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10889 lines
426 KiB
10889 lines
426 KiB
%% ******************************************************************************** |
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%% AUTHOR: Raj Dahya |
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%% CREATED: November 2020 |
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%% EDITED: — |
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%% TYPE: Notizen |
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%% TITLE: Lösungen zu diversen Aufgaben im Kurs |
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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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%% ******************************************************************************** |
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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-localmacros.tex; |
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%% | |
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%% ---- front/index.tex; |
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%% | |
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%% ---- front/title.tex; |
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%% | |
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%% ---- front/foreword.tex; |
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%% | |
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%% ---- front/contents.tex; |
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%% | |
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%% ---- body/index.tex; |
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%% | |
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%% ---- body/uebung/ueb1.tex; |
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%% | |
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%% ---- body/uebung/ueb2.tex; |
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%% | |
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%% ---- body/uebung/ueb3.tex; |
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%% | |
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%% ---- body/uebung/ueb4.tex; |
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%% | |
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%% ---- body/uebung/ueb5.tex; |
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%% | |
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%% ---- body/uebung/ueb6.tex; |
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%% | |
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%% ---- body/uebung/ueb7.tex; |
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%% | |
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%% ---- body/uebung/ueb8.tex; |
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%% | |
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%% ---- body/uebung/ueb9.tex; |
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%% | |
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%% ---- body/uebung/ueb10.tex; |
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%% | |
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%% ---- body/uebung/ueb11.tex; |
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%% | |
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%% ---- body/ska/ska4.tex; |
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%% | |
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%% ---- body/ska/ska5.tex; |
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%% | |
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%% ---- body/ska/ska6.tex; |
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%% | |
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%% ---- body/quizzes/quiz1.tex; |
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%% | |
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%% ---- body/quizzes/quiz2.tex; |
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%% | |
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%% ---- body/quizzes/quiz3.tex; |
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%% | |
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%% ---- body/quizzes/quiz4.tex; |
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%% | |
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%% ---- body/quizzes/quiz5.tex; |
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%% | |
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%% ---- body/quizzes/quiz6.tex; |
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%% | |
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%% ---- body/quizzes/quiz7.tex; |
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%% | |
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%% ---- body/quizzes/quiz8.tex; |
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%% | |
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%% ---- body/quizzes/quiz9.tex; |
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%% | |
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%% ---- body/quizzes/quiz10.tex; |
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%% | |
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%% ---- body/quizzes/quiz11.tex; |
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%% | |
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%% ---- back/index.tex; |
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%% | |
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%% ---- ./back/quelle.bib; |
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%% |
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%% DOCUMENT-RANDOM-SEED: 5637845 |
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%% ******************************************************************************** |
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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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%% ******************************************************************************** |
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%% FILE: parameters.tex |
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%% ******************************************************************************** |
