1656 lines
59 KiB
TeX
1656 lines
59 KiB
TeX
%% ********************************************************************************
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%% AUTHOR: Raj Dahya
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%% CREATED: 26. März 2020
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%% EDITED: März-April 2020
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%% TYPE: Notizen
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%% TITLE: Musterlösung Klausur2 WiSe 2020/2021, A5
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%% DOI: —
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%% DEPARTMENT: Fakultät for Mathematik und Informatik
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%% INSTITUTE: Universität Leipzig
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%% ********************************************************************************
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%% ********************************************************************************
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%% DOCUMENT STRUCTURE:
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%% ~~~~~~~~~~~~~~~~~~~
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%%
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%% - root.tex;
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%% |
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%% ---- parameters.tex;
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%% |
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%% ---- srclocal/index.tex;
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%% |
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%% ---- src/setup-type.tex;
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%% |
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%% ---- src/setup-packages.tex;
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%% |
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%% ---- src/setup-parameters.tex;
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%% |
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%% ---- src/setup-macros.tex;
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%% |
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%% ---- src/setup-environments.tex;
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%% |
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%% ---- src/setup-layout.tex;
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%% |
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%% ---- srclocal/setup-locallayout.tex;
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%% |
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%% ---- srclocal/setup-localmacros.tex;
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%% |
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%% ---- body/index.tex;
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%% |
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%% ---- body/A5.tex;
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%% |
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%% ---- body/A5a.tex;
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%% |
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%% ---- body/A5b.tex;
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%%
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%% DOCUMENT-RANDOM-SEED: 5637845
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%% ********************************************************************************
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%% ********************************************************************************
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%% FILE: root.tex
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%% ********************************************************************************
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%% ********************************************************************************
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%% FILE: parameters.tex
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%% ********************************************************************************
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%% ********************************************************************************
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%% FILE: srclocal/index.tex
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%% ********************************************************************************
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\makeatletter
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%% ********************************************************************************
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%% FILE: src/setup-type.tex
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%% ********************************************************************************
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\documentclass[
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10pt,
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a4paper,
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oneside,
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openright,
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center,
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chapterbib,
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||
crosshair,
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||
fleqn,
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||
headcount,
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||
headline,
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||
indent,
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||
indentfirst=false,
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||
portrait,
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||
phonetic,
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||
oldernstyle,
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onecolumn,
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sfbold,
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upper,
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]{scrbook}
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%% ********************************************************************************
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%% FILE: src/setup-packages.tex
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%% ********************************************************************************
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\PassOptionsToPackage{T2A,OT1}{fontenc} % T1,OT1,T2A,OT2
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\PassOptionsToPackage{utf8}{inputenc} % utf8
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\PassOptionsToPackage{british,english,ngerman,russian}{babel}
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||
\PassOptionsToPackage{
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||
english,
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||
ngerman,
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||
russian,
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||
capitalise,
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||
}{cleveref}
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||
\PassOptionsToPackage{
|
||
bookmarks=true,
|
||
bookmarksopen=false,
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||
bookmarksopenlevel=0,
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||
bookmarkstype=toc,
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||
colorlinks=false,
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||
raiselinks=true,
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||
hyperfigures=true,
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}{hyperref}
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\PassOptionsToPackage{
|
||
reset,
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||
left=1in,
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||
right=1in,
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||
top=20mm,
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||
bottom=20mm,
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||
heightrounded,
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||
}{geometry}
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||
\PassOptionsToPackage{
|
||
framemethod=TikZ,
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||
}{mdframed}
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||
\PassOptionsToPackage{normalem}{ulem}
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||
\PassOptionsToPackage{
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amsmath,
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||
thmmarks,
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||
}{ntheorem}
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||
\PassOptionsToPackage{table}{xcolor}
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||
\PassOptionsToPackage{
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all,
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||
color,
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||
curve,
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||
frame,
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||
import,
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||
knot,
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||
line,
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||
movie,
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||
rotate,
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||
textures,
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||
tile,
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||
tips,
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||
web,
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||
xdvi,
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||
}{xy}
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||
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\usepackage{amsfonts}
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\usepackage{amsmath}
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\usepackage{amssymb}
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\usepackage{ntheorem} % <— muss nach den ams* Packages vorkommen!!
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\usepackage{array}
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\usepackage{babel}
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\usepackage{bbding}
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\usepackage{bbm}
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\usepackage{calc}
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\usepackage{sectsty}
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\usepackage{titlesec}
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\usepackage{fancyhdr}
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||
\usepackage{footmisc}
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\usepackage{geometry}
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\usepackage{graphicx}
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\usepackage{ifpdf}
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||
\usepackage{ifthen}
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||
\usepackage{ifnextok}
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\usepackage{longtable}
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||
\usepackage{multicol}
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\usepackage{multirow}
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\usepackage{nameref}
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\usepackage{nowtoaux}
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\usepackage{paralist}
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\usepackage{enumerate} %% nach [paralist]
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\usepackage{pgf}
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\usepackage{arydshln} %% nach [pgf]!
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\usepackage{pgfplots}
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\usepackage{proof}
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\usepackage{refcount}
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\usepackage{relsize}
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\usepackage{savesym}
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\usepackage{stmaryrd}
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\usepackage{subfigure}
