From 1faa1ce9c39f80abd95924cb0e4ac1aaccc6ec9b Mon Sep 17 00:00:00 2001
From: raj_mathe <raj.dahya@math.uni-leipzig.de>
Date: Thu, 4 Feb 2021 20:37:34 +0100
Subject: [PATCH] master > master: Aufgaben mit dim(U)=dim(V), weitere
 Korrekturen.

---
 docs/zusatz.pdf |  Bin 181593 -> 196969 bytes
 docs/zusatz.tex | 1825 ++++++++++++-----------------------------------
 2 files changed, 470 insertions(+), 1355 deletions(-)

diff --git a/docs/zusatz.pdf b/docs/zusatz.pdf
index ab5f25e3908112fdef2945c19e1ef8848df8a750..76070198a96a4c6ec5fec9e8400656669d46863b 100644
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zumHK2Nk<ovwT^rxn~-7#<3L3u6Tjeg+3I_c$?B_&Cm*OpT_O`tu@ta+>fLif&(NzD
zgLg}5NUR(=Gy`H>ea_k(f_&pW;!W0}R&zO-wYgx2cxq+XfX<mdI8HcDjB+FS;2AJ~
zgGfk7BvrzXq8^Oh)u7}`Cnvn@y~H0<&9%_>T1deE$A)cFaItp($25=8cI-fe@c!FK
zGJz3p+AZH&&Zl*?3)KsDMy>i~!@$02^CErt*Cgc}txc68-}+YdBG7t?YF5i9bJr;m
z^KD-m^og%76`ZCOCkd;8y&aRH{J}k&BBe-{?!+P5sV+5~=9U|MtG>NJOz8xS9A1s|
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gt0~vJ0_l4N<0*om;P?ofjO_n(qR7a^6vW~F3nX~_fdBvi

diff --git a/docs/zusatz.tex b/docs/zusatz.tex
index 5873f54..08ed7e1 100644
--- a/docs/zusatz.tex
+++ b/docs/zusatz.tex
@@ -17,23 +17,23 @@
 %%    |
 %%    ---- 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;
 %%        |
@@ -62,1272 +62,6 @@
 %% 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{<sectionclassname>}
-%%	{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
 
@@ -1482,9 +216,8 @@ gelten.
     \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
+    Betrachten wir die $\reell$-Vektorräume $U:=\reell^{4}$ und $V:=\reell^{2}$
+    und die Vektoren $u_{1},u_{2},u_{3}\in U$ und $v_{1},v_{2},v_{3}\in V$
 
         \begin{mathe}[mc]{ccc}
             u_{1} = \begin{svector} 0\\ 0\\ 1\\ 1\\\end{svector},
@@ -1557,7 +290,7 @@ gelten.
                 \end{mathe}
 
             gelten. Insbesondere sind Bedingungen {i) + ii)} erfüllt.
-            Also lautet wie oben argumentiert, die Antwort auf die Originalfrage: »Ja«!
+            Also lautet wie oben argumentiert, die Antwort auf die Originalfrage: \fbox{Ja}!
         \end{soln*}
 
     \begin{qstn}
@@ -1583,11 +316,11 @@ gelten.
             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«,
+            Also lauten die Antworten auf \textbf{b)}, \textbf{d)} \fbox{Nein},
             und da es mindestens eine lineare Ausdehnung existiert,
-            lautet die Antwort auf b') und d') »Ja«.
+            lautet die Antwort auf \textbf{b')} und \textbf{d')} \fbox{Ja}.
 