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|
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%% ******************************************************************************** |
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%% FILE: srclocal/index.tex |
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%% ******************************************************************************** |
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|
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\makeatletter |
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|
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%% ******************************************************************************** |
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%% FILE: src/setup-type.tex |
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%% ******************************************************************************** |
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|
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\documentclass[ |
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12pt, |
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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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%% ******************************************************************************** |
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%% FILE: src/setup-packages.tex |
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%% ******************************************************************************** |
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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{ |
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bookmarks=true, |
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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{ |
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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{ |
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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{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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|
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\pgfplotsset{compat=newest} |
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|
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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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|
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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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|
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\ifpdf |
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\usepackage{pdfcolmk} |
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\fi |
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|
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\usepackage{mdframed} |
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|
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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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|
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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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%% ******************************************************************************** |
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%% FILE: src/setup-parameters.tex |
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%% ******************************************************************************** |
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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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|
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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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|
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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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|
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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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%% **************************************************************** |
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%% TEX: |
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%% **************************************************************** |
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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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|
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\def\nichtzeigen#1{\phantom{#1}} |
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|
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%% **************************************************************** |
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%% SPACING: |
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%% **************************************************************** |
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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}} |
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\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}} |
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\def\erlaubeplatz{\relax\ifmmode\else\@\xspace\fi} |
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\def\entferneplatz{\relax\ifmmode\else\expandafter\@gobble\fi} |
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|
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%% **************************************************************** |
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%% TAGS / BEZEICHNUNGEN / LABELLING: |
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%% **************************************************************** |
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|
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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}} |