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\usepackage{yfonts} %% <— Altgotische Fonts
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\usepackage{tikz}
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\usepackage{xy}
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\usepackage{undertilde}
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\usepackage{ulem} %% <– f\"ur besseren \underline-Befehl (\ul)
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\usepackage{xcolor}
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\usepackage{xspace}
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\usepackage{xstring}
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\usepackage{hyperref}
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\usepackage{cleveref} % must vor hyperref geladen werden.
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\pgfplotsset{compat=newest}
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\usetikzlibrary{
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angles,
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arrows,
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||
automata,
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||
calc,
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||
decorations,
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||
decorations.pathmorphing,
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||
decorations.pathreplacing,
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||
math,
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||
positioning,
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||
patterns,
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||
quotes,
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||
snakes,
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||
}
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%% \var ≈ alter Befehl
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%% \xvar ≈ wie das neue Package \var interpretieren soll.
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\savesymbol{Diamond}
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\savesymbol{emptyset}
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\savesymbol{ggg}
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\savesymbol{int}
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\savesymbol{lll}
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||
\savesymbol{RectangleBold}
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\savesymbol{langle}
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\savesymbol{rangle}
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\savesymbol{hookrightarrow}
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\savesymbol{hookleftarrow}
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\savesymbol{Asterisk}
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\usepackage{mathabx}
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\usepackage{wasysym}
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||
\let\varemptyset=\emptyset
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\restoresymbol{x}{Diamond}
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\restoresymbol{x}{emptyset}
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||
\restoresymbol{x}{ggg}
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\restoresymbol{x}{int}
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||
\restoresymbol{x}{lll}
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\restoresymbol{x}{RectangleBold}
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\restoresymbol{x}{langle}
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||
\restoresymbol{x}{rangle}
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||
\restoresymbol{x}{hookrightarrow}
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||
\restoresymbol{x}{hookleftarrow}
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||
\restoresymbol{x}{Asterisk}
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||
|
||
\ifpdf
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||
\usepackage{pdfcolmk}
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||
\fi
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|
||
\usepackage{mdframed}
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||
|
||
%% Force-Import aus MnSymbol
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\DeclareFontFamily{U}{MnSymbolA}{}
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\DeclareFontShape{U}{MnSymbolA}{m}{n}{
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<-6> MnSymbolA5
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<6-7> MnSymbolA6
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<7-8> MnSymbolA7
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<8-9> MnSymbolA8
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<9-10> MnSymbolA9
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<10-12> MnSymbolA10
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<12-> MnSymbolA12
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}{}
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\DeclareFontShape{U}{MnSymbolA}{b}{n}{
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<-6> MnSymbolA-Bold5
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<6-7> MnSymbolA-Bold6
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<7-8> MnSymbolA-Bold7
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<8-9> MnSymbolA-Bold8
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<9-10> MnSymbolA-Bold9
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<10-12> MnSymbolA-Bold10
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<12-> MnSymbolA-Bold12
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}{}
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\DeclareSymbolFont{MnSyA}{U}{MnSymbolA}{m}{n}
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\DeclareMathSymbol{\lcirclearrowright}{\mathrel}{MnSyA}{252}
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\DeclareMathSymbol{\lcirclearrowdown}{\mathrel}{MnSyA}{255}
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\DeclareMathSymbol{\rcirclearrowleft}{\mathrel}{MnSyA}{250}
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\DeclareMathSymbol{\rcirclearrowdown}{\mathrel}{MnSyA}{251}
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\DeclareFontFamily{U}{MnSymbolC}{}
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\DeclareSymbolFont{MnSyC}{U}{MnSymbolC}{m}{n}
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\DeclareFontShape{U}{MnSymbolC}{m}{n}{
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<-6> MnSymbolC5
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<6-7> MnSymbolC6
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<7-8> MnSymbolC7
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<8-9> MnSymbolC8
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<9-10> MnSymbolC9
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<10-12> MnSymbolC10
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<12-> MnSymbolC12%
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}{}
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\DeclareMathSymbol{\powerset}{\mathord}{MnSyC}{180}
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%% ********************************************************************************
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%% FILE: src/setup-parameters.tex
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%% ********************************************************************************
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\def\boolwahr{true}
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\def\boolfalsch{false}
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||
\def\boolleer{}
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||
\let\documenttwosided\boolfalsch
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||
\let\boolinappendix\boolfalsch
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\let\boolinmdframed\boolfalsch
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||
\let\eqtagset\boolfalsch
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\let\eqtaglabel\boolleer
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||
\let\eqtagsymb\boolleer
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||
\newcount\bufferctr
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||
\newcount\bufferreplace
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||
\newcounter{columnanzahl}
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||
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||
\newlength\rtab
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||
\newlength\gesamtlinkerRand
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||
\newlength\gesamtrechterRand
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||
\newlength\ownspaceabovethm
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||
\newlength\ownspacebelowthm
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||
\setlength{\rtab}{0.025\textwidth}
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||
\setlength{\ownspaceabovethm}{0.5\baselineskip}
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||
\setlength{\ownspacebelowthm}{0.5\baselineskip}
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||
\setlength{\gesamtlinkerRand}{0pt}
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||
\setlength{\gesamtrechterRand}{0pt}
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\def\secnumberingpt{$\cdot$}
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||
\def\secnumberingseppt{.}
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||
\def\subsecnumberingseppt{}
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||
\def\thmnumberingpt{$\cdot$}
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||
\def\thmnumberingseppt{}
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||
\def\thmForceSepPt{.}
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||
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||
\definecolor{leer}{gray}{1}
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||
\definecolor{hellgrau}{gray}{0.85}
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||
\definecolor{dunkelgrau}{gray}{0.5}
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||
\definecolor{maroon}{rgb}{0.6901961,0.1882353,0.3764706}
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||
\definecolor{dunkelgruen}{rgb}{0.015625,0.363281,0.109375}
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||
\definecolor{dunkelrot}{rgb}{0.5450980392,0,0}
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||
\definecolor{dunkelblau}{rgb}{0,0,0.5450980392}
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||
\definecolor{blau}{rgb}{0,0,1}
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||
\definecolor{newresult}{rgb}{0.6,0.6,0.6}
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||
\definecolor{improvedresult}{rgb}{0.9,0.9,0.9}
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||
\definecolor{hervorheben}{rgb}{0,0.9,0.7}
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||
\definecolor{starkesblau}{rgb}{0.1019607843,0.3176470588,0.8156862745}
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||
\definecolor{achtung}{rgb}{1,0.5,0.5}
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||
\definecolor{frage}{rgb}{0.5,1,0.5}
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||
\definecolor{schreibweise}{rgb}{0,0.7,0.9}
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||
\definecolor{axiom}{rgb}{0,0.3,0.3}
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||
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%% ********************************************************************************
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%% FILE: src/setup-macros.tex
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%% ********************************************************************************
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%% ****************************************************************
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%% TEX:
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||
%% ****************************************************************
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\def\let@name#1#2{\expandafter\let\csname #1\expandafter\endcsname\csname #2\endcsname\relax}
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||
\DeclareRobustCommand\crfamily{\fontfamily{ccr}\selectfont}
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||
\DeclareTextFontCommand{\textcr}{\crfamily}
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\def\nichtzeigen#1{\phantom{#1}}
