******************************************************************************** + +\begin{thebibliography}{Wal16} + +\bibitem[Jec97]{jech1997} +Thomas Jech. +\newblock {\em {Set Theory}}. +\newblock Springer-Verlag, 1997. + +\bibitem[Sin20]{sinn2020} +Rainer Sinn. +\newblock {Lineare Algebra I: Skript zur Veranstaltung Universit\"at Leipzig}. +\newblock Vorlesungsskript, 2020. + +\bibitem[Wal16]{waldmann2016} +Stefan Waldmann. +\newblock {\em {Lineare Algebra 1: Die Grundlagen f\"ur Studierende der + Mathematik und Physik}}. +\newblock Springer Berlin Heidelberg, 2016. + +\end{thebibliography} +\egroup + +\end{document}