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\def\rlabel@aux#1[#2]#3#4#5{% |
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\send@toaux{\newlabel{#1}{{\@currentlabel}{\thepage}{{\unexpanded{#5}}}{#2.\csname the#2\endcsname}{}}}\relax% |
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} |
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|
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%% \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]{}}} |
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\def\tag@rawscheme@@#1#2[#3]#4#5[#6]#7{% |
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\ifthenelse{\equal{#6}{*}}{% |
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\ifthenelse{\equal{#7}{\boolleer}}{\refstepcounter{#3}#4\csname the#3\endcsname#5}{#4#7#5}% |
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}{% |
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\refstepcounter{#3}#4% |
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\ifthenelse{\equal{#7}{\boolleer}}{\rlabel{#6}[#3]{#1}{#2}{\csname the#3\endcsname}}{\rlabel{#6}[#3]{#1}{#2}{#7}}% |
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#5% |
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}% |
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} |
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%% \tag@scheme{CREF-SHORT}{CREF-LONG}[CTR] [LABEL]{DISPLAYTEXT} |
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\def\tag@scheme#1#2[#3]{\tag@rawscheme{#1}{#2}[#3]{\upshape(}{\upshape)}} |
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|
||
%% \eqtag[LABEL]{DISPLAYTEXT} |
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\def\eqtag@post#1{\makebox[0pt][r]{#1}} |
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\def\eqtag@pre{\tag@scheme{Eq}{Equation}[Xe]} |
||
\def\eqtag{\@ifnextchar[{\eqtag@}{\eqtag@[*]}} |
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\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{% |
||
\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}[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-localmacros.tex |
||
%% ******************************************************************************** |
||
|
||
%% **************************************************************** |
||
%% MATHE: |
||
%% **************************************************************** |
||
|
||
\def\cal#1{\mathcal{#1}} |
||
\def\reell{\mathbb{R}} |
||
\def\kmplx{\mathbb{C}} |
||
\def\Torus{\mathbb{T}} |
||
\def\rtnl{\mathbb{Q}} |
||
\def\intgr{\mathbb{Z}} |
||
|
||
\def\ntrl{\mathbb{N}} |
||
\def\ntrlpos{\mathbb{N}} |
||
\def\ntrlzero{\mathbb{N}_{0}} |
||
\def\reellNonNeg{\reell_{+}} |
||
|
||
\def\imageinh{\imath} |
||
\def\ReTeil{\mathop{\mathfrak{R}\text{\upshape e}}} |
||
\def\ImTeil{\mathop{\mathfrak{I}\text{\upshape m}}} |
||
|
||
\def\leer{\emptyset} |
||
\def\restr#1{\vert_{#1}} |
||
\def\ohne{\mathbin{\setminus}} |
||
\def\Pot{\mathop{\mathcal{P}}} |
||
\def\einser{\mathbf{1}} |
||
\def\supp{\mathop{\mathrm{supp}}} |
||
|
||
\def\brkt#1{\langle{}#1{}\rangle} |
||
\def\lsim{\mathop{\sim}} |
||
\def\lneg{\mathop{\neg}} |
||
\def\land{\mathop{\wedge}} |
||
\def\lor{\mathop{\vee}} |
||
|
||
\def\eps{\varepsilon} |
||
\let\altphi\phi |
||
\let\altvarphi\varphi |
||
\def\phi{\altvarphi} |
||
\def\varphi{\altphi} |
||
|
||
\def\vectorspacespan{\mathop{\text{\upshape Lin}}} |
||
\def\dim{\mathop{\text{\upshape dim}}} |
||
\def\rank{\mathop{\text{\upshape Rang}}} |
||
\def\onematrix{\text{\upshape\bfseries I}} |
||
\def\zeromatrix{\text{\upshape\bfseries 0}} |
||
\def\zerovector{\text{\upshape\bfseries 0}} |
||
|
||
\def\graph{\mathop{\text{\upshape Gph}}} |
||
\def\domain{\mathop{\text{\upshape dom}}} |
||
\def\range{\mathop{\text{\upshape Bild}}} |
||
\def\ker{\mathop{\text{\upshape Kern}}} |
||
\def\functionspace{\mathop{\text{\upshape Abb}}} |
||
\def\id{\text{\upshape id}} |
||
\def\modfn{\mathop{\text{\upshape mod}}} |
||
\def\divides{\mathbin{\mid}} |
||
\def\ndivides{\mathbin{\nmid}} |
||
\def\ggT{\mathop{\text{\upshape ggT}}} |
||
\def\choose#1#2{\begin{smatrix}#1\\#2\\\end{smatrix}} |
||
|
||
\makeatother |
||
|
||
\begin{document} |
||
\startdocumentlayoutoptions |
||
|
||
%% FRONTMATTER: |
||
\thispagestyle{plain} |
||
|
||
%% ******************************************************************************** |
||
%% FILE: front/index.tex |
||
%% ******************************************************************************** |
||
|
||
%% ******************************************************************************** |
||
%% FILE: front/title.tex |
||
%% ******************************************************************************** |
||
|
||
\begin{titlepage} |
||
\null |
||
|
||
\vraum |
||
|
||
\noindent\rule{\linewidth}{2pt} |
||
|
||
{\hraum\LARGE Lineare Algebra I\hraum}\\ |
||
{\hraum\LARGE $\oast$\,\rule[0.175\baselineskip]{0.65\linewidth}{1pt}\,$\oast$ \hraum}\\ |
||
{\hraum\Large Lösungen zu diversen Aufgaben im Kurs\hraum} |
||
|
||
\noindent\rule{\linewidth}{2pt} |
||
|
||
\vraum |
||
|
||
\noindent |
||
\hraum{\footnotesize Raj Dahya}\hraum\\ |
||
\hraum{\small \itshape Fakultät für Mathematik und Informatik}\hraum\\ |
||
\hraum{\small \itshape Universität Leipzig.}\hraum\\ |
||
\hraum{\small Wintersemester 2020/2021 }\hraum |
||
\end{titlepage} |
||
|
||
%% ******************************************************************************** |
||
%% FILE: front/foreword.tex |
||
%% ******************************************************************************** |
||
|
||
\chapter*{Vorwort} |
||
|
||
Dieses Dokument enthält Lösungsansätze zu den Übungsserien, Selbstkontrollenaufgaben, und Quizzes. |
||
(Diese werden natürlich \emph{nach} Abgabefristen hochgeladen.) |
||
Der Zweck dieser Lösungen besteht darin, Ansätze zu präsentieren, |
||
mit denen man seine eigenen Versuche vergleichen kann. |
||
|
||
%% ******************************************************************************** |
||
%% FILE: front/contents.tex |
||
%% ******************************************************************************** |
||
|
||
\kopfzeiledefault |
||
\footnotesize |
||
\setcounter{tocdepth}{1} |
||
\def\contentsname{Inhaltsverzeichnis} |
||
|
||
\tableofcontents |
||
|
||
%% HAUPTTEXT: |
||
|
||
%% ******************************************************************************** |