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%% ****************************************************************
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%% SPACING:
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||
%% ****************************************************************
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\def\ifthenelseleer#1#2#3{\ifthenelse{\equal{#1}{}}{#2}{#1#3}}
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||
\def\bedingtesspaceexpand#1#2#3{\ifthenelseleer{\csname #1\endcsname}{#3}{#2#3}}
|
||
\def\voritemise{\leavevmode\nvraum{1}}
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||
\def\hraum{\null\hfill\null}
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||
\def\vraum{\null\vfill\null}
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||
\def\nvraum{\@ifnextchar\bgroup{\nvraum@c}{\nvraum@bes}}
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\def\nvraum@c#1{\vspace*{-#1\baselineskip}}
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\def\nvraum@bes{\vspace*{-\baselineskip}}
|
||
\def\erlaubeplatz{\relax\ifmmode\else\@\xspace\fi}
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||
\def\entferneplatz{\relax\ifmmode\else\expandafter\@gobble\fi}
|
||
\def\forceindent{\hspace*{20pt}} %% * nötig, damit am Anfang/Ende einer Zeile nicht ignoriert wird
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||
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%% ****************************************************************
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||
%% TAGS / BEZEICHNUNGEN / LABELLING:
|
||
%% ****************************************************************
|
||
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||
\def\send@toaux#1{\@bsphack\protected@write\@auxout{}{\string#1}\@esphack}
|
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|
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%% \rlabel{LABEL}[CTR]{CREF-SHORT}{CREF-LONG}{DISPLAYTEXT}
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||
\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}
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||
\def\tag@rawscheme#1#2[#3]#4#5{\@ifnextchar[{\tag@rawscheme@{#1}{#2}[#3]{#4}{#5}}{\tag@rawscheme@{#1}{#2}[#3]{#4}{#5}[*]}}
|
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\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}%
|
||
}{%
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||
\refstepcounter{#3}#4%
|
||
\ifthenelse{\equal{#7}{\boolleer}}{\rlabel{#6}[#3]{#1}{#2}{\csname the#3\endcsname}}{\rlabel{#6}[#3]{#1}{#2}{#7}}%
|
||
#5%
|
||
}%
|
||
}
|
||
%% \tag@scheme{CREF-SHORT}{CREF-LONG}[CTR] [LABEL]{DISPLAYTEXT}
|
||
\def\tag@scheme#1#2[#3]{\tag@rawscheme{#1}{#2}[#3]{\upshape(}{\upshape)}}
|
||
|
||
%% \eqtag[LABEL]{DISPLAYTEXT}
|
||
\def\eqtag@post#1{\makebox[0pt][r]{#1}}
|
||
\def\eqtag@pre{\tag@scheme{Eq}{Equation}[Xe]}
|
||
\def\eqtag{\@ifnextchar[{\eqtag@}{\eqtag@[*]}}
|
||
\def\eqtag@[#1]{\@ifnextchar\bgroup{\eqtag@@[#1]}{\eqtag@@[#1]{}}}
|
||
\def\eqtag@@[#1]#2{\eqtag@post{\eqtag@pre[#1]{#2}}}
|
||
|
||
\def\eqcref#1{\text{(\ref{#1})}}
|
||
\def\ptcref#1{\ref{#1}}
|
||
\def\punktlabel#1{\label{it:#1:\beweislabel}}
|
||
\def\punktcref#1{\eqcref{it:#1:\beweislabel}}
|
||
\def\crefit#1#2{\cref{#1}~\eqcref{it:#2:#1}}
|
||
\def\Crefit#1#2{\Cref{#1}~\eqcref{it:#2:#1}}
|
||
|
||
%% UNDER/OVERSET BEFEHLE
|
||
\def\opfromto[#1]_#2^#3{\underset{#2}{\overset{#3}{#1}}}
|
||
\def\textoverset#1#2{\overset{\text{#1}}{#2}}
|
||
\def\textunderset#1#2{\underset{#2}{\text{#1}}}
|
||
\def\crefoverset#1#2{\textoverset{\cref{#1}}{#2}}
|
||
\def\Crefoverset#1#2{\textoverset{\Cref{#1}}{#2}}
|
||
\def\crefunderset#1#2{\textunderset{#2}{\cref{#1}}}
|
||
\def\Crefunderset#1#2{\textunderset{#2}{\Cref{#1}}}
|
||
\def\eqcrefoverset#1#2{\textoverset{\eqcref{#1}}{#2}}
|
||
\def\eqcrefunderset#1#2{\textunderset{#2}{\eqcref{#1}}}
|
||
\def\mathclap#1{#1}
|
||
\def\oberunterset#1{\@ifnextchar^{\oberunterset@oben{#1}}{\oberunterset@unten{#1}}}
|
||
\def\oberunterset@oben#1^#2_#3{\underset{\mathclap{#3}}{\overset{\mathclap{#2}}{#1}}}
|
||
\def\oberunterset@unten#1_#2^#3{\underset{\mathclap{#2}}{\overset{\mathclap{#3}}{#1}}}
|
||
\def\breitunderbrace#1_#2{\underbrace{#1}_{\mathclap{#2}}}
|
||
\def\breitoverbrace#1^#2{\overbrace{#1}^{\mathclap{#2}}}
|
||
\def\breitunderbracket#1_#2{\underbracket{#1}_{\mathclap{#2}}}
|
||
\def\breitoverbracket#1^#2{\overbracket{#1}^{\mathclap{#2}}}
|
||
|
||
\def\generatenestedsecnumbering#1#2#3{%
|
||
\expandafter\gdef\csname thelong#3\endcsname{%
|
||
\expandafter\csname the#2\endcsname%
|
||
\secnumberingpt%
|
||
\expandafter\csname #1\endcsname{#3}%
|
||
}%
|
||
\expandafter\gdef\csname theshort#3\endcsname{%
|
||
\expandafter\csname #1\endcsname{#3}%
|
||
}%
|
||
}
|
||
\def\generatenestedthmnumbering#1#2#3{%
|
||
\expandafter\gdef\csname the#3\endcsname{%
|
||
\expandafter\csname the#2\endcsname%
|
||
\thmnumberingpt%
|
||
\expandafter\csname #1\endcsname{#3}%
|
||
}%
|
||
\expandafter\gdef\csname theshort#3\endcsname{%
|
||
\expandafter\csname #1\endcsname{#3}%
|
||
}%
|
||
}
|
||
|
||
%% ****************************************************************
|
||
%% ALLG. MACROS:
|
||
%% ****************************************************************
|
||
|
||
\def\+#1{\addtocounter{#1}{1}}
|
||
\def\setcounternach#1#2{\setcounter{#1}{#2}\addtocounter{#1}{-1}}
|
||
\def\textsubscript#1{${}_{\textup{#1}}$}
|
||
\def\rome#1{\overline{\underline{#1}}}
|
||
\def\textTODO{\text{[{\large\textcolor{red}{More work needed!}}]}}
|
||
\def\hlineEIGENpt{\hdashline[0.5pt/5pt]}
|
||
\def\clineEIGENpt#1{\cdashline{#1}[0.5pt/5pt]}
|
||
|
||
\def\forcepunkt#1{#1\IfEndWith{#1}{.}{}{.}}
|
||
\def\lateinabkuerzung#1#2{%
|
||
\expandafter\gdef\csname #1\endcsname{\emph{#2}\@ifnextchar.{\entferneplatz}{\erlaubeplatz}}
|
||
}
|
||
\def\deutscheabkuerzung#1#2{%
|
||
\expandafter\gdef\csname #1\endcsname{{#2}\@ifnextchar.{\entferneplatz}{\erlaubeplatz}}
|
||
}
|
||
|
||
%% ****************************************************************
|
||
%% MATHE
|
||
%% ****************************************************************
|
||
|
||
\def\matrix#1{\left(\begin{array}{#1}}
|
||
\def\endmatrix{\end{array}\right)}
|
||
\def\smatrix{\left(\begin{smallmatrix}}
|
||
\def\endsmatrix{\end{smallmatrix}\right)}
|
||
|
||
\def\multiargrekursiverbefehl#1#2#3#4#5#6#7#8{%
|
||
\expandafter\gdef\csname#1\endcsname #2##1#4{\csname #1@anfang\endcsname##1#3\egroup}
|
||
\expandafter\def\csname #1@anfang\endcsname##1#3{#5##1\@ifnextchar\egroup{\csname #1@ende\endcsname}{#7\csname #1@mitte\endcsname}}
|
||
\expandafter\def\csname #1@mitte\endcsname##1#3{#6##1\@ifnextchar\egroup{\csname #1@ende\endcsname}{#7\csname #1@mitte\endcsname}}
|
||
\expandafter\def\csname #1@ende\endcsname##1{#8}
|
||
}
|
||
\multiargrekursiverbefehl{svektor}{[}{;}{]}{\begin{smatrix}}{}{\\}{\\\end{smatrix}}
|
||
\multiargrekursiverbefehl{vektor}{[}{;}{]}{\begin{matrix}{c}}{}{\\}{\\\end{matrix}}
|
||
\multiargrekursiverbefehl{vektorzeile}{}{,}{;}{}{&}{}{}
|
||
\multiargrekursiverbefehl{matlabmatrix}{[}{;}{]}{\begin{smatrix}\vektorzeile}{\vektorzeile}{;\\}{;\end{smatrix}}
|
||
|
||
\def\cases[#1]#2{\left\{\begin{array}[#1]{#2}}
|
||
\def\endcases{\end{array}\right.}
|
||
|
||
\def\BeweisRichtung[#1]{\@ifnextchar\bgroup{\@BeweisRichtung@c[#1]}{\@BeweisRichtung@bes[#1]}}
|
||
\def\@BeweisRichtung@bes[#1]{{\bfseries(#1).~}}
|
||
\def\@BeweisRichtung@c[#1]#2#3{{\bfseries(#2#1#3).~}}
|
||
\def\erzeugeBeweisRichtungBefehle#1#2{
|
||
\expandafter\gdef\csname #1text\endcsname##1##2{\BeweisRichtung[#2]{##1}{##2}}
|
||
\expandafter\gdef\csname #1\endcsname{%
|
||
\@ifnextchar\bgroup{\csname #1@\endcsname}{\csname #1text\endcsname{}{}}%
|
||
}
|
||
\expandafter\gdef\csname #1@\endcsname##1##2{%
|
||
\csname #1text\endcsname{\punktcref{##1}}{\punktcref{##2}}%
|
||
}
|
||
}
|
||
\erzeugeBeweisRichtungBefehle{hinRichtung}{$\Longrightarrow$}
|
||
\erzeugeBeweisRichtungBefehle{herRichtung}{$\Longleftarrow$}
|
||
\erzeugeBeweisRichtungBefehle{hinherRichtung}{$\Longleftrightarrow$}
|
||
|
||
\def\cal#1{\mathcal{#1}}
|
||
\def\brkt#1{\langle{}#1{}\rangle}
|
||
\def\mathfrak#1{\mbox{\usefont{U}{euf}{m}{n}#1}}
|
||
\def\kurs#1{\textit{#1}}
|
||
\def\rectangleblack{\text{\RectangleBold}}
|
||
\def\rectanglewhite{\text{\Rectangle}}
|
||
\def\squareblack{\blacksquare}
|
||
\def\squarewhite{\Box}
|
||
|
||
%% ********************************************************************************
|
||
%% FILE: src/setup-environments.tex
|
||
%% ********************************************************************************
|
||
|
||
%% **********************************************************************
|
||
%% CLEVEREF: ************************************************************
|
||
|
||
\def\crefname@full#1#2#3{\crefname{#1}{#2}{#3}\Crefname{#1}{#2}{#3}}
|
||
\crefname@full{chapter}{Kapitel}{Kapitel}
|
||
\crefname@full{section}{Abschnitt}{Abschnitte}
|
||
\crefname@full{figure}{Fig.}{Fig.}
|
||
\crefname@full{subfigure}{Fig.}{Fig.}
|
||
|
||
\crefname@full{proof}{Beweis}{Beweise}
|
||
\crefname@full{thm}{Theorem}{Theoreme}
|
||
\crefname@full{satz}{Satz}{Sätze}
|
||
\crefname@full{claim}{Behauptung}{Behauptungen}
|
||
\crefname@full{lemm}{Lemma}{Lemmata}
|
||
\crefname@full{cor}{Korollar}{Korollarien}
|
||
\crefname@full{folg}{Folgerung}{Folgerungen}
|
||
\crefname@full{prop}{Proposition}{Propositionen}
|
||
\crefname@full{defn}{Definition}{Definitionen}
|
||
\crefname@full{conv}{Konvention}{Konventionen}
|
||
\crefname@full{fact}{Fakt}{Fakten}
|
||
\crefname@full{rem}{Bemerkung}{Bemerkungen}
|
||
\crefname@full{qstn}{Frage}{Fragen}
|
||
\crefname@full{e.g.}{Beipsiel}{Beipsiele}
|
||
|
||
%% ****************************************************************
|
||
%% THEOREME:
|
||
%% ****************************************************************
|
||
|
||
\def\qedEIGEN#1{\@ifnextchar[{\qedEIGEN@c{#1}}{\qedEIGEN@bes{#1}}}%]
|
||
\def\qedEIGEN@bes#1{%
|
||
\bgroup%
|
||
\parfillskip=0pt% % so \par doesnt push \square to left
|
||
\widowpenalty=10000% % so we dont break the page before \square
|
||
\displaywidowpenalty=10000% % ditto
|
||
\finalhyphendemerits=0% % TeXbook exercise 14.32
|
||
\leavevmode% % \nobreak means lines not pages
|
||
\unskip% % remove previous space or glue
|
||
\nobreak% % don’t break lines
|
||
\hfil% % ragged right if we spill over
|
||
\penalty50% % discouragement to do so
|
||
\hskip.2em% % ensure some space
|
||
\null% % anchor following \hfill
|
||
\hfill% % push \square to right
|
||
#1% % the end-of-proof mark
|
||
\par%
|
||
\egroup%
|
||
}
|
||
\def\qedEIGEN@c#1[#2]{%
|
||
\bgroup%
|
||
\parfillskip=0pt% % so \par doesnt push \square to left
|
||
\widowpenalty=10000% % so we dont break the page before \square
|
||
\displaywidowpenalty=10000% % ditto
|
||
\finalhyphendemerits=0% % TeXbook exercise 14.32
|
||
\leavevmode% % \nobreak means lines not pages
|
||
\unskip% % remove previous space or glue
|
||
\nobreak% % don’t break lines
|
||
\hfil% % ragged right if we spill over
|
||
\penalty50% % discouragement to do so
|
||
\hskip.2em% % ensure some space
|
||
\null% % anchor following \hfill
|
||
\hfill% % push \square to right
|
||
{#1~{\smaller\bfseries\upshape (#2)}}%
|
||
\par%
|
||
\egroup%
|
||
}
|
||
\def\qedVARIANT#1#2{
|
||
\expandafter\def\csname ennde#1Sign\endcsname{#2}
|
||
\expandafter\def\csname ennde#1\endcsname{\@ifnextchar[{\qedEIGEN@c{#2}}{\qedEIGEN@bes{#2}}} %]
|
||
}
|
||
\qedVARIANT{OfProof}{$\squareblack$}
|
||
\qedVARIANT{OfWork}{\rectangleblack}
|
||
\qedVARIANT{OfSomething}{$\dashv$}
|
||
\qedVARIANT{OnNeutral}{$\lozenge$} % \lozenge \bigcirc \blacklozenge
|
||
\def\qedsymbol{\enndeOfProofSign}
|
||
\def\proofSymbol{\enndeOfProofSign}
|
||
|
||
\def\ra@pretheoremwork{
|
||
\setlength{\theorempreskipamount}{\ownspaceabovethm}
|
||
}
|
||
\def\rathmtransfer#1#2{
|
||
\expandafter\def\csname #2\endcsname{\csname #1\endcsname}
|
||
\expandafter\def\csname end#2\endcsname{\csname end#1\endcsname}
|
||
}
|
||
|
||
\def\ranewthm#1#2#3[#4]{
|
||
%% FOR \BEGIN{THM}
|
||
\theoremstyle{\current@theoremstyle}
|
||
\theoremseparator{\current@theoremseparator}
|
||
\theoremprework{\ra@pretheoremwork}
|
||
\@ifundefined{#1@basic}{\newtheorem{#1@basic}[#4]{#2}}{\renewtheorem{#1@basic}[#4]{#2}}
|
||
%% FOR \BEGIN{THM}[...]
|
||
\theoremstyle{\current@theoremstyle}
|
||
\theoremseparator{\thmForceSepPt}
|
||
\theoremprework{\ra@pretheoremwork}
|
||
\@ifundefined{#1@withName}{\newtheorem{#1@withName}[#4]{#2}}{\renewtheorem{#1@withName}[#4]{#2}}
|
||
%% FOR \BEGIN{THM*}
|
||
\theoremstyle{nonumberplain}
|
||
\theoremseparator{\thmForceSepPt}
|
||
\theoremprework{\ra@pretheoremwork}
|
||
\@ifundefined{#1@star@basic}{\newtheorem{#1@star@basic}[Xdisplaynone]{#2}}{\renewtheorem{#1@star@basic}[Xdisplaynone]{#2}}
|
||
%% FOR \BEGIN{THM*}[...]