-            Es bleiben nur noch c) und c') zu bestimmen.
+            Es bleiben nur noch \textbf{c)} und \textbf{c')} zu bestimmen.
             Sei ${\phi:U\to V}$ eine lineare Ausdehnung von i)--iii).
             Dann wegen Bedingungen {i) + ii)} und Linearität gilt
 
@@ -1600,15 +333,16 @@ gelten.
                 \end{mathe}
 
             Die letzte Gleichung gilt, weil $\{v_{1},v_{2}\}$
-            linear unabhängig ist\footnote{
+            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,
+            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«.
+            Darum lautet die Antwort auf \textbf{c)} \fbox{Ja} und auf \textbf{c')} \fbox{Nein}.
         \end{soln*}
 
 %% AUFGABE 2
@@ -1619,9 +353,8 @@ gelten.
     \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
+    Betrachten wir die $\reell$-Vektorräume $U:=\reell^{4}$ und $V:=\reell^{2}$
+    und die Vektoren $u_{1},u_{2},u_{3}\in U$ und $v_{1},v_{2},v_{3}\in V$
 
         \begin{mathe}[mc]{cccc}
             u_{1} = \begin{svector} 0\\ 0\\ 1\\ 1\\\end{svector},
@@ -1672,7 +405,8 @@ gelten.
             weil wegen der o.\,s. Verhältnisse {iii) + iv)}
             für lineare Abbildungen automatisch mit erfüllt werden.
 
-            Wir \uline{erweitern} nun die lineare unabhängige Menge $\{u_{1},u_{2}\}$
+            Wir \uline{erweitern} nun die lineare unabhängige Menge
+                $\{u_{1},u_{2}\}$
             zu einer Basis
                 $\{u_{1},u_{2},u'_{3},u'_{4}\}$
             von $U$.
@@ -1692,7 +426,7 @@ gelten.
                 \end{mathe}
 
             gelten. Insbesondere sind Bedingungen {i) + ii)} erfüllt.
-            Also lautet wie oben argumentiert, die Antwort auf die Originalfrage: »Ja«!
+            Also lautet wie oben argumentiert, die Antwort auf die Originalfrage: \fbox{Ja}!
         \end{soln*}
 
     \begin{qstn}
@@ -1716,15 +450,15 @@ gelten.
             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«,
+            Also lauten die Antworten auf \textbf{b)}, \textbf{d)} \fbox{Nein},
             und da es mindestens eine lineare Ausdehnung existiert,
-            lautet die Antwort auf b') und d') »Ja«.
+            lautet die Antwort auf \textbf{b')} und \textbf{d')} \fbox{Ja}.
 
-            Es bleiben nur noch c) und c') zu bestimmen.
+            Es bleiben nur noch \textbf{c)} und \textbf{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 $v'_{3}:=\begin{svector} 1\\ 0\\\end{svector}$
+            Zu \textbf{c)} wähle bspw. $v'_{3}:=\begin{svector} 1\\ 0\\\end{svector}$
             und $v'_{4}:=\zerovector$
             und sei ${\phi_{1}:U\to V}$ die lineare Abbildung
             im o.\,s. Beweis mit diesen Vektoren in der Konstruktion.
@@ -1736,25 +470,17 @@ gelten.
                         &= &\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}\}\\
+                        &\supseteq &\vectorspacespan\{v_{1},v'_{3}\}\\
+                        &= &V\\
                 \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«.
+            Die letzte Gleichung gilt,
+            weil \uline{per Wahl} $\{v_{1},v'_{3}\}$ linear unabhängig ist
+            und somit eine Basis des $2$-dimenionalen Vektorraums, $V$ ist.
+            Da $\range(\phi_{1})\supseteq V$, ist $\phi_{1}$ surjektiv.
+            Die Antwort auf \textbf{c)} lautet also \fbox{Ja}.
 
-            Zu c') wähle $v'_{3},v'_{4}:=\zerovector$
+            Zu \textbf{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
@@ -1765,13 +491,13 @@ gelten.
                     =\dim(\vectorspacespan\{v_{1}\})
                     \leq 1
                 $,
-            da \uline{per Wahl} $v_{2},v_{3},v_{4}\in\vectorspacespan(\{v_{1}\})$
+            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«.
+            Die Antwort auf \textbf{c')} lautet also \fbox{Ja}.
         \end{soln*}
 