||
%% FILE: body/index.tex |
||
%% ******************************************************************************** |
||
|
||
\setcounternach{part}{1} |
||
\part{Übungsserien} |
||
|
||
\def\chaptername{Übungsserie} |
||
|
||
%% ******************************************************************************** |
||
%% FILE: body/uebung/ueb1.tex |
||
%% ******************************************************************************** |
||
|
||
\setcounternach{chapter}{1} |
||
\chapter[Woche 1]{Woche 1} |
||
\label{ueb:1} |
||
|
||
\textbf{ACHTUNG.} |
||
Diese Lösungen dienen \emph{nicht} als Musterlösungen sondern eher als Referenz. |
||
Hier wird eingehender gearbeitet, als generell verlangt wird. |
||
Das Hauptziel hier ist, eine Variant anzubieten, gegen die man seine Versuche vergleichen kann. |
||
|
||
%% AUFGABE 1-1 |
||
\let\altsectionname\sectionname |
||
\def\sectionname{Aufgabe} |
||
\section[Aufgabe 1]{} |
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\label{ueb:1:ex:1} |
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\let\sectionname\altsectionname |
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|
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Zu bestimmen ist die Lösungsmenge |
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|
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\begin{mathe}[mc]{rcl} |
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L_{\alpha,\beta} &:= &\{ |
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\mathbf{x}\in\reell^{n} |
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\mid A_{\alpha}\mathbf{x}=\mathbf{b}_{\beta} |
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\}\\ |
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\end{mathe} |
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|
||
für $\alpha,\beta\in\reell$, |
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wobei $m=3$ und $n=4$, und |
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$A_{\alpha}\in\reell^{m\times n}$ und $\mathbf{b}_{\beta}\in\reell^{m}$ |
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durch |
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|
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\begin{mathe}[mc]{rclqrcl} |
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A_{\alpha} &:= &\begin{matrix}{cccc} |
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1 &7 &2 &-1\\ |
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1 &8 &6 &-3\\ |
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2 &14 &\alpha &-2\\ |
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\end{matrix} |
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&\mathbf{b}_{\beta} &:= &\begin{vector} 4\\ 0\\ \beta\\\end{vector} |
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\end{mathe} |
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|
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gegeben sind. |
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Um die Lösungsmenge zu bestimmen führen wir das Gaußverfahren aus: |
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|
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\begin{algorithm}[\rtab][\rtab] |
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Ursprüngliches LGS $(A_{\alpha}|b_{\beta})$: |
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|
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\begin{mathe}[mc]{c} |
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\begin{matrix}{cccc|c} |
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1 &7 &2 &-1 &4\\ |
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1 &8 &6 &-3 &0\\ |
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2 &14 &\alpha &-2 &\beta\\ |
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\end{matrix}\\ |
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\end{mathe} |
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|
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Wende die Zeilentransformationen |
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|
||
{\footnotesize |
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\begin{mathe}[mc]{rcl} |
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Z_{2} &\leftsquigarrow &Z_{2}-Z_{1}\\ |
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Z_{3} &\leftsquigarrow &Z_{3}-2\cdot Z_{1}\\ |
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\end{mathe}} |
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|
||
an: |
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|
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\begin{mathe}[mc]{c} |
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\begin{matrix}{cccc|c} |
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\boxed{1} &7 &2 &-1 &4\\ |
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0 &\boxed{1} &4 &-2 &-4\\ |
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0 &0 &\boxed{\alpha - 4} &0 &\beta - 8\\ |
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\end{matrix}\\ |
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\end{mathe} |
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|
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\end{algorithm} |
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|
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Die eingezeichneten Einträge markieren die ersten Einträge der Stufen. |
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Es gibt also $2$ oder $3$ Stufen, je nachdem, ob ${\alpha - 4=0}$. |
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Dies führt zu einem Fallunterschied: |
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|
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\begin{enumerate}{\bfseries {Fall} 1.} |
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%% FALL 1 |
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\item $\alpha-4=0$. Das heißt, $\alpha=4$. |
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In diesem Falle hat das augmentierte System genau $2$ Stufen |
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und sieht wie folgt aus: |
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|
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\begin{mathe}[mc]{c} |