|
||
\theoremstyle{nonumberplain}
|
||
\theoremseparator{\thmForceSepPt}
|
||
\theoremprework{\ra@pretheoremwork}
|
||
\@ifundefined{#1@star@withName}{\newtheorem{#1@star@withName}[Xdisplaynone]{#2}}{\renewtheorem{#1@star@withName}[Xdisplaynone]{#2}}
|
||
%% GENERATE ENVIRONMENTS:
|
||
\umbauenenv{#1}{#3}[#4]
|
||
\umbauenenv{#1@star}{#3}[Xdisplaynone]
|
||
%% TRANSFER *-DEFINITION
|
||
\rathmtransfer{#1@star}{#1*}
|
||
}
|
||
|
||
\def\umbauenenv#1#2[#3]{%
|
||
%% \BEGIN{THM}...
|
||
\expandafter\def\csname #1\endcsname{\relax%
|
||
\@ifnextchar[{\csname #1@\endcsname}{\csname #1@\endcsname[*]}%
|
||
}
|
||
%% \BEGIN{THM}[ANFANG]...
|
||
\expandafter\def\csname #1@\endcsname[##1]{\relax%
|
||
\@ifnextchar[{\csname #1@@\endcsname[##1]}{\csname #1@@\endcsname[##1][*]}%
|
||
}
|
||
%% \BEGIN{THM}[ANFANG][SCHLUSS]
|
||
\expandafter\def\csname #1@@\endcsname[##1][##2]{%
|
||
\ifx*##1%
|
||
\def\enndeOfBlock{\csname end#1@basic\endcsname}
|
||
\csname #1@basic\endcsname%
|
||
\else%
|
||
\def\enndeOfBlock{\csname end#1@withName\endcsname}
|
||
\csname #1@withName\endcsname[##1]%
|
||
\fi%
|
||
\def\makelabel####1{%
|
||
\gdef\beweislabel{####1}%
|
||
\label{\beweislabel}%
|
||
}%
|
||
\ifx*##2%
|
||
\def\enndeSymbol{\qedEIGEN{#2}}
|
||
\else%
|
||
\def\enndeSymbol{\qedEIGEN{#2}[##2]}
|
||
\fi
|
||
}
|
||
%% \END{THM}
|
||
\expandafter\gdef\csname end#1\endcsname{\enndeSymbol\enndeOfBlock}
|
||
}
|
||
|
||
%% NEWTHEOREM EINSTELLUNGSOPTIONEN:
|
||
%% F\"UR \theoremstyle
|
||
%% plain Emulates original LATEX defin, except uses param \theorem...skipamount.
|
||
%% break Header followed by line break.
|
||
%% change Header, Number and Text are interchanged, without a line break.
|
||
%% changebreak =change, but with a line break after Header.
|
||
%% margin Number in left margin, without a line break.
|
||
%% marginbreak =margin, but with a line break after the header.
|
||
%% nonumberplain =plain, without number.
|
||
%% nonumberbreak =break, without number.
|
||
%% empty No number, no name. Only the optional argument is typeset.
|
||
%% \theoremclass \theoremnumbering
|
||
%% \theorempreskip \theorempostkip \theoremindent
|
||
%% \theoremprework \theorempostwork
|
||
|
||
\def\current@theoremstyle{plain}
|
||
\def\current@theoremseparator{\thmnumberingseppt}
|
||
\theoremstyle{\current@theoremstyle}
|
||
\theoremseparator{\current@theoremseparator}
|
||
\theoremsymbol{}
|
||
|
||
\newtheorem{X}{X}[chapter] % for most theorems
|
||
\newtheorem{Xe}{Xe}[chapter] % for equations
|
||
\newtheorem*{Xdisplaynone}{Xdisplaynone}[chapter] % a dummy counter, that will never be displayed.
|
||
\newtheorem{Xsp}{Xsp}[chapter] % for special theorems
|
||
\generatenestedthmnumbering{arabic}{chapter}{X}
|
||
\generatenestedthmnumbering{arabic}{chapter}{Xe}
|
||
\generatenestedthmnumbering{Roman}{chapter}{Xsp}
|
||
\let\theXsp\theshortXsp
|
||
|
||
\theoremheaderfont{\upshape\bfseries}
|
||
\theorembodyfont{\slshape}
|
||
|
||
\ranewthm{thm}{Theorem}{\enndeOnNeutralSign}[X]
|
||
\ranewthm{satz}{Satz}{\enndeOnNeutralSign}[X]
|
||
\ranewthm{claim}{Behauptung}{\enndeOnNeutralSign}[X]
|
||
\ranewthm{lemm}{Lemma}{\enndeOnNeutralSign}[X]
|
||
\ranewthm{cor}{Korollar}{\enndeOnNeutralSign}[X]
|
||
\ranewthm{folg}{Folgerung}{\enndeOnNeutralSign}[X]
|
||
\ranewthm{prop}{Proposition}{\enndeOnNeutralSign}[X]
|
||
|
||
\theorembodyfont{\upshape}
|
||
|
||
\ranewthm{defn}{Definition}{\enndeOnNeutralSign}[X]
|
||
\ranewthm{conv}{Konvention}{\enndeOnNeutralSign}[X]
|
||
\ranewthm{obs}{Beobachtung}{\enndeOnNeutralSign}[X]
|
||
\ranewthm{e.g.}{Beipsiel}{\enndeOnNeutralSign}[X]
|
||
\ranewthm{fact}{Fakt}{\enndeOnNeutralSign}[X]
|
||
\ranewthm{rem}{Bemerkung}{\enndeOnNeutralSign}[X]
|
||
\ranewthm{qstn}{Frage}{\enndeOnNeutralSign}[X]
|
||
\ranewthm{exer}{Aufgabe}{\enndeOnNeutralSign}[X]
|
||
\ranewthm{soln}{Lösung}{\enndeOnNeutralSign}[X]
|
||
|
||
\theoremheaderfont{\itshape\bfseries}
|
||
\theorembodyfont{\upshape}
|
||
|
||
\ranewthm{proof@tmp}{Beweis}{\enndeOfProofSign}[Xdisplaynone]
|
||
\rathmtransfer{proof@tmp*}{proof}
|
||
|
||
\def\behauptungbeleg@claim{%
|
||
\iflanguage{british}{Claim}{%
|
||
\iflanguage{english}{Claim}{%
|
||
\iflanguage{ngerman}{Behauptung}{%
|
||
\iflanguage{russian}{Утверждение}{%
|
||
Claim%
|
||
}}}}%
|
||
}
|
||
\def\behauptungbeleg@pf@kurz{%
|
||
\iflanguage{british}{Pf}{%
|
||
\iflanguage{english}{Pf}{%
|
||
\iflanguage{ngerman}{Bew}{%
|
||
\iflanguage{russian}{Доказательство}{%
|
||
Pf%
|
||
}}}}%
|
||
}
|
||
\def\behauptungbeleg{\@ifnextchar\bgroup{\behauptungbeleg@c}{\behauptungbeleg@bes}}
|
||
\def\behauptungbeleg@c#1{\item[{\bfseries \behauptungbeleg@claim\erlaubeplatz #1.}]}
|
||
\def\behauptungbeleg@bes{\item[{\bfseries \behauptungbeleg@claim.}]}
|
||
\def\belegbehauptung{\item[{\bfseries\itshape\behauptungbeleg@pf@kurz.}]}
|
||
|
||
%% ****************************************************************
|
||
%% ALTE UMGEBUNGEN:
|
||
%% ****************************************************************
|
||
|
||
\newcolumntype{\RECHTS}[1]{>{\raggedleft}p{#1}}
|
||
\newcolumntype{\LINKS}[1]{>{\raggedright}p{#1}}
|
||
\newcolumntype{m}{>{$}l<{$}}
|
||
\newcolumntype{C}{>{$}c<{$}}
|
||
\newcolumntype{L}{>{$}l<{$}}
|
||
\newcolumntype{R}{>{$}r<{$}}
|
||
\newcolumntype{0}{@{\hspace{0pt}}}
|
||
\newcolumntype{\LINKSRAND}{@{\hspace{\@totalleftmargin}}}
|
||
\newcolumntype{h}{@{\extracolsep{\fill}}}
|
||
\newcolumntype{i}{>{\itshape}}
|
||
\newcolumntype{t}{@{\hspace{\tabcolsep}}}
|
||
\newcolumntype{q}{@{\hspace{1em}}}
|
||
\newcolumntype{n}{@{\hspace{-\tabcolsep}}}
|
||
\newcolumntype{M}[2]{%
|
||
>{\begin{minipage}{#2}\begin{math}}%
|
||
{#1}%
|
||
<{\end{math}\end{minipage}}%
|
||
}
|
||
\newcolumntype{T}[2]{%
|
||
>{\begin{minipage}{#2}}%
|
||
{#1}%
|
||
<{\end{minipage}}%
|
||
}
|
||
\setlength{\LTpre}{\baselineskip}
|
||
\setlength{\LTpost}{0pt}
|
||
\def\center{\centering}
|
||
\def\endcenter{}
|
||
|
||
\def\punkteumgebung@genbefehl#1#2#3{
|
||
\punkteumgebung@genbefehl@{#1}{#2}{#3}{}{}
|
||
\punkteumgebung@genbefehl@{multi#1}{#2}{#3}{
|
||
\setlength{\columnsep}{10pt}%
|
||
\setlength{\columnseprule}{0pt}%
|
||
\begin{multicols}{\thecolumnanzahl}%
|
||
}{\end{multicols}\nvraum{1}}
|
||
}
|
||
\def\punkteumgebung@genbefehl@#1#2#3#4#5{
|
||
\expandafter\gdef\csname #1\endcsname{
|
||
\@ifnextchar\bgroup{\csname #1@c\endcsname}{\csname #1@bes\endcsname}
|
||
}%]
|
||
\expandafter\def\csname #1@c\endcsname##1{
|
||
\@ifnextchar[{\csname #1@c@\endcsname{##1}}{\csname #1@c@\endcsname{##1}[\z@]}
|
||
}%]
|
||
\expandafter\def\csname #1@c@\endcsname##1[##2]{
|
||
\@ifnextchar[{\csname #1@c@@\endcsname{##1}[##2]}{\csname #1@c@@\endcsname{##1}[##2][\z@]}