 %% AUFGABE 3
@@ -1782,9 +508,8 @@ gelten.
     \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
+    Betrachten wir die $\reell$-Vektorräume $U:=\reell^{2}$ und $V:=\reell^{4}$
+    und die Vektoren $u_{1},u_{2},u_{3}\in U$ und $v_{1},v_{2},v_{3}\in V$
 
         \begin{mathe}[mc]{ccc}
             u_{1} = \begin{svector} 1\\ 1\\\end{svector},
@@ -1829,7 +554,7 @@ gelten.
             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«.
+            Darum lautet die Antwort \fbox{Nein}.
             Es gibt keine lineare Ausdehnung.
         \end{soln*}
 
@@ -1841,9 +566,8 @@ gelten.
     \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
+    Betrachten wir die $\reell$-Vektorräume $U:=\reell^{2}$ und $V:=\reell^{4}$
+    und die Vektoren $u_{1},u_{2},u_{3}\in U$ und $v_{1},v_{2},v_{3}\in V$
 
         \begin{mathe}[mc]{ccc}
             u_{1} = \begin{svector} 1\\ 1\\\end{svector},
@@ -1905,7 +629,7 @@ gelten.
 
             gelten. Insbesondere sind Bedingung {i) + iii)} erfüllt.
             Also lautet wie oben argumentiert,
-            die Antwort auf die Originalfrage: »Ja«!
+            die Antwort auf die Originalfrage: \fbox{Ja}!
         \end{soln*}
 
     \begin{qstn}
@@ -1929,20 +653,20 @@ gelten.
             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«,
+            Also lauten die Antworten auf \textbf{c)}, \textbf{d)} \fbox{Nein},
             und da es mindestens eine lineare Ausdehnung existiert,
-            lautet die Antwort auf c') und d') »Ja«.
+            lautet die Antwort auf \textbf{c')} und \textbf{d')} \fbox{Ja}.
 
-            Es bleiben nur noch b) und b') zu bestimmen.
-            Sei $\phi$ eine lineare Ausdehnung, die i)---iii) erfüllt.
-            Dann wegen Linearität von $\phi_{1}$
+            Es bleiben nur noch \textbf{b)} und \textbf{b')} zu bestimmen.
+            Sei $\phi$ eine lineare Ausdehnung, die i)--iii) erfüllt.
+            Dann wegen Linearität von $\phi$
             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})\}
+                    \range(\phi)
+                        &= &\phi(\vectorspacespan\{u_{1},u_{3}\})
+                        &= &\vectorspacespan\{\phi(u_{1}),\phi(u_{3})\}
                         &= &\vectorspacespan\{v_{1},v_{3}\}
                 \end{mathe}
 
@@ -1963,8 +687,8 @@ gelten.
             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«.
+            Darum lautet die Antwort auf \textbf{b)} \fbox{Nein}
+            und auf \textbf{b')} \fbox{Ja}.
         \end{soln*}
 
 %% AUFGABE 5
@@ -1975,9 +699,8 @@ gelten.
     \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
+    Betrachten wir die $\reell$-Vektorräume $U:=\reell^{2}$ und $V:=\reell^{4}$
+    und die Vektoren $u_{1},u_{2},u_{3}\in U$ und $v_{1},v_{2},v_{3}\in V$
 
         \begin{mathe}[mc]{ccc}
             u_{1} = \begin{svector} 1\\ 1\\\end{svector},
@@ -2040,7 +763,7 @@ gelten.
 
             gelten. Insbesondere sind Bedingung {i)} erfüllt.
             Also lautet wie oben argumentiert,
-            die Antwort auf die Originalfrage: »Ja«!
+            die Antwort auf die Originalfrage: \fbox{Ja}!
         \end{soln*}
 
     \begin{qstn}
@@ -2064,26 +787,26 @@ gelten.
             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«,
+            Also lauten die Antworten auf \textbf{c)}, \textbf{d)} \fbox{Nein},
             und da es mindestens eine lineare Ausdehnung existiert,
-            lautet die Antwort auf c') und d') »Ja«.
+            lautet die Antwort auf \textbf{c')} und \textbf{d')} \fbox{Ja}.
 