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\begin{matrix}{cccc|c} |
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\boxed{1} &7 &2 &-1 &4\\ |
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0 &\boxed{1} &4 &-2 &-4\\ |
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0 &0 &0 &0 &\beta - 8\\ |
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\end{matrix}\\ |
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\end{mathe} |
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|
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Dies führt zu zwei weiteren Fällen, denn die $3$. Gleichung ist jetzt genau dann lösbar, |
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wenn $\beta-8=0$. |
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|
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\begin{enumerate}{\bfseries {Fall 1}a.} |
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%% FALL 1a |
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\item $\beta-8\neq 0$. Das heißt, $\beta\neq 8$. |
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Dann ist die $3$. Gleichung und damit das LGS nicht lösbar. |
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Darum erhalten wir $\boxed{L_{\alpha,\beta}=\leer}$. |
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|
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%% FALL 1b |
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\item $\beta-8=0$. Das heißt, $\beta=8$. |
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Dann ist die $3$. Gleichung trivialerweise erfüllt. |
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Das augmentierte System sieht wird zum |
||
|
||
\begin{mathe}[mc]{c} |
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\begin{matrix}{cccc|c} |
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\boxed{1} &7 &2 &-1 &4\\ |
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0 &\boxed{1} &4 &-2 &-4\\ |
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0 &0 &0 &0 &0\\ |
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\end{matrix}\\ |
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\end{mathe} |
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|
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und kann jetzt aufgelöst werden. |
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Wir arbeiten von unten nach oben: |
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|
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\begin{algorithm}[2\rtab][\rtab] |
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Aus der ganzen Zeilenstufenform erschließt sich |
||
|
||
\begin{mathe}[mc]{c} |
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x_{3},\, x_{4}\,\text{sind frei}\\ |
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\end{mathe} |
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|
||
Aus der Stufenform von Gleichungen $2$ und $1$ erschließt sich |
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|
||
\begin{mathe}[mc]{rcl} |
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x_{2} &= &-4 - 4x_{3} + 2x_{4}\\ |
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x_{1} &= &4 - 7x_{2} - 2x_{3} + x_{4}\\ |
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&= &4 - 7(-4 - 4x_{3} + 2x_{4}) - 2x_{3} + x_{4}\\ |
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&= &32 + 26x_{3} + -13x_{4}\\ |
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\end{mathe} |
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|
||
Zusammengefasst erhalten wir die allgemeine Form der Lösung: |
||
|
||
\begin{mathe}[mc]{rcl} |
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\mathbf{x} &= &\begin{svector} x_{1}\\ x_{2}\\ x_{3}\\ x_{4}\\\end{svector}\\ |
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&= &\begin{svector} 32 + 26x_{3} + -13x_{4}\\ -4 - 4x_{3} + 2x_{4}\\ x_{3}\\ x_{4}\\\end{svector}\\ |
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&= &\begin{svector} 32 + 26x_{3} + -13x_{4}\\ -4 - 4x_{3} + 2x_{4}\\ 0 + 1x_{3} + 0x_{4}\\ 0 + 0x_{3} + 1x_{4}\\\end{svector}\\ |
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&= &\begin{svector} 32\\ -4\\ 0\\ 0\\\end{svector} |
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+ \begin{svector} 26x_{3}\\ -4x_{3}\\ 1x_{3}\\ 0x_{3}\\\end{svector} |
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+ \begin{svector} -13x_{4}\\ 2x_{4}\\ 1x_{4}\\ 1x_{4}\\\end{svector}\\ |
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&= &\begin{svector} 32\\ -4\\ 0\\ 0\\\end{svector} |
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+ x_{3}\cdot\begin{svector} 26\\ -4\\ 1\\ 0\\\end{svector} |
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+ x_{4}\cdot\begin{svector} -13\\ 2\\ 1\\ 1\\\end{svector}\\ |
||
\end{mathe} |
||
|
||
mit $x_{3}$, $x_{4}$ frei wählbar. |
||
\end{algorithm} |
||
|
||
Also erhalten wird in diesem Falle |
||
$\boxed{ |
||
L_{\alpha,\beta}=\left\{ |
||
\begin{svector} 32\\ -4\\ 0\\ 0\\\end{svector} |
||
+ t_{1}\cdot\begin{svector} 26\\ -4\\ 1\\ 0\\\end{svector} |
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+ t_{2}\cdot\begin{svector} -13\\ 2\\ 1\\ 1\\\end{svector} |
||
\mid t_{1}, t_{2}\in\reell |
||
\right\} |
||
}$, |
||
oder etwas kompakter formuliert, |
||
${L_{\alpha,\beta}=\begin{svector} 32\\ -4\\ 0\\ 0\\\end{svector} + \vectorspacespan\left\{\begin{svector} 26\\ -4\\ 1\\ 0\\\end{svector}, \begin{svector} -13\\ 2\\ 1\\ 1\\\end{svector}\right\}}$. |
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\end{enumerate} |
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|
||
%% FALL 2 |
||
\item $\alpha-4\neq 0$. Das heißt, $\alpha\neq 4$. |
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In diesem Falle hat das augmentierte System genau $3$ Stufen und diesmal ist nur $x_{4}$ frei. |
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Man beachte, dass dies im Grunde genau wie Fall 1b ist, nur dass wir zusätzlich Gleichung 3 beachten und $x_{3}$ bestimmen müssen. |
||
|
||
\begin{algorithm}[2\rtab][\rtab] |
||
Aus der Stufenform von Gleichungen $3$ ergibt sich |
||
|
||
\begin{mathe}[mc]{rcl} |
||
x_{3} &= &\frac{\beta-8}{\alpha-4}\\ |
||
\end{mathe} |
||
|
||
Der Rest der Lösung des Gleichungssystems verhält sich genau wie im Fall 3b, |