|
||
}%]
|
||
\expandafter\def\csname #1@c@@\endcsname##1[##2][##3]{
|
||
\let\alterlinkerRand\gesamtlinkerRand
|
||
\let\alterrechterRand\gesamtrechterRand
|
||
\addtolength{\gesamtlinkerRand}{##2}
|
||
\addtolength{\gesamtrechterRand}{##3}
|
||
\advance\linewidth -##2%
|
||
\advance\linewidth -##3%
|
||
\advance\@totalleftmargin ##2%
|
||
\parshape\@ne \@totalleftmargin\linewidth%
|
||
#4
|
||
\begin{#2}[\upshape ##1]%
|
||
\setlength{\parskip}{0.5\baselineskip}\relax%
|
||
\setlength{\topsep}{\z@}\relax%
|
||
\setlength{\partopsep}{\z@}\relax%
|
||
\setlength{\parsep}{\parskip}\relax%
|
||
\setlength{\itemsep}{#3}\relax%
|
||
\setlength{\listparindent}{\z@}\relax%
|
||
\setlength{\itemindent}{\z@}\relax%
|
||
}
|
||
\expandafter\def\csname #1@bes\endcsname{
|
||
\@ifnextchar[{\csname #1@bes@\endcsname}{\csname #1@bes@\endcsname[\z@]}
|
||
}%]
|
||
\expandafter\def\csname #1@bes@\endcsname[##1]{
|
||
\@ifnextchar[{\csname #1@bes@@\endcsname[##1]}{\csname #1@bes@@\endcsname[##1][\z@]}
|
||
}%]
|
||
\expandafter\def\csname #1@bes@@\endcsname[##1][##2]{
|
||
\let\alterlinkerRand\gesamtlinkerRand
|
||
\let\alterrechterRand\gesamtrechterRand
|
||
\addtolength{\gesamtlinkerRand}{##1}
|
||
\addtolength{\gesamtrechterRand}{##2}
|
||
\advance\linewidth -##1%
|
||
\advance\linewidth -##2%
|
||
\advance\@totalleftmargin ##1%
|
||
\parshape\@ne \@totalleftmargin\linewidth%
|
||
#4
|
||
\begin{#2}%
|
||
\setlength{\parskip}{0.5\baselineskip}\relax%
|
||
\setlength{\topsep}{\z@}\relax%
|
||
\setlength{\partopsep}{\z@}\relax%
|
||
\setlength{\parsep}{\parskip}\relax%
|
||
\setlength{\itemsep}{#3}\relax%
|
||
\setlength{\listparindent}{\z@}\relax%
|
||
\setlength{\itemindent}{\z@}\relax%
|
||
}
|
||
\expandafter\gdef\csname end#1\endcsname{%
|
||
\end{#2}#5
|
||
\setlength{\gesamtlinkerRand}{\alterlinkerRand}
|
||
\setlength{\gesamtlinkerRand}{\alterrechterRand}
|
||
}
|
||
}
|
||
|
||
\def\ritempunkt{{\Large\textbullet}} % \textbullet, $\sqbullet$, $\blacktriangleright$
|
||
\setdefaultitem{\ritempunkt}{\ritempunkt}{\ritempunkt}{\ritempunkt}
|
||
\punkteumgebung@genbefehl{itemise}{compactitem}{\parskip}{}{}
|
||
\punkteumgebung@genbefehl{kompaktitem}{compactitem}{\z@}{}{}
|
||
\punkteumgebung@genbefehl{enumerate}{compactenum}{\parskip}{}{}
|
||
\punkteumgebung@genbefehl{kompaktenum}{compactenum}{\z@}{}{}
|
||
|
||
\let\ALTthebibliography\thebibliography
|
||
\renewenvironment{thebibliography}[1]{%
|
||
\begin{ALTthebibliography}{#1}
|
||
\addcontentsline{toc}{part}{\bibname}
|
||
}{%
|
||
\end{ALTthebibliography}
|
||
}
|
||
|
||
%% ****************************************************************
|
||
%% NEUE UMGEBUNGEN:
|
||
%% ****************************************************************
|
||
|
||
\def\matrix#1{\left(\begin{array}[mc]{#1}}
|
||
\def\endmatrix{\end{array}\right)}
|
||
\def\smatrix{\left(\begin{smallmatrix}}
|
||
\def\endsmatrix{\end{smallmatrix}\right)}
|
||
\def\vector{\begin{matrix}{c}}
|
||
\def\endvector{\end{matrix}}
|
||
\def\svector{\begin{smatrix}}
|
||
\def\endsvector{\end{smatrix}}
|
||
|
||
\def\multiargrekursiverbefehl#1#2#3#4#5#6#7#8{%
|
||
\expandafter\gdef\csname#1\endcsname #2##1#4{\csname #1@anfang\endcsname##1#3\egroup}
|
||
\expandafter\def\csname #1@anfang\endcsname##1#3{#5##1\@ifnextchar\egroup{\csname #1@ende\endcsname}{#7\csname #1@mitte\endcsname}}
|
||
\expandafter\def\csname #1@mitte\endcsname##1#3{#6##1\@ifnextchar\egroup{\csname #1@ende\endcsname}{#7\csname #1@mitte\endcsname}}
|
||
\expandafter\def\csname #1@ende\endcsname##1{#8}
|
||
}
|
||
\multiargrekursiverbefehl{svektor}{[}{;}{]}{\begin{smatrix}}{}{\\}{\\\end{smatrix}}
|
||
\multiargrekursiverbefehl{vektor}{[}{;}{]}{\begin{matrix}{c}}{}{\\}{\\\end{matrix}}
|
||
\multiargrekursiverbefehl{vektorzeile}{}{,}{;}{}{&}{}{}
|
||
\multiargrekursiverbefehl{matlabmatrix}{[}{;}{]}{\begin{smatrix}\vektorzeile}{\vektorzeile}{;\\}{;\end{smatrix}}
|
||
|
||
\def\underbracenodisplay#1{%
|
||
\mathop{\vtop{\m@th\ialign{##\crcr
|
||
$\hfil\displaystyle{#1}\hfil$\crcr
|
||
\noalign{\kern3\p@\nointerlineskip}%
|
||
\upbracefill\crcr\noalign{\kern3\p@}}}}\limits%
|
||
}
|
||
|
||
\def\mathe[#1]#2{%
|
||
\ifthenelse{\equal{\boolinmdframed}{\boolwahr}}{}{\begin{escapeeinzug}}
|
||
\noindent%
|
||
\let\eqtagset\boolfalsch
|
||
\let\eqtaglabel\boolleer
|
||
\let\eqtagsymb\boolleer
|
||
\let\alteqtag\eqtag
|
||
\def\eqtag{\@ifnextchar[{\eqtag@loc@}{\eqtag@loc@[*]}}%
|
||
\def\eqtag@loc@[##1]{\@ifnextchar\bgroup{\eqtag@loc@@[##1]}{\eqtag@loc@@[##1]{}}}%
|
||
\def\eqtag@loc@@[##1]##2{%
|
||
\gdef\eqtagset{\boolwahr}
|
||
\gdef\eqtaglabel{##1}
|
||
\gdef\eqtagsymb{##2}
|
||
}%
|
||
\def\verticalalign{}%
|
||
\IfBeginWith{#1}{t}{\def\verticalalign{t}}{}%
|
||
\IfBeginWith{#1}{m}{\def\verticalalign{c}}{}%
|
||
\IfBeginWith{#1}{b}{\def\verticalalign{b}}{}%
|
||
\def\horizontalalign{\null\hfill\null}%
|
||
\IfEndWith{#1}{l}{}{\null\hfill\null}%
|
||
\IfEndWith{#1}{r}{\def\horizontalalign{}}{}%
|
||
\begin{math}
|
||
\begin{array}[\verticalalign]{0#2}%
|
||
}
|
||
\def\endmathe{%
|
||
\end{array}
|
||
\end{math}\horizontalalign%
|
||
\let\eqtag\alteqtag
|
||
\ifthenelse{\equal{\eqtagset}{\boolwahr}}{\eqtag[\eqtaglabel]{\eqtagsymb}}{}
|
||
\ifthenelse{\equal{\boolinmdframed}{\boolwahr}}{}{\end{escapeeinzug}}%
|
||
}
|
||
|
||
\def\longmathe[#1]#2{\relax
|
||
\let\altarraystretch\arraystretch
|
||
\renewcommand\arraystretch{1.2}\relax
|
||
\begin{longtable}[#1]{\LINKSRAND #2}
|
||
}
|
||
\def\endlongmathe{
|
||
\end{longtable}
|
||
\renewcommand\arraystretch{\altarraystretch}
|
||
}
|
||
|
||
\def\einzug{\@ifnextchar[{\indents@}{\indents@[\z@]}}%]
|
||
\def\indents@[#1]{\@ifnextchar[{\indents@@[#1]}{\indents@@[#1][\z@]}}%]
|
||
\def\indents@@[#1][#2]{%
|
||
\begin{list}{}{\relax
|
||
\setlength{\topsep}{\z@}\relax
|
||
\setlength{\partopsep}{\z@}\relax
|
||
\setlength{\parsep}{\parskip}\relax
|
||
\setlength{\listparindent}{\z@}\relax
|
||
\setlength{\itemindent}{\z@}\relax
|
||
\setlength{\leftmargin}{#1}\relax
|
||
\setlength{\rightmargin}{#2}\relax
|
||
\let\alterlinkerRand\gesamtlinkerRand
|
||
\let\alterrechterRand\gesamtrechterRand
|
||
\addtolength{\gesamtlinkerRand}{#1}
|
||
\addtolength{\gesamtrechterRand}{#2}
|
||
}\relax
|
||
\item[]\relax
|
||
}
|
||
\def\endeinzug{%
|
||
\setlength{\gesamtlinkerRand}{\alterlinkerRand}
|
||
\setlength{\gesamtlinkerRand}{\alterrechterRand}
|
||
\end{list}%
|
||
}
|
||
|
||
\def\escapeeinzug{\begin{einzug}[-\gesamtlinkerRand][-\gesamtrechterRand]}
|
||
\def\endescapeeinzug{\end{einzug}}
|
||
|
||
\def\programmiercode{
|
||
\modulolinenumbers[1]
|
||
\begin{einzug}[\rtab][\rtab]%
|
||
\begin{linenumbers}%