-            Es bleiben nur noch b) und b') zu bestimmen.
+            Es bleiben nur noch \textbf{b)} und \textbf{b')} zu bestimmen.
             Beachte, dass in der Konstruktion von $\phi$ im o.\,s. Beweis
             wir $v'_{2}$ beliebig auswählen konnten.
 
-            Zu b) wähle $v'_{2}:=\begin{svector} 0\\ 1\\ 0\\ 0\\\end{svector}$
+            Zu \textbf{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}
+                \begin{mathe}[mc]{rcccccccl}
                     \range(\phi_{1})
                         &= &\phi_{1}(\vectorspacespan\{u_{1},u'_{2}\})
                         &= &\vectorspacespan\{\phi_{1}(u_{1}),\phi_{1}(u'_{2})\}
                         &= &\vectorspacespan\{v_{1},v'_{2}\}
+                        &= &V,
                 \end{mathe}
 
             und damit
@@ -2098,9 +821,9 @@ gelten.
             Folglich ist $\phi_{1}$
             laut \cite[Korollar~6.3.15(1)]{sinn2020}
             injektiv.
-            Die Antwort auf b) lautet also »Ja«.
+            Die Antwort auf \textbf{b)} lautet also \fbox{Ja}.
 
-            Zu b') wähle $v'_{2}:=\zerovector$
+            Zu \textbf{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
@@ -2109,15 +832,14 @@ gelten.
                     \textoverset{Defn}{=}\dim(\range(\phi_{2}))
                     =\dim(\vectorspacespan\{v_{1},v'_{2}\})
                     =\dim(\vectorspacespan\{v_{1}\})
-                    =1
+                    \leq 1
                 $,
-            da \uline{per Wahl} $v'_{2}\in\vectorspacespan(\{v_{1}\})$
-            und $v_{1}\neq\zerovector$.
+            da \uline{per Wahl} $v'_{2}\in\vectorspacespan\{v_{1}\}$.
             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«.
+            Die Antwort auf \textbf{b')} lautet also \fbox{Ja}.
         \end{soln*}
 
 %% AUFGABE 6
@@ -2128,31 +850,424 @@ gelten.
     \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.
-}
+    Betrachten wir die $\reell$-Vektorräume $U=V:=\reell^{4}$
+    und die Vektoren $u_{1},u_{2},u_{3},u_{4}\in U$ und $v_{1},v_{2},v_{3},v_{4}\in V$
+
+        \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} 0\\ 3\\ 0\\ 0\\\end{svector},
+            &v_{2} = \begin{svector} 1\\ 3\\ 0\\ 0\\\end{svector},
+            &v_{3} = \begin{svector} 10\\ 0\\ 0\\ 0\\\end{svector},
+            &v_{4} = \begin{svector} 1\\ 6\\ 0\\ 0\\\end{svector}.\\
+        \end{mathe}
+
+    \begin{qstn}
+        \makelabel{qstn:6:ch:lin-ext}
+        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*}
+            Zunächst beobachte, dass $\{u_{1},u_{2}\}$ linear unabhängig ist,
+            und dass $u_{3},u_{4}\in\vectorspacespan\{u_{1},u_{2}\}$,
+            da ${u_{3}=10u_{2}-10u_{1}}$ und ${u_{4}=u_{1}+u_{2}}$.
+            Beachte auch, dass sich diese Verhältnisse in den Outputvektoren wiederspiegeln:
+
+                \begin{mathe}[mc]{rcccl}
+                    10v_{2}-10v_{1}
+                        &= &\begin{svector} 10\\ 0\\ 0\\ 0\\\end{svector} &= &v_{3},\\
+                    v_{1}+v_{2}
+                        &= &\begin{svector} 1\\ 6\\ 0\\ 0\\\end{svector} &= &v_{4}.\\
+                \end{mathe}
+
+            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.
+
+            \uline{Erweitere} nun die linear 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: \fbox{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 $\dim(U)=\dim(V)$,
+            sind \textbf{b)}, \textbf{c)}, \textbf{d)} äquivalent
+            und genauso sind \textbf{b')}, \textbf{c')}, \textbf{d)} äquivalent,
+            da für lineare Abbildung zwischen endlichdimensionalen Räumen gleicher Dimensionen
+                Injektivität, Surjektivität, und Bijektivität
+            äquivalent sind.
+            Darum reicht es aus, nur \textbf{c)} und \textbf{c')} zu behandeln.
+
+            Zu \textbf{c)}, da $\{v_{1},v_{2}\}\subseteq V$ linear unabhängig sind,
+            wähle $v'_{3},v'_{4}\in V$ so,
+            dass $\{v_{1},v_{2},v'_{3},v'_{4}\}$ eine Basis
+            des $4$-dimensionalen Raums, $V$, bildet.
+            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 $\range(\phi_{1})=V$,
+            da \uline{per Wahl} $\{v_{1},v_{2},v'_{3},v'_{4}\}$ eine Basis von $V$ ist.
+            Folglich ist $\phi_{1}$ surjektiv.
+            Die Antwort auf \textbf{c)} (und \textbf{b)} und \textbf{d)}), lautet also \fbox{Ja}.
+
+            Zu \textbf{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},v_{2}\})
+                    \leq 2
+                $,
+            da \uline{per Wahl} $v'_{3},v'_{4}\in\vectorspacespan\{v_{1},v_{2}\}$.
+            Also, $\rank(\phi_{2})<4=\dim(V)$.
+            Folglich ist $\phi_{2}$
+            laut \cite[Korollar~6.3.15(2)]{sinn2020}
+            nicht-surjektiv.
+            Die Antwort auf \textbf{c')} (und \textbf{b')} und \textbf{d')}) lautet also \fbox{Ja}.
+        \end{soln*}
 