||
das heißt |
||
|
||
\begin{mathe}[mc]{rcl} |
||
\mathbf{x} &= &\begin{svector} 32\\ -4\\ 0\\ 0\\\end{svector} |
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+ x_{3}\cdot\begin{svector} 26\\ -4\\ 1\\ 0\\\end{svector} |
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+ x_{4}\cdot\begin{svector} -13\\ 2\\ 1\\ 1\\\end{svector}\\ |
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&= &\begin{svector} 32\\ -4\\ 0\\ 0\\\end{svector} |
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+ \frac{\beta-8}{\alpha-4}\cdot\begin{svector} 26\\ -4\\ 1\\ 0\\\end{svector} |
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+ x_{4}\cdot\begin{svector} -13\\ 2\\ 1\\ 1\\\end{svector},\\ |
||
\end{mathe} |
||
|
||
wobei $x_{4}$ frei wählbar ist. |
||
\end{algorithm} |
||
|
||
Also erhalten wird in diesem Falle |
||
$\boxed{ |
||
L_{\alpha,\beta}=\left\{ |
||
\begin{svector} 32\\ -4\\ 0\\ 0\\\end{svector} |
||
+ \frac{\beta-8}{\alpha-4}\cdot\begin{svector} 26\\ -4\\ 1\\ 0\\\end{svector} |
||
+ t\cdot\begin{svector} -13\\ 2\\ 1\\ 1\\\end{svector} |
||
\mid t\in\reell |
||
\right\} |
||
}$, |
||
oder etwas kompakter formuliert, |
||
${L_{\alpha,\beta}=\begin{svector} 32\\ -4\\ 0\\ 0\\\end{svector} + \frac{\beta-8}{\alpha-4}\cdot\begin{svector} 26\\ -4\\ 1\\ 0\\\end{svector} + \vectorspacespan\left\{\begin{svector} -13\\ 2\\ 1\\ 1\\\end{svector}\right\}}$. |
||
\end{enumerate} |
||
|
||
Wir fassen die Lösung für alle Fälle zusammen: |
||
|
||
\begin{mathe}[mc]{rcl} |
||
L_{\alpha,\beta} &= &\begin{cases}[m]{lcl} |
||
\leer &: &\alpha=4,\,\beta\neq 8\\ |
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\mathbf{u} + \vectorspacespan\{\mathbf{v},\mathbf{w}\} &: &\alpha=4,\,\beta=8\\ |
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\mathbf{u} + \frac{\alpha-4}{\beta-8}\mathbf{v} + \vectorspacespan\{\mathbf{w}\} &: &\alpha\neq 4\\ |
||
\end{cases} |
||
\end{mathe} |
||
|
||
für alle $\alpha,\beta\in\reell$, |
||
wobei |
||
$\mathbf{u} = \begin{svector} 32\\ -4\\ 0\\ 0\\\end{svector}$, |
||
$\mathbf{v} = \begin{svector} 26\\ -4\\ 1\\ 0\\\end{svector}$, |
||
$\mathbf{w} = \begin{svector} -13\\ 2\\ 1\\ 1\\\end{svector}$. |
||
|
||
%% AUFGABE 1-2 |
||
\let\altsectionname\sectionname |
||
\def\sectionname{Aufgabe} |
||
\section[Aufgabe 2]{} |
||
\label{ueb:1:ex:2} |
||
\let\sectionname\altsectionname |
||
|
||
\begin{schattierteboxdunn} |
||
\begin{satz} |
||
\makelabel{satz:main:ueb:1:ex:2} |
||
Angewandt auf die erweiterte Koeffizientenmatrix eines linearen Gleichungssystems |
||
verändern |
||
die elementaren Zeilenumformungen vom Typ (I), (II) und (III) |
||
die Menge der Lösungen nicht. |
||
\end{satz} |
||
\end{schattierteboxdunn} |
||
|
||
Wir beweisen \Cref{satz:main:ueb:1:ex:2} mithilfe der folgenden Teilergebnisse. |
||
|
||
\begin{lemm} |
||
\makelabel{lemm:1:ueb:1:ex:2} |
||
Seien $m,n\in\ntrlpos$ und $A\in\reell^{m\times n}$ und $\mathbf{b}\in\reell^{m}$. |
||
Für $i,j\in\{1,2,\ldots,m\}$ mit $i\neq j$ bezeichne mit |
||
|
||
\begin{mathe}[mc]{rcl} |
||
(A|\mathbf{b}) &\overset{I;i,j}{\rightsquigarrow} &(A'|\mathbf{b}')\\ |
||
\end{mathe} |
||
|
||
die Anwendung von Zeilentransformation (I) auf $(A|\mathbf{b})$, |
||
wobei Zeile${}_{i}$ und Zeile${}_{j}$ umgetauscht werden, |
||
was in $(A'|\mathbf{b}')$ resultiert. |
||
Dann für alle ${\mathbf{x}\in\reell^{n}}$, |
||
falls $\mathbf{x}$ eine Lösung für $(A|\mathbf{b})$ ist, |
||
dann ist $\mathbf{x}$ eine Lösung für $(A'|\mathbf{b}')$. |
||
\end{lemm} |
||
|
||
\begin{einzug}[\rtab][\rtab] |
||
\begin{proof} |
||
Betrachte den Fall $i<j$. |
||
Es gilt |
||
|
||
\begin{longtable}[mc]{RL} |
||
&\text{$\mathbf{x}$ eine Lösung für $(A|\mathbf{b})$}\\ |
||
\Longrightarrow |
||
&{\scriptsize |
||
\left\{ |
||
\begin{array}[m]{crccccclcl} |
||
&(a_{1,1}x_{1} &+ &a_{1,2}x_{2} &+ &\cdots &+ &a_{1,n}x_{n} &= &b_{1})\\ |
||
\text{und} &(a_{2,1}x_{1} &+ &a_{2,2}x_{2} &+ &\cdots &+ &a_{2,n}x_{n} &= &b_{2})\\ |
||
\cdots\\ |
||
\text{und} &(a_{i,1}x_{1} &+ &a_{i,2}x_{2} &+ &\cdots &+ &a_{i,n}x_{n} &= &b_{i})\\ |
||
\cdots\\ |
||
\text{und} &(a_{j,1}x_{1} &+ &a_{j,2}x_{2} &+ &\cdots &+ &a_{j,n}x_{n} &= &b_{j})\\ |
||
\cdots\\ |
||
\text{und} &(a_{m,1}x_{1} &+ &a_{m,2}x_{2} &+ &\cdots &+ &a_{m,n}x_{n} &= &b_{m}) |
||
\end{array} |
||
\right.}\\ |
||
\\ |
||
\Longrightarrow |
||
&{\scriptsize |
||
\left\{ |
||
\begin{array}[m]{crccccclcl} |
||
&(a_{1,1}x_{1} &+ &a_{1,2}x_{2} &+ &\cdots &+ &a_{1,n}x_{n} &= &b_{1})\\ |
||
\text{und} &(a_{2,1}x_{1} &+ &a_{2,2}x_{2} &+ &\cdots &+ &a_{2,n}x_{n} &= &b_{2})\\ |
||
\cdots\\ |
||
\text{und} &(a_{j,1}x_{1} &+ &a_{j,2}x_{2} &+ &\cdots &+ &a_{j,n}x_{n} &= &b_{j})\\ |
||
\cdots\\ |
||
\text{und} &(a_{i,1}x_{1} &+ &a_{i,2}x_{2} &+ &\cdots &+ &a_{i,n}x_{n} &= &b_{i})\\ |
||
\cdots\\ |
||
\text{und} &(a_{m,1}x_{1} &+ &a_{m,2}x_{2} &+ &\cdots &+ &a_{m,n}x_{n} &= &b_{m}) |
||
\end{array} |
||
\right.}\\ |
||
\\ |
||
&\text{da lediglich zwei Aussagen in einer Konjunktion umgetauscht werden}\\ |
||
\\ |
||
\Longrightarrow |
||
&\text{$\mathbf{x}$ eine Lösung für $(A'|\mathbf{b})'$, da $(A|\mathbf{b})\overset{I;i,j}{\rightsquigarrow}(A'|\mathbf{b}')$.}\\ |
||
\end{longtable} |
||
|
||
Der Fall $i>j$ lässt sich analog zeigen. |
||
Falls $i=j$ bleibt das System unverändert, sodass die Behauptung trivialerweise gilt. |
||
\end{proof} |
||
\end{einzug} |
||
|
||
\begin{lemm} |
||
\makelabel{lemm:2:ueb:1:ex:2} |
||
Seien $m,n\in\ntrlpos$ und $A\in\reell^{m\times n}$ und $\mathbf{b}\in\reell^{m}$. |
||
Für ${i\in\{1,2,\ldots,m\}}$ und ${\alpha\in\reell\ohne\{0\}}$ bezeichne mit |
||
|
||
\begin{mathe}[mc]{rcl} |
||
(A|\mathbf{b}) &\overset{II;i,\alpha}{\rightsquigarrow} &(A'|\mathbf{b}')\\ |
||
\end{mathe} |
||
|
||
die Anwendung von Zeilentransformation (II) auf $(A|\mathbf{b})$, |
||
wobei Zeile${}_{i}$ durch $\alpha\cdot$Zeile${}_{i}$ ersetzt wird, |
||
was in $(A'|\mathbf{b}')$ resultiert. |
||
Dann für alle ${\mathbf{x}\in\reell^{n}}$, |
||
falls $\mathbf{x}$ eine Lösung für $(A|\mathbf{b})$ ist, |
||
dann ist $\mathbf{x}$ eine Lösung für $(A'|\mathbf{b}')$. |
||
\end{lemm} |
||
|
||
\begin{einzug}[\rtab][\rtab] |
||
\begin{proof} |
||
Es gilt |
||
|
||
\begin{longtable}[mc]{RL} |
||
&\text{$\mathbf{x}$ eine Lösung für $(A|\mathbf{b})$}\\ |
||
\Longrightarrow |
||
&{\scriptsize |
||
\left\{ |
||