|
||
\fontfamily{cmtt}\fontseries{m}\fontshape{u}\selectfont%
|
||
\setlength{\parskip}{1\baselineskip}%
|
||
\setlength{\parindent}{0pt}%
|
||
}
|
||
\def\endprogrammiercode{
|
||
\end{linenumbers}
|
||
\end{einzug}
|
||
}
|
||
|
||
\def\schattiertebox@genbefehl#1#2#3{
|
||
\expandafter\gdef\csname #1\endcsname{%
|
||
\@ifnextchar[{\csname #1@args\endcsname}{\csname #1@args\endcsname[#3]}%]%
|
||
}
|
||
\expandafter\def\csname #1@args\endcsname[##1]{%
|
||
\@ifnextchar[{\csname #1@args@l\endcsname[##1]}{\csname #1@args@n\endcsname[##1]}%]%
|
||
}
|
||
\expandafter\def\csname #1@args@l\endcsname[##1][##2]{%
|
||
\@ifnextchar[{\csname #1@args@l@r\endcsname[##1][##2]}{\csname #1@args@l@n\endcsname[##1][##2]}%]%
|
||
}
|
||
\expandafter\def\csname #1@args@n\endcsname[##1]{%
|
||
\let\boolinmdframed\boolwahr
|
||
\begin{mdframed}[#2leftmargin=0,rightmargin=0,outermargin=0,innermargin=0,##1]
|
||
}
|
||
\expandafter\def\csname #1@args@l@n\endcsname[##1][##2]{%
|
||
\let\boolinmdframed\boolwahr
|
||
\begin{mdframed}[#2leftmargin=##2/2,rightmargin=##2/2,outermargin=##2/2,innermargin=##2/2,##1]
|
||
}
|
||
\expandafter\def\csname #1@args@l@r\endcsname[##1][##2][##3]{%
|
||
\let\boolinmdframed\boolwahr
|
||
\begin{mdframed}[#2leftmargin=##2,rightmargin=##3,outermargin=##2,innermargin=##3,##1]
|
||
}
|
||
\expandafter\gdef\csname end#1\endcsname{%
|
||
\end{mdframed}
|
||
\let\boolinmdframed\boolfalsch
|
||
}
|
||
}
|
||
\schattiertebox@genbefehl{schattiertebox}{
|
||
splittopskip=0,%
|
||
splitbottomskip=0,%
|
||
frametitleaboveskip=0,%
|
||
frametitlebelowskip=0,%
|
||
skipabove=1\baselineskip,%
|
||
skipbelow=1\baselineskip,%
|
||
linewidth=2pt,%
|
||
linecolor=black,%
|
||
roundcorner=4pt,%
|
||
}{
|
||
backgroundcolor=leer,%
|
||
nobreak=true,%
|
||
}
|
||
|
||
\schattiertebox@genbefehl{schattierteboxdunn}{
|
||
splittopskip=0,%
|
||
splitbottomskip=0,%
|
||
frametitleaboveskip=0,%
|
||
frametitlebelowskip=0,%
|
||
skipabove=1\baselineskip,%
|
||
skipbelow=1\baselineskip,%
|
||
linewidth=1pt,%
|
||
linecolor=black,%
|
||
roundcorner=2pt,%
|
||
}{
|
||
backgroundcolor=leer,%
|
||
nobreak=true,%
|
||
}
|
||
|
||
\def\algorithm{\schattiertebox[backgroundcolor=hellgrau,nobreak=false]}
|
||
\def\endalgorithm{\endschattiertebox}
|
||
|
||
\def\tikzsetzenode#1{%
|
||
\tikz[remember picture,baseline,overlay]{\node #1;}%
|
||
}
|
||
\def\tikzsetzepfeil#1{%
|
||
\begin{tikzpicture}[remember picture,overlay,>=latex]%
|
||
\draw #1;%
|
||
\end{tikzpicture}%
|
||
}
|
||
\def\tikzsetzeoverlay#1{%
|
||
\begin{tikzpicture}[remember picture,overlay,>=latex]%
|
||
#1%
|
||
\end{tikzpicture}%
|
||
}
|
||
\def\tikzsetzekreise[#1]#2#3{%
|
||
\tikzsetzepfeil{%
|
||
[rounded corners,#1]%
|
||
([shift={(-\tabcolsep,0.75\baselineskip)}]#2)%
|
||
rectangle%
|
||
([shift={(\tabcolsep,-0.5\baselineskip)}]#3)
|
||
}%
|
||
}
|
||
|
||
\tikzset{
|
||
>=stealth,
|
||
auto,
|
||
thick,
|
||
main node/.style={
|
||
circle,draw,font=\sffamily\Large\bfseries,minimum size=0pt
|
||
},
|
||
}
|
||
|
||
%% ********************************************************************************
|
||
%% FILE: src/setup-layout.tex
|
||
%% ********************************************************************************
|
||
|
||
\pagestyle{fancyplain}
|
||
|
||
\@ifundefined{setcitestyle}{%
|
||
%% do nothing
|
||
}{%
|
||
\setcitestyle{numeric-comp,open={[},close={]}}
|
||
}
|
||
\def\crefpairconjunction{ und }
|
||
\def\crefmiddleconjunction{, }
|
||
\def\creflastconjunction{, und }
|
||
|
||
\raggedbottom %% <- pushes footers up
|
||
\sloppy
|
||
\def\headrulewidth{0pt}
|
||
\def\footrulewidth{0pt}
|
||
\setlength{\columnsep}{20pt}
|
||
\setlength{\columnseprule}{1pt}
|
||
\setlength{\headheight}{11pt}
|
||
\setlength{\partopsep}{0pt}
|
||
\setlength{\topsep}{\baselineskip}
|
||
\setlength{\topskip}{0.5\baselineskip}
|
||
\setlength{\footskip}{-1\baselineskip}
|
||
\setlength{\maxdepth}{0pt}
|
||
\renewcommand{\baselinestretch}{1}
|
||
\renewcommand{\arraystretch}{1}
|
||
\setcounter{LTchunksize}{\infty}
|
||
\setlength{\abovedisplayskip}{0pt}
|
||
\setlength{\parskip}{1\baselineskip}
|
||
\def\firstparagraph{\noindent}
|
||
\def\continueparagraph{\noindent}
|
||
|
||
\hypersetup{
|
||
hidelinks=true,
|
||
}
|
||
|
||
\@addtoreset{chapter}{part} %% nötig für Hyperref.
|
||
|
||
\def\partfont{\documentfont\fontseries{bx}\Huge\selectfont}
|
||
\def\chapterfont{\documentfont\fontseries{bx}\huge\selectfont}
|
||
\def\sectionfont{\documentfont\fontseries{bx}\Large\selectfont}
|
||
\def\subsectionfont{\documentfont\fontseries{bx}\large\selectfont}
|
||
|
||
\def\thepart{\Roman{part}}
|
||
\generatenestedsecnumbering{arabic}{part}{chapter}
|
||
\generatenestedsecnumbering{arabic}{chapter}{section}
|
||
\generatenestedsecnumbering{arabic}{section}{subsection}
|
||
\generatenestedsecnumbering{arabic}{subsection}{subsubsection}
|
||
\def\theunitnamepart{\thepart}
|
||
\def\theunitnamechapter{\theshortchapter}
|
||
\def\theunitnamesection{\thelongsection}
|
||
\def\theunitnamesubsection{\thelongsubsection}
|
||
\def\theunitnamesubsubsection{\thelongsubsubsection}
|
||
|
||
\def\partname{Teil\erlaubeplatz}
|
||
\def\chaptername{Kapitel\erlaubeplatz}
|
||
\def\sectionname{\S\erlaubeplatz}
|
||
\def\subsectionname{}
|
||
\def\subsubsectionname{}
|
||
|
||
\let\appendix@orig\appendix
|
||
\def\appendix{%
|
||
\appendix@orig%
|
||
\let\boolinappendix\boolwahr
|
||
\addcontentsline{toc}{part}{\appendixname}%
|
||
\addtocontents{toc}{\protect\setcounter{tocdepth}{0}}
|
||
\def\sectionname{Appendix}%
|
||
\def\theunitnamesection{\Alph{section}}%
|
||
}
|
||
\def\notappendix{%
|
||
\let\boolinappendix\boolfalse
|
||
\addtocontents{toc}{\protect\setcounter{tocdepth}{1 }}
|
||
\def\sectionname{}%
|
||
\def\theunitnamesection{\arabic{section}}%
|
||
}
|
||
|
||
%% \titlespacing{<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: srclocal/setup-locallayout.tex
|
||
%% ********************************************************************************
|
||
|
||
\def\theunitnamesection{\theshortsection}
|
||
|
||
%% ********************************************************************************
|
||
%% FILE: srclocal/setup-localmacros.tex
|
||
%% ********************************************************************************
|
||
|
||
%% ****************************************************************
|
||
%% MATHE:
|
||
%% ****************************************************************
|
||
|
||
\def\cal#1{\mathcal{#1}}
|
||
\def\reell{\mathbb{R}}
|
||
\def\kmplx{\mathbb{C}}
|
||
\def\Torus{\mathbb{T}}
|
||
\def\rtnl{\mathbb{Q}}
|
||
\def\intgr{\mathbb{Z}}
|
||
|
||
\def\ntrl{\mathbb{N}}