 %% AUFGABE 7
+\clearpage
 \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.
-}
+    Betrachten wir die $\reell$-Vektorräume $U=V:=\reell^{4}$
+    und die Vektoren $u_{1},u_{2},u_{3},u_{4}\in U$ und $v_{1},v_{2},v_{3},v_{4}\in V$
+
+        \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} 0\\ 3\\ 0\\ 0\\\end{svector},
+            &v_{2} = \begin{svector} 0\\ 6\\ 0\\ 0\\\end{svector},
+            &v_{3} = \begin{svector} 0\\ 30\\ 0\\ 0\\\end{svector},
+            &v_{4} = \begin{svector} 0\\ 9\\ 0\\ 0\\\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*}
+            Zunächst beobachte, dass $\{u_{1},u_{2}\}$ linear unabhängig ist,
+            und dass $u_{3},u_{4}\in\vectorspacespan\{u_{1},u_{2}\}$,
+            da ${u_{3}=10u_{2}-10u_{1}}$ und ${u_{4}=u_{1}+u_{2}}$.
+            Beachte auch, dass sich diese Verhältnisse in den Outputvektoren wiederspiegeln:
+
+                \begin{mathe}[mc]{rcccl}
+                    10v_{2}-10v_{1}
+                        &= &\begin{svector} 0\\ 30\\ 0\\ 0\\\end{svector} &= &v_{3},\\
+                    v_{1}+v_{2}
+                        &= &\begin{svector} 0\\ 9\\ 0\\ 0\\\end{svector} &= &v_{4}.\\
+                \end{mathe}
+
+            Der Rest dieser Aufgabe lässt sich nun genauso wie
+            bei \Cref{qstn:6:ch:lin-ext} erledigen.
+            Die Antwort hier lautet also wieder: \fbox{Ja},
+            es gibt eine lineare Ausdehnung, die i)--iv) erfüllt.
+        \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 $\dim(U)=\dim(V)$,
+            sind \textbf{b)}, \textbf{c)}, \textbf{d)} äquivalent
+            und genauso sind \textbf{b')}, \textbf{c')}, \textbf{d)} äquivalent,
+            da für lineare Abbildung zwischen endlichdimensionalen Räumen gleicher Dimensionen
+                Injektivität, Surjektivität, und Bijektivität
+            äquivalent sind.
+            Darum reicht es aus, nur \textbf{b)} und \textbf{b')} zu behandeln.
+
+            Sei nun ${\phi:U\to V}$ eine beliebige lineare Abbildung,
+            die i)--iv) erfüllt (laut der letzten Aufgabe existiert mindestens eine).
+            Da $\{u_{1},u_{2}\}$ linear unabhängig ist,
+            können wir dies zu einer Basis
+                $\{u_{1},u_{2},u'_{3},u'_{4}\}$
+            von $U$ erweitern.
+            Da $\phi$ eine Ausdehnung und linear ist,
+            gilt
+
+                \begin{mathe}[mc]{rcl}
+                    \range(\phi)
+                        &= &\phi(\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},\phi(u'_{3}),\phi(u'_{4})\}.\\
+                \end{mathe}
+
+            Darum gilt
+                $%
+                    \rank(\phi)
+                    \textoverset{Defn}{=}\dim(\range(\phi))
+                    =\dim(\vectorspacespan\{v_{1},v_{2},\phi(u'_{3}),\phi(u'_{4})\})
+                    =\dim(\vectorspacespan\{v_{1},\phi(u'_{3}),\phi(u'_{4})\})
+                    \leq 3
+                $,
+            da $v_{2}\in\vectorspacespan\{v_{1}\}$.
+            Also, $\rank(\phi)<4=\dim(V)$.
+            Folglich ist $\phi$
+            laut \cite[Korollar~6.3.15(2)]{sinn2020}
+            nicht-surjektiv.
+            Da $\phi$ beliebig war,
+            haben wir tatsächlich gezeigt,
+            dass alle lineare Ausdehnungen von i)--iv) nicht-surjektiv sind.
+            Darum lautet die Antwort auf
+                \textbf{c)} (und \textbf{b)} und \textbf{d)}) \fbox{Nein}
+            und auf \textbf{c')} (und \textbf{b')} und \textbf{d')}) \fbox{Ja}.
+        \end{soln*}
 