\begin{array}[m]{crccccclcl} |
||
&(a_{1,1}x_{1} &+ &a_{1,2}x_{2} &+ &\cdots &+ &a_{1,n}x_{n} &= &b_{1})\\ |
||
\text{und} &(a_{2,1}x_{1} &+ &a_{2,2}x_{2} &+ &\cdots &+ &a_{2,n}x_{n} &= &b_{2})\\ |
||
\cdots\\ |
||
\text{und} &(a_{i,1}x_{1} &+ &a_{i,2}x_{2} &+ &\cdots &+ &a_{i,n}x_{n} &= &b_{i})\\ |
||
\cdots\\ |
||
\text{und} &(a_{m,1}x_{1} &+ &a_{m,2}x_{2} &+ &\cdots &+ &a_{m,n}x_{n} &= &b_{m}) |
||
\end{array} |
||
\right.}\\ |
||
\\ |
||
\Longrightarrow |
||
&{\scriptsize |
||
\left\{ |
||
\begin{array}[m]{crccccclcl} |
||
&(a_{1,1}x_{1} &+ &a_{1,2}x_{2} &+ &\cdots &+ &a_{1,n}x_{n} &= &b_{1})\\ |
||
\text{und} &(a_{2,1}x_{1} &+ &a_{2,2}x_{2} &+ &\cdots &+ &a_{2,n}x_{n} &= &b_{2})\\ |
||
\cdots\\ |
||
\text{und} &(\alpha\cdot (a_{i,1}x_{1} &+ &a_{i,2}x_{2} &+ &\cdots &+ &a_{i,n}x_{n}) &= &\alpha\cdot b_{i})\\ |
||
\cdots\\ |
||
\text{und} &(a_{m,1}x_{1} &+ &a_{m,2}x_{2} &+ &\cdots &+ &a_{m,n}x_{n} &= &b_{m}) |
||
\end{array} |
||
\right.}\\ |
||
\\ |
||
\Longrightarrow |
||
&{\scriptsize |
||
\left\{ |
||
\begin{array}[m]{crccccclcl} |
||
&(a_{1,1}x_{1} &+ &a_{1,2}x_{2} &+ &\cdots &+ &a_{1,n}x_{n} &= &b_{1})\\ |
||
\text{und} &(a_{2,1}x_{1} &+ &a_{2,2}x_{2} &+ &\cdots &+ &a_{2,n}x_{n} &= &b_{2})\\ |
||
\cdots\\ |
||
\text{und} &(\alpha\cdot a_{i,1}x_{1} &+ &\alpha\cdot a_{i,2}x_{2} &+ &\cdots &+ &\alpha\cdot a_{i,n}x_{n} &= &\alpha\cdot b_{i})\\ |
||
\cdots\\ |
||
\text{und} &(a_{m,1}x_{1} &+ &a_{m,2}x_{2} &+ &\cdots &+ &a_{m,n}x_{n} &= &b_{m}) |
||
\end{array} |
||
\right.}\\ |
||
\\ |
||
&\text{$\mathbf{x}$ eine Lösung für $(A'|\mathbf{b})'$, da $(A|\mathbf{b})\overset{II;i,\alpha}{\rightsquigarrow}(A'|\mathbf{b}')$.} |
||
\end{longtable} |
||
|
||
Also gilt die Behauptung. |
||
\end{proof} |
||
\end{einzug} |
||
|
||
\begin{lemm} |
||
\makelabel{lemm:3:ueb:1:ex:2} |
||
Seien $m,n\in\ntrlpos$ und $A\in\reell^{m\times n}$ und $\mathbf{b}\in\reell^{m}$. |
||
Für ${i,j\in\{1,2,\ldots,m\}}$ mit $i\neq j$ und $\alpha\in\reell$ bezeichne mit |
||
|
||
\begin{mathe}[mc]{rcl} |
||
(A|\mathbf{b}) &\overset{III;i,j,\alpha}{\rightsquigarrow} &(A'|\mathbf{b}')\\ |
||
\end{mathe} |
||
|
||
die Anwendung von Zeilentransformation (III) auf $(A|\mathbf{b})$, |
||
wobei Zeile${}_{i}$ durch die Addition von Zeile${}_{i}$ mit $\alpha\cdot$Zeile${}_{j}$ ersetzt wird, |
||
was in $(A'|\mathbf{b}')$ resultiert. |
||
Dann für alle ${\mathbf{x}\in\reell^{n}}$, |
||
falls $\mathbf{x}$ eine Lösung für $(A|\mathbf{b})$ ist, |
||
dann ist $\mathbf{x}$ eine Lösung für $(A'|\mathbf{b}')$. |
||
\end{lemm} |
||
|
||
\begin{einzug}[\rtab][\rtab] |
||
\begin{proof} |
||
Es gilt |
||
|
||
\begin{longtable}[mc]{RL} |
||
&\text{$\mathbf{x}$ eine Lösung für $(A|\mathbf{b})$}\\ |
||
\Longrightarrow |
||
&{\scriptsize |
||
\left\{ |
||
\begin{array}[m]{crccccclcl} |
||
&(a_{1,1}x_{1} &+ &a_{1,2}x_{2} &+ &\cdots &+ &a_{1,n}x_{n} &= &b_{1})\\ |
||
\text{und} &(a_{2,1}x_{1} &+ &a_{2,2}x_{2} &+ &\cdots &+ &a_{2,n}x_{n} &= &b_{2})\\ |
||
\cdots\\ |
||
\text{und} &(a_{i,1}x_{1} &+ &a_{i,2}x_{2} &+ &\cdots &+ &a_{i,n}x_{n} &= &b_{i})\\ |
||
\cdots\\ |
||
\text{und} &(a_{m,1}x_{1} &+ &a_{m,2}x_{2} &+ &\cdots &+ &a_{m,n}x_{n} &= &b_{m}) |
||
\end{array} |
||
\right.}\\ |
||
\\ |
||
\Longrightarrow |
||
&{\scriptsize |
||
\left\{ |
||
\begin{array}[m]{crccccclcl} |
||
&(a_{1,1}x_{1} &+ &a_{1,2}x_{2} &+ &\cdots &+ &a_{1,n}x_{n} &= &b_{1})\\ |
||
\text{und} &(a_{2,1}x_{1} &+ &a_{2,2}x_{2} &+ &\cdots &+ &a_{2,n}x_{n} &= &b_{2})\\ |
||
\cdots\\ |
||
\text{und} &(a_{i,1}x_{1} &+ &a_{i,2}x_{2} &+ &\cdots &+ &a_{i,n}x_{n} + \alpha\cdot b_{j} &= &b_{i} + \alpha\cdot b_{j})\\ |
||
\cdots\\ |
||
\text{und} &(a_{m,1}x_{1} &+ &a_{m,2}x_{2} &+ &\cdots &+ &a_{m,n}x_{n} &= &b_{m}) |
||
\end{array} |
||
\right.}\\ |
||
\\ |
||
\Longrightarrow |
||
&{\scriptsize |
||
\left\{ |
||
\begin{array}[m]{crccccclcl} |
||
&(a_{1,1}x_{1} &+ &a_{1,2}x_{2} &+ &\cdots &+ &a_{1,n}x_{n} &= &b_{1})\\ |
||
\text{und} &(a_{2,1}x_{1} &+ &a_{2,2}x_{2} &+ &\cdots &+ &a_{2,n}x_{n} &= &b_{2})\\ |
||
\cdots\\ |
||
\text{und} &(a_{i,1}x_{1} &+ &a_{i,2}x_{2} &+ &\cdots &+ &a_{i,n}x_{n}\\ |
||
&+\alpha\cdot a_{j,1}x_{1} &+ &\alpha\cdot a_{j,2}x_{2} &+ &\cdots &+ &\alpha\cdot a_{j,n}x_{n} &= &b_{i} + \alpha\cdot b_{j})\\ |
||
\cdots\\ |
||
\text{und} &(a_{m,1}x_{1} &+ &a_{m,2}x_{2} &+ &\cdots &+ &a_{m,n}x_{n} &= &b_{m}) |
||
\end{array} |
||
\right.}\\ |
||
\\ |
||
&\text{da laut der $j$-ten Gleichung gilt ${b_{j}=\sum_{k=1}^{m}a_{j,k}x_{k}}$}\\ |
||
\\ |
||
\Longrightarrow |
||
&{\scriptsize |
||
\left\{ |
||
\begin{array}[m]{crccccclcl} |
||
&(a_{1,1}x_{1} &+ &a_{1,2}x_{2} &+ &\cdots &+ &a_{1,n}x_{n} &= &b_{1})\\ |
||
\text{und} &(a_{2,1}x_{1} &+ &a_{2,2}x_{2} &+ &\cdots &+ &a_{2,n}x_{n} &= &b_{2})\\ |
||
\cdots\\ |
||
\text{und} &(a'_{i,1}x_{1} &+ &a'_{i,2}x_{2} &+ &\cdots &+ &a'_{i,n}x_{n} &= &b'_{i})\\ |
||
\cdots\\ |
||
\text{und} &(a_{m,1}x_{1} &+ &a_{m,2}x_{2} &+ &\cdots &+ &a_{m,n}x_{n} &= &b_{m}), |
||
\end{array} |
||
\right.}\\ |
||
\\ |
||
&\text{wobei $a'_{i,k}=a_{i,k}+\alpha\cdot a_{j,k}$ für alle $k$ und $b'_{i}=b_{i}+\alpha\cdot b_{j}$}\\ |
||
\\ |
||
\Longrightarrow |
||
&\text{$\mathbf{x}$ eine Lösung für $(A'|\mathbf{b})'$, da $(A|\mathbf{b})\overset{III;i,j,\alpha}{\rightsquigarrow}(A'|\mathbf{b}')$.} |
||
\end{longtable} |
||
|
||
Also gilt die Behauptung. |
||
\end{proof} |
||
\end{einzug} |
||
|
||
Endlich können wir \Cref{satz:main:ueb:1:ex:2} beweisen: |
||
|
||
\begin{proof}[von \Cref{satz:main:ueb:1:ex:2}] |
||
Seien $m,n\in\ntrlpos$ und $A\in\reell^{m\times n}$ und $\mathbf{b}\in\reell^{m}$. |
||
Seien $A'\in\reell^{m\times n}$ und $\mathbf{b}'\in\reell^{m}$, |
||
so dass $(A|\mathbf{b})$ durch eine Transformation der Art (I), (II) oder (III) |
||
aus $(A|\mathbf{b})$ entsteht. |
||
Das heißt, entweder |
||
|
||
\begin{mathe}[mc]{lrcl} |
||
\eqtag[eq:0:\beweislabel] |
||
&(A|\mathbf{b}) &\overset{I;i,j}{\rightsquigarrow} &(A'|\mathbf{b}')\\ |
||
\text{oder} &(A|\mathbf{b}) &\overset{I;i,\alpha}{\rightsquigarrow} &(A'|\mathbf{b}')\\ |
||
\text{oder} &(A|\mathbf{b}) &\overset{III;i,j,\alpha}{\rightsquigarrow} &(A'|\mathbf{b}')\\ |
||
\end{mathe} |
||
|
||
gilt, für ein $i,j\in\{1,2,\ldots,m\}$ mit $i\neq j$ und $\alpha\in\reell\ohne\{0\}$.\\ |
||
\textbf{Zu zeigen:} |
||
|
||
\begin{mathe}[mc]{rcl} |
||
\eqtag[eq:1:\beweislabel] |
||