|
||
\def\ntrlpos{\mathbb{N}}
|
||
\def\ntrlzero{\mathbb{N}_{0}}
|
||
\def\reellNonNeg{\reell_{+}}
|
||
|
||
\def\imageinh{\imath}
|
||
\def\ReTeil{\mathop{\mathfrak{R}\text{\upshape e}}}
|
||
\def\ImTeil{\mathop{\mathfrak{I}\text{\upshape m}}}
|
||
|
||
\def\leer{\emptyset}
|
||
\def\restr#1{\vert_{#1}}
|
||
\def\ohne{\mathbin{\setminus}}
|
||
\def\Pot{\mathop{\mathcal{P}}}
|
||
\def\einser{\mathbf{1}}
|
||
\def\supp{\mathop{\mathrm{supp}}}
|
||
|
||
\def\brkt#1{\langle{}#1{}\rangle}
|
||
\def\lsim{\mathop{\sim}}
|
||
\def\lneg{\mathop{\neg}}
|
||
\def\land{\mathop{\wedge}}
|
||
\def\lor{\mathop{\vee}}
|
||
|
||
\def\eps{\varepsilon}
|
||
\let\altphi\phi
|
||
\let\altvarphi\varphi
|
||
\def\phi{\altvarphi}
|
||
\def\varphi{\altphi}
|
||
|
||
\def\vectorspacespan{\mathop{\text{\upshape Lin}}}
|
||
\def\dim{\mathop{\text{\upshape dim}}}
|
||
\def\det{\mathop{\text{\upshape det}}}
|
||
\def\rank{\mathop{\text{\upshape Rang}}}
|
||
\def\onematrix{\text{\upshape\bfseries I}}
|
||
\def\zeromatrix{\text{\upshape\bfseries 0}}
|
||
\def\zerovector{\text{\upshape\bfseries 0}}
|
||
|
||
\def\graph{\mathop{\text{\upshape Gph}}}
|
||
\def\domain{\mathop{\text{\upshape dom}}}
|
||
\def\range{\mathop{\text{\upshape Bild}}}
|
||
\def\ker{\mathop{\text{\upshape Kern}}}
|
||
\def\functionspace{\mathop{\text{\upshape Abb}}}
|
||
\def\id{\text{\upshape id}}
|
||
\def\modfn{\mathop{\text{\upshape mod}}}
|
||
\def\divides{\mathbin{\mid}}
|
||
\def\ndivides{\mathbin{\nmid}}
|
||
\def\ggT{\mathop{\text{\upshape ggT}}}
|
||
\def\choose#1#2{\begin{smatrix}#1\\#2\\\end{smatrix}}
|
||
|
||
\def\punktschema{
|
||
\footnotesize
|
||
\hraum
|
||
\begin{tabular}[mc]{|p{0.1\textwidth}|p{0.7\textwidth}|}
|
||
\hline
|
||
\multicolumn{2}{|l|}{{\bfseries NOTENSCHEMA}}\\
|
||
Punkte &Beschreibung\\
|
||
\hline
|
||
\hline
|
||
}
|
||
\def\endpunktschema{
|
||
\hline
|
||
\end{tabular}
|
||
\hraum
|
||
}
|
||
|
||
\def\headingTeilaufgabe#1{
|
||
\uwave{{\bfseries\large Aufgabe #1}}
|
||
}
|
||
|
||
\makeatother
|
||
|
||
\begin{document}
|
||
\startdocumentlayoutoptions
|
||
|
||
%% HAUPTTEXT:
|
||
|
||
%% ********************************************************************************
|
||
%% FILE: body/index.tex
|
||
%% ********************************************************************************
|
||
|
||
%% ********************************************************************************
|
||
%% FILE: body/A5.tex
|
||
%% ********************************************************************************
|
||
|
||
%% AUFGABE 5
|
||
\setcounter{chapter}{5}
|
||
\let\altsectionname\sectionname
|
||
\def\sectionname{Aufgabe}
|
||
\setcounternach{section}{5}
|
||
\section[]{(Klausur${}_{2}$, WiSe 2020/2021)}
|
||
\label{ueb:5}
|
||
\let\sectionname\altsectionname
|
||
|
||
Es sei $V$ ein endlich dimensionaler Vektorraum über einem Körper, $K$.
|
||
Gegeben ist eine lineare Abbildung ${\phi:V\to V}$
|
||
mit der Eigenschaft $\phi\circ\phi=\phi$.
|
||
|
||
%% ********************************************************************************
|
||
%% FILE: body/A5a.tex
|
||
%% ********************************************************************************
|
||
|
||
%% AUFGABE 5a
|
||
\headingTeilaufgabe{5a}
|
||
|
||
\begin{claim*}
|
||
$\ker(\phi)\cap\range(\phi)=\{\zerovector\}$.
|
||
\end{claim*}
|
||
|
||
Es gibt hierfür ein paar Ansätze.
|
||
Zuerst der direkte Ansatz:
|
||
|
||
\begin{enumerate}{\bfseries {Ansatz} I.}
|
||
\setcounternach{enumi}{1}
|
||
\item
|
||
\textbf{Zu zeigen:}
|
||
|
||
\begin{kompaktenum}{\bfseries (i)}
|
||
\item triviale Inklusion: $\{\zerovector\}\subseteq\ker(\phi)\cap\range(\phi)$;
|
||
\item nichttriviale Inklusion: $\ker(\phi)\cap\range(\phi)\subseteq\{\zerovector\}$.
|
||
\end{kompaktenum}
|
||
|
||
Zu (i):
|
||
Da $\ker(\phi)$ und $\range(\phi)$ Untervektorräume sind, liegt der Nullvektor in beiden und somit im Schnitt.
|
||
|
||
Zu (ii): Sei $v$ ein Vektor im Schnitt. Wir müssen zeigen, dass $v=\zerovector$.\\
|
||
Aus $v\in\range(\phi)$ folgt $v=\phi(w)$ für ein $w\in V$.\\
|
||
Da \uline{auch} $v\in\ker(\phi)$, erhalten wir
|
||
|
||
\begin{mathe}[mc]{rclql}
|
||
v &= &\phi(w)
|
||
&\text{per Wahl von $w$}\\
|
||
&= &\phi(\phi(w))
|
||
&\text{da per Annahme $\phi\circ\phi=\phi$}\\
|
||
&= &\phi(v)
|
||
&\text{per Wahl von $w$}\\
|
||
&= &\zerovector
|
||
&\text{da $v\in\ker(\phi)$.}\\
|
||
\end{mathe}
|
||
|
||
Also gelten (i) + (ii), woraus sich die behauptete Gleichung ergibt.
|
||
\enndeOfProof
|
||
\end{enumerate}
|
||
|
||
Für die folgenden Ansätze benutzen wir das Resultat aus (b).\\
|
||
\textbf{ACHTUNG:} Dafür aber dürfen wir später (b) das Ergebnis aus (a) \uline{nicht} verwenden,
|
||
denn sonst wäre dies ein \textbf{Zirkelschluss}.
|
||
|
||
\begin{enumerate}{\bfseries {Ansatz} I.}
|
||
\setcounternach{enumi}{2}
|
||
\item
|
||
Laut (b) gilt ($\ast$)~$V=\ker(\phi)+\range(\phi)$.\\
|
||
Fixiere die Untervektorräume, $U_{1}:=\ker(\phi)$ und $U_{2}:=\range(\phi)$, von $V$.\\
|
||
Laut der \emph{Dimensionsformel für lineare Abbildungen} gilt:
|
||
|
||
\begin{mathe}[mc]{rcl}
|
||
\eqtag[eq:A5a:ansatz:2:1]
|
||
\dim(V) &= &\dim(\ker(\phi)) + \dim(\range(\phi)).\\
|
||
\end{mathe}
|
||
|
||
Aus ($\ast$) und der \emph{Dimensionsformel für Untervektorräume}
|
||
und da $V$ endlich dimensional ist,
|
||
erhalten wir außerdem:
|
||
|
||
\begin{mathe}[mc]{rcl}
|
||
\eqtag[eq:A5a:ansatz:2:2]
|
||
\dim(V) &= &\dim(\ker(\phi) + \range(\phi))\\
|
||
&= &\dim(U_{1} + U_{2})\\
|
||
&= &\dim(U_{1}) + \dim(U_{2}) - \dim(U_{1}\cap U_{2})\\
|
||
&= &\dim(\ker(\phi)) + \dim(\range(\phi))
|
||
- \dim(\ker(\phi)\cap\range(\phi)).\\
|
||
\end{mathe}
|
||
|
||
Da all diese Zahlen endlich sind, folgt aus
|
||
\eqcref{eq:A5a:ansatz:2:1}
|
||
und
|
||
\eqcref{eq:A5a:ansatz:2:2},
|
||
dass $\dim(\ker(\phi)\cap\range(\phi))=0$.\\
|
||
Da $\ker(\phi)\cap\range(\phi)$ ein Untervektorraum von $V$ ist
|
||
und $0$-dimensional ist,
|
||
gilt ${\ker(\phi)\cap\range(\phi)=\{\zerovector\}}$.
|
||
\enndeOfProof
|
||
\item
|
||
Laut (b) gilt $V=\ker(\phi)+\range(\phi)$.\\
|
||
Fixiere die Untervektorräume, $U_{1}:=\ker(\phi)$ und $U_{2}:=\range(\phi)$, von $V$.
|
||
Auf $U_{1}$ verhält sich $\phi$ wie $\zeromatrix$ (die Nullabbildung)
|
||
und auf $U_{2}$ verhält sich $\phi$ wie $\onematrix$ (die Identitätsabbildung),
|
||
weil $\phi\circ\phi=\phi$.
|
||
|
||
Für $x\in\ker(\phi)\cap\range(\phi)$ gilt $x\in U_{1}\cap U_{2}$.
|
||
Da sich $\phi$ wie die Null- bzw. Identitätsabbildung auf $U_{1}$ bzw. $U_{2}$ verhält,
|
||
gilt $x=\onematrix(x)=\phi(x)=\zeromatrix(x)=\zerovector$.
|
||
Also $\ker(\phi)\cap\range(\phi)\subseteq\{\zerovector\}$.