 %% AUFGABE 8
+\clearpage
 \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.
-}
+    Betrachten wir die $\reell$-Vektorräume $U=V:=\reell^{3}$
+    und die Vektoren $u_{1},u_{2},u_{3},u_{4}\in U$ und $v_{1},v_{2},v_{3},v_{4}\in V$
+
+        \begin{mathe}[mc]{cccc}
+            u_{1} = \begin{svector} 0\\ 0\\ 1\\\end{svector},
+            &u_{2} = \begin{svector} 3\\ 0\\ 1\\\end{svector},
+            &u_{3} = \begin{svector} 30\\ 2\\ 0\\\end{svector},
+            &u_{4} = \begin{svector} 0\\ 2\\ 0\\\end{svector},\\
+        \end{mathe}
+
+        \begin{mathe}[mc]{cccc}
+            v_{1} = \begin{svector} 0\\ 3\\ 0\\\end{svector},
+            &v_{2} = \begin{svector} 1\\ 3\\ 0\\\end{svector},
+            &v_{3} = \begin{svector} 10\\ 0\\ 0\\\end{svector},
+            &v_{4} = \begin{svector} 1\\ 1\\ 1\\\end{svector}.\\
+        \end{mathe}
+
+    \begin{qstn}
+        \makelabel{ex:8:ch:lin-ext}
+        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*}
+            Zunächst beobachte, dass $\{u_{1},u_{2},u_{4}\}$ linear unabhängig ist,
+            und dass $u_{3}\in\vectorspacespan\{u_{1},u_{2},u_{4}\}$,
+            da ${u_{3}=10u_{2}-10u_{1}+u_{4}}$.
+            Beachte auch, dass sich dieses Verhältnis in den Outputvektoren wiederspiegelt:
+
+                \begin{mathe}[mc]{rcccl}
+                    10v_{2}-10v_{1}+v_{4}
+                        &= &\begin{svector} 11\\ 1\\ 1\\\end{svector} &= &v_{3}.\\
+                \end{mathe}
+
+            Darum können wir die Frage auf Bedingungen {i)--iii)} reduzieren,
+            weil wegen des o.\,s. Verhältnisses {iv)}
+            für lineare Abbildungen automatisch mit erfüllt wird.
+
+            Hier müssen wir nun im Gegensatz zu den anderen Aufgaben \uline{nichts hinzufügen}!
+            Da $\{u_{1},u_{2},u_{3}\}$ linear unabhängig ist,
+            bildet dies bereits eine Basis des $3$-dimenionalen Raums $U$.
+            Da $\{u_{1},u_{2},u_{3}\}$ eine Basis von $U$ ist
+            und $v_{1},v_{2},v_{3}\in V$,
+            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}
+                \end{mathe}
+
+            gelten. Insbesondere sind Bedingungen {i)--iii)} erfüllt.
+            Also lautet wie oben argumentiert, die Antwort auf die Originalfrage: \fbox{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 $\dim(U)=\dim(V)$,
+            sind \textbf{b)}, \textbf{c)}, \textbf{d)} äquivalent
+            und genauso sind \textbf{b')}, \textbf{c')}, \textbf{d)} äquivalent,