\{\mathbf{x}\in\reell^{n}\mid\mathbf{x}\text{ eine Lösung für }(A|\mathbf{b})\} |
||
&= &\{\mathbf{x}\in\reell^{n}\mid\mathbf{x}\text{ eine Lösung für }(A|\mathbf{b})\}.\\ |
||
\end{mathe} |
||
|
||
Wir zeigen dies in zwei Teile: |
||
|
||
\uline{\bfseries ($\subseteq$.)}\\ |
||
Sei $\mathbf{x}\in\reell^{n}$ ein beliebiges Element aus der linken Menge, |
||
d.\,h. $\mathbf{x}$ ist eine Lösung zu $(A|\mathbf{b})$. |
||
Laut \Cref{lemm:1:ueb:1:ex:2} + \Cref{lemm:2:ueb:1:ex:2} + \Cref{lemm:3:ueb:1:ex:2} |
||
und wegen \eqcref{eq:0:\beweislabel} |
||
erhalten wir, dass $\mathbf{x}$ eine Lösung zu $(A'|\mathbf{b}')$ ist, |
||
d.\,h. $\mathbf{x}$ liegt in der rechten Menge. |
||
Also ist die linke Menge in der rechten enthalten. |
||
|
||
\uline{\bfseries ($\supseteq$.)}\\ |
||
Man beachte zuerst, dass sich die Transformation in \eqcref{eq:0:\beweislabel} umkehren lässt---\text{und zwar durch Elementartransformationen}. |
||
Es ist einfach zu sehen, dass entweder |
||
|
||
\begin{mathe}[mc]{lrcl} |
||
&(A'|\mathbf{b}') &\overset{I;i,j}{\rightsquigarrow} &(A|\mathbf{b})\\ |
||
\text{oder} &(A'|\mathbf{b}') &\overset{I;i,\alpha^{-1}}{\rightsquigarrow} &(A|\mathbf{b})\\ |
||
\text{oder} &(A'|\mathbf{b}') &\overset{III;i,j,-\alpha}{\rightsquigarrow} &(A|\mathbf{b}).\\ |
||
\end{mathe} |
||
|
||
Die Situation ist also analog zum $\subseteq$-Teil. |
||
Darum gilt die $\supseteq$-Inklusion in \eqcref{eq:1:\beweislabel}. |
||
\end{proof} |
||
|
||
\clearpage |
||
%% AUFGABE 1-3 |
||
\let\altsectionname\sectionname |
||
\def\sectionname{Aufgabe} |
||
\section[Aufgabe 3]{} |
||
\label{ueb:1:ex:3} |
||
\let\sectionname\altsectionname |
||
|
||
Für diese Aufgabe wird das Konzept der \emph{linearen Unabhängigkeit} aus Kapitel 5 angewandt. |
||
|
||
\begin{defn} |
||
Seien $m,n\in\ntrlpos$ mit $m>n$ |
||
und seien $A\in\reell^{m\times n}$, $\mathbf{b}\in\reell^{m}$, |
||
und $I\subseteq\{1,2,\ldots,m\}$. |
||
Bezeichne mit $(A|\mathbf{b})_{I}$ die erweiterte Koeffizientenmatrix $(A|\mathbf{b})$, |
||
die auf die Zeilen mit Indexes aus $I$ (in bspw. aufsteigender Reihenfolge) reduziert ist. |
||
\end{defn} |
||
|
||
\begin{e.g.} |
||
Für $(A|\mathbf{b})$ gleich |
||
|
||
{\scriptsize |
||
\begin{mathe}[mc]{c} |
||
\begin{matrix}{ccc|c} |
||
-5 &0 &0 &-7\\ |
||
4 &-6 &-10 &6\\ |
||
-2 &-6 &-6 &9\\ |
||
-7 &4 &-1 &-5\\ |
||
4 &-5 &2 &-9\\ |
||
-5 &8 &-7 &-5\\ |
||
\end{matrix} |
||
\end{mathe}} |
||
|
||
und $I=\{2,5,6\}$ ist $(A|\mathbf{b})_{I}$ gleich |
||
|
||
{\scriptsize |
||
\begin{mathe}[bc]{c} |
||
\begin{matrix}{ccc|c} |
||
4 &-6 &-10 &6\\ |
||
4 &-5 &2 &-9\\ |
||
-5 &8 &-7 &-5\\ |
||
\end{matrix}. |
||
\end{mathe}} |
||
|
||
\nvraum{1} |
||
|
||
\end{e.g.} |
||
|
||
Mit diesem Mittel können wir nun die Hauptaussage in der Aufgabe formulieren: |
||
|
||
\begin{schattierteboxdunn} |
||
\begin{satz} |
||
\makelabel{satz:main:ueb:1:ex:3} |
||
Seien $m,n\in\ntrlpos$ mit $m>n$ |
||
und seien $A\in\reell^{m\times n}$ und $\mathbf{b}\in\reell^{m}$. |
||
Falls $(A|\mathbf{b})$ unlösbar ist, |
||
dann existiert $I\subseteq\{1,2,\ldots,m\}$ mit $|I|=n+1$, |
||
so dass $(A|\mathbf{b})_{I}$ unlösbar ist. |
||
\end{satz} |
||
\end{schattierteboxdunn} |
||
|
||
\begin{einzug}[\rtab][\rtab] |
||
\begin{proof}[*][\Cref{\beweislabel}] |
||
Es stehen nun die \emph{Zeilen} der Matrix $A$ im Fokus. |
||
Wir verwandeln diese in Vektoren, d.\,h. setze |
||
|
||
\begin{mathe}[mc]{c} |
||
\mathbf{z}^{(i)}\in\reell^{n}\,\text{die $i$-te Zeile von $A$ als Vektor geschrieben} |
||
\end{mathe} |
||
|
||
für $i\in\{1,2,\ldots,m\}$. |
||
Da ${\mathbf{z}^{(1)},\mathbf{z}^{(2)},\ldots,\mathbf{z}^{(m)}\in\reell^{n}}$, |
||
können wir eine \emph{maximale Menge} ${I_{0}\subseteq\{1,2,\ldots,m\}}$ finden, |
||
so dass $(\mathbf{z}^{(i)})_{i\in I_{0}}$ aus linear unabhängigen Vektoren besteht. |
||
Wegen der Dimension von $\reell^{n}$ gilt ${|I|\leq\min\{m,n\}=n}$. |
||
Sei ${k\in\{1,2,\ldots,m\}\ohne I_{0}}$ beliebig. |
||
Wegen Maximalität muss $(\mathbf{z}^{(i)})_{i\in I_{0}\cup\{k\}}$ \emph{linear abhängig} sein. |
||
Und wegen der linearen Unabhängigkeit von $(\mathbf{z}^{(i)})_{i\in I_{0}}$ |
||
existieren (eindeutige) Koeffizienten $c_{k,i}\in\reell$ für $i\in I_{0}$ so dass |
||
|
||
\begin{mathe}[mc]{rcl} |
||
\eqtag[eq:1:\beweislabel] |
||
\mathbf{z}^{(k)} &= &\sum_{i\in I_{0}:~}c_{k,i}\mathbf{z}^{(i)}\\ |
||
\end{mathe} |
||
|
||
gilt. |
||
|
||
Um nun die Hauptaussage zu zeigen, nehmen wir an, dass $(A|\mathbf{b})$ unlösbar ist. |
||
\textbf{Zu zeigen:} Es gibt eine Teilmenge ${I\subseteq\{1,2,\ldots,m\}}$ mit ${|I|=n+1}$, |
||
so dass $(A|\mathbf{b})_{I}$ unlösbar ist. |
||
\fbox{Angenommen, dies sei nicht der Fall.} |
||
Aus dieser Annahme leiten wir folgende Behauptungen ab: |
||
|
||
\begin{kompaktitem}[\rtab][\rtab] |
||
\behauptungbeleg{1} |
||
Die Verhältnisse zwischen den Zeilenvektoren in \eqcref{eq:1:\beweislabel} gelten auch für die Einträge aus $\mathbf{b}$. |
||
Das heißt |
||
|
||
\begin{mathe}[mc]{rcl} |
||
\eqtag[eq:2:\beweislabel] |
||
b_{k} &= &\sum_{i\in I_{0}:~}c_{k,i}b_{i}\\ |
||
\end{mathe} |
||
|
||
für alle ${k\in\{1,2,\ldots,m+1\}\ohne I_{0}}$.\\ |
||
\voritemise |
||
\belegbehauptung |
||
Sei $k\in\{1,2,\ldots,m+1\}\ohne I_{0}$ beliebig. |
||
Da $|I_{0}|\leq n<n+1$ lässt sich eine Teilmenge $I\subseteq\{1,2,\ldots,m\}$ wählen, |
||
mit $I\supseteq I_{0}\cup\{k\}$ und $|I|=n+1$. |
||
Dann per \emph{Annahme} ist $(A|\mathbf{b})_{I}$ lösbar. |
||
Das heißt, $\mathbf{x}\in\reell^{n}$ existiert, so dass |
||
|
||
\begin{mathe}[mc]{rcl} |
||
\eqtag[eq:3:\beweislabel] |
||
b_{i} &= &\sum_{j=1}^{n}a_{i,j}x_{j}\\ |
||
\end{mathe} |
||
|
||
für alle $i\in I$ gilt. |
||
Da $k\in I$ und $I_{0}\subseteq I$ und wegen \eqcref{eq:1:\beweislabel} erhalten wir |
||
nun das Verhältnis |
||
|
||
\begin{longmathe}[mc]{RCL} |
||
b_{k} &= &\sum_{j=1}^{n}a_{k,j}x_{j}\\ |
||
&= &\sum_{j=1}^{n}(\mathbf{z}^{(k)})_{j}x_{j}\\ |
||
&&\quad\text{da die Einträge der $k$-ten Zeile den Einträgen von $\mathbf{z}^{(k)}$ entsprechen}\\ |
||
&\eqcrefoverset{eq:1:\beweislabel}{=} |
||
&\sum_{j=1}^{n}(\sum_{i\in I_{0}}c_{k,i}\mathbf{z}^{(i)})_{j}x_{j}\\ |
||
&= &\sum_{j=1}^{n}\sum_{i\in I_{0}}c_{k,i}z^{(i)}_{j}x_{j}\\ |
||
&= &\sum_{i\in I_{0}}c_{k,i}\sum_{j=1}^{n}z^{(i)}_{j}x_{j}\\ |
||
&= &\sum_{i\in I_{0}}c_{k,i}\sum_{j=1}^{n} |