|
||
Da außerdem $\zerovector$ in allen Untervektorräumen liegt,
|
||
erhalten wir ${\ker(\phi)\cap\range(\phi)=\{\zerovector\}}$.
|
||
\enndeOfProof
|
||
\end{enumerate}
|
||
|
||
\begin{punktschema}
|
||
5 &Argument vollständig (=ausführlich) und logisch gültig.\\
|
||
\hline
|
||
4 &Nichttriviale Inklusion wurde vollständig bewiesen, aber die triviale Inklusion fehlte.\\
|
||
\hdashline
|
||
4 &Ansatz II/III gemacht aber fehlte zu erwähnen, dass $V$ endlich dimensional.\\
|
||
\hline
|
||
3 &Triviale Inklusion bewiesen und nichttriviale Inklusion lückenhaft (aber richtig) bewiesen.\\
|
||
\hdashline
|
||
3 &Argument baut auf falscher Annahme auf, aber Annahme „rechtfertigt“.\\
|
||
\hdashline
|
||
3 &Ansatz II/III gemacht, aber mehrere lücken.\\
|
||
\hline
|
||
2 &Nur triviale Inklusion bewiesen.\\
|
||
\hdashline
|
||
2 &Argument baut auf falscher Annahme auf und Annahme kaum rechtfertigt.\\
|
||
\hline
|
||
0 &sonst.\\
|
||
\end{punktschema}
|
||
|
||
{\footnotesize
|
||
\textbf{Bemerkungen.}
|
||
Da es sich hier um die Bewertung von Argumentationen handelt,
|
||
kann man in Wirklichkeit kein Schema festlegen.
|
||
Stattdessen musste über die Qualität Urteile getroffen werden.
|
||
In erster Linie kriegt man volle Punkte, wenn man
|
||
vollständig (idealerweise auch ausführlich) + gültig + überzeugend
|
||
argumentierte.
|
||
Ab dann wurden anhand unterschiedlicher Defizite empirische Graduierungen implementiert.
|
||
Wegen wichtiger Lücken wurden Punkte abgezogen.
|
||
Wenn man auf einer falschen Annahme aufbaute
|
||
(z.\,B. $\phi$ sei injektiv/bijektiv od. $\phi=\onematrix$ od. in Ansatz III eine falsche Dichotomie)
|
||
war entscheiden, ob man zumindest versuchte,
|
||
die falsche Annahme zu rechtfertigen.
|
||
Wenn man auf einer falschen Annahme das Argument aufbaute
|
||
und diese Annahme alles trivial machen ließ
|
||
und diese Annahme nicht genügend rechtfertigte,
|
||
dann resultierte dies in $0$ Punkten.
|
||
Wenn man Ansatz II/III implementierte aber im (b) mit dem Ergebnis aus (a) argumentierte,
|
||
war dies ein Kreisargument und somit wurden auch Punkte abgezogen.
|
||
}
|
||
|
||
\clearpage
|
||
|
||
%% ********************************************************************************
|
||
%% FILE: body/A5b.tex
|
||
%% ********************************************************************************
|
||
|
||
%% AUFGABE 5b
|
||
\headingTeilaufgabe{5b}
|
||
|
||
\begin{claim*}
|
||
Es gilt $V=\ker(\phi)+\range(\phi)$.
|
||
\end{claim*}
|
||
|
||
Es gab hier grundsätzlich zwei Ansätze:
|
||
|
||
\begin{enumerate}{\bfseries {Ansatz} I.}
|
||
\item
|
||
\textbf{Zu zeigen:}
|
||
|
||
\begin{kompaktenum}{\bfseries (i)}
|
||
\item triviale Inklusion: ${\ker(\phi)+\range(\phi)\subseteq V}$;
|
||
\item nichttriviale Inklusion: ${V\subseteq\ker(\phi)+\range(\phi)}$.
|
||
\end{kompaktenum}
|
||
|
||
Zu (i):
|
||
Da $\ker(\phi),\range(\phi)\subseteq V$ und $V$ unter Addition abgeschlossen ist,
|
||
gilt ${\ker(\phi)+\range(\phi)\subseteq V}$.
|
||
(Alternative: die Summe aus Untervektorräumen ist i.\,A. ein Untervektorraum.)
|
||
|
||
Zu (ii):
|
||
Sei $v\in V$ beliebig.
|
||
Beachte zunächst, dass $v-\phi(v)\in\ker(\phi)$, da
|
||
|
||
\begin{mathe}[mc]{rcccl}
|
||
\phi(v-\phi(v)) &= &\phi(v)-\phi(\phi(v)) &\textoverset{($\ast$)}{=} &\phi(v) - \phi(v).\\
|
||
\end{mathe}
|
||
|
||
wobei ($\ast$) wegen der Hauptannahme $\phi\circ\phi=\phi$ gilt.
|
||
Daraus folgt
|
||
|
||
\begin{mathe}[mc]{rcccl}
|
||
v &= &\underbrace{
|
||
(v-\phi(v))
|
||
}_{\in\ker(\phi)}
|
||
+
|
||
\underbrace{
|
||
\phi(v)
|
||
}_{\in\range(\phi)}
|
||
&\in &\ker(\phi)+\range(\phi).\\
|
||
\end{mathe}
|
||
|
||
Also gelten (i) + (ii), woraus sich die behauptete Gleichung ergibt.
|
||
\item
|
||
Da $U_{1}:=\ker(\phi),U_{2}:=\range(\phi)\subseteq V$
|
||
und damit auch $\ker(\phi)+\range(\phi)\subseteq V$
|
||
Untervektorräume sind,
|
||
und da $V$ \uline{endlich dimensional} ist,
|
||
reicht es auch \textbf{zu zeigen}, dass
|
||
|
||
\begin{mathe}[mc]{rcl}
|
||
\eqtag[eq:A5b:ansatz:2:0]
|
||
\dim(\ker(\phi)+\range(\phi)) &= &\dim(V).\\
|
||
\end{mathe}
|
||
|
||
Laut der \emph{Dimensionsformel für lineare Abbildungen} gilt:
|
||
|
||
\begin{mathe}[mc]{rcl}
|
||
\eqtag[eq:A5b:ansatz:2:1]
|
||
\dim(V) &= &\dim(\ker(\phi)) + \dim(\range(\phi)).\\
|
||
\end{mathe}
|
||
|
||
Laut der \emph{Dimensionsformel für Untervektorräume}
|
||
gilt außerdem:
|
||
|
||
\begin{mathe}[mc]{rcl}
|
||
\eqtag[eq:A5b:ansatz:2:2]
|
||
\dim(\ker(\phi)) + \dim(\range(\phi))
|
||
&= &\dim(U_{1}) + \dim(U_{2})\\
|
||
&= &\dim(U_{1} + U_{2}) + \dim(U_{1}\cap U_{2})\\
|
||
&= &\dim(\ker(\phi) + \range(\phi)) \\
|
||
&&\qquad + \dim(\ker(\phi)\cap \range(\phi))\\
|
||
&\textoverset{($\ast$)}{=} &\dim(\ker(\phi) + \range(\phi)) + 0,\\
|
||
\end{mathe}
|
||
|
||
wobei ($\ast$) gilt, weil laut A5 a)
|
||
$\ker(\phi)\cap \range(\phi)=\{\zerovector\}$
|
||
ein $0$-dimensionaler Untervektorraum ist.
|
||
Aus \eqcref{eq:A5b:ansatz:2:1} + \eqcref{eq:A5b:ansatz:2:2}
|
||
folgt nun \eqcref{eq:A5b:ansatz:2:0}.
|
||
|
||
Also stimmt die Dimension des Untervektorraums, $\ker(\phi)+\range(\phi)$,
|
||
mit der des \uline{endlichdimensionalen} Vektorraums, $V$, überein.
|
||
Wie oben erzählt folgt daraus: $\ker(\phi)+\range(\phi)=V$.
|
||
\enndeOfProof
|
||
\end{enumerate}
|
||
|
||
\begin{punktschema}
|
||
5 &Argument vollständig (=ausführlich) und logisch gültig.\\
|
||
\hline
|
||
4 &Ansatz I: Nichttriviale Inklusion vollständig, aber triviale fehlte.\\
|
||
\hdashline
|
||
4 &Ansatz II: Argument vollständig aber eine Lücke, z.\,B. man benutzt die Annahme der Endlichdimensionalität von $V$ nicht.\\
|
||
\hdashline
|
||
4 &Ansatz I/II: Argument vollständig kleine Fehler.\\
|
||
\hline
|
||
3 &Ein paar Lücken/Fehler, aber generell überzeugend.\\
|
||
\hdashline
|
||
3 &Argument baut auf falscher Annahme auf, aber Annahme „rechtfertigt“.\\
|
||
\hline
|
||
2 &Nur triviale Inklusion bewiesen.\\
|
||
\hdashline
|
||
2 &Argument baut auf falscher Annahme auf und Annahme kaum rechtfertigt.\\
|
||
\hline
|
||
1 &Man fing mit ein paar richtigen Schritt von einem Ansatz an aber es fehlte der Rest.\\
|
||
\hline
|
||
0 &sonst.\\
|
||
\end{punktschema}
|
||
|
||
{\footnotesize
|
||
\textbf{Bemerkungen.}
|
||
Auf folgende Qualitätsmerkmale wurde geschaut:
|
||
Wichtig war, dass der ganze Text aus (a) + (b) eine kohärente Erzählung bildete.
|
||
Bezeichnungen und Ausdrücke mussten sinnvoll sein.
|
||
Man darf Symbole nicht willkürlich hinschreiben.
|
||
Es darf keine Kreisschlüsse oder sonstige logische Täuschungen geben.
|
||
Es darf nicht sein, dass dem Text zu entnehmen ist, dann man sich seiner eigenen Argumentation nicht bewusst ist.
|
||
In der gesamten Aufgabe musste die Hauptannahme der Aufgabe
|
||
(in diesem Falle $\phi\circ\phi=\phi$)
|
||
gebraucht (oder zumindest angesprochen) werden.
|
||
}
|
||
|
||
\end{document}
|