+            da für lineare Abbildung zwischen endlichdimensionalen Räumen gleicher Dimensionen
+                Injektivität, Surjektivität, und Bijektivität
+            äquivalent sind.
+            Darum reicht es aus, nur \textbf{c)} und \textbf{c')} zu behandeln.
+
+            Sei nun ${\phi:U\to V}$ eine beliebige lineare Abbildung,
+            die i)--iv) erfüllt (laut der letzten Aufgabe existiert mindestens eine).
+            Da $\phi$ eine Ausdehnung und linear ist
+            und da $\{u_{1},u_{2},u_{3}\}$ eine Basis von $U$ ist,
+            gilt
+
+                \begin{mathe}[mc]{rcl}
+                    \range(\phi)
+                        &= &\phi(\vectorspacespan\{u_{1},u_{2},u_{3}\})\\
+                        &= &\vectorspacespan\{\phi(u_{1}),\phi(u_{2}),\phi(u_{3})\}\\
+                        &= &\vectorspacespan\{v_{1},v_{2},v_{3}\}.\\
+                \end{mathe}
+
+            Nun ist $\{v_{1},v_{2},v_{3}\}\subseteq V$ linear unabhängig
+            und bildet somit eine Basis des $3$-dimenionalen Vektorraums, $V$.
+            Darum gilt $\range(\phi)\supseteq V$,
+            sodass $\phi$ surjektiv ist.
+            Da $\phi$ beliebig war,
+            haben wir tatsächlich gezeigt,
+            dass alle lineare Ausdehnungen von i)--iv) surjektiv sind.
+            Darum lautet die Antwort auf
+                \textbf{c)} (und \textbf{b)} und \textbf{d)}) \fbox{Ja}
+            und auf \textbf{c')} (und \textbf{b')} und \textbf{d')}) \fbox{Nein}.
+        \end{soln*}
+
+    \begin{rem}
+        Laut \cite[Satz~6.1.13]{sinn2020}
+        die für \Cref{ex:8:ch:lin-ext} konstruierte lineare Abbildung, $\phi$,
+        eindeutig.
+        Da wir keine freie Wahl trafen, gibt es also \uline{exakt eine}
+        lineare Abbildung, die i)--iv) erfüllt.
+        Darum ist die letzte Frage eigentlich
+            »Ist \uline{die} lineare Ausdehnung injektiv/nicht-injektiv/...?«.
+    \end{rem}
+
+    \begin{rem}
+        Wir hätten die o.\,s. Aufgabe so aufstellen können,
+        dass $\{v_{1},v_{2},v_{3}\}$ linear abhängig wäre.
+        Dann hätte die lineare Ausdehnung den Rang $<3$.
+        Die Antworten auf \textbf{c)}, \textbf{b)}, \textbf{d)}) würden dann »Nein« lauten,
+        und auf \textbf{c')}, \textbf{b')}, \textbf{d')} »Ja«.
+    \end{rem}
 
 %% ********************************************************************************
 %% FILE: back/index.tex