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%
% gaushomobasis.tex -- Bestimmung einer Basis der Homologiegruppen
%
% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
%
\documentclass[tikz]{standalone}
\usepackage{amsmath}
\usepackage{times}
\usepackage{txfonts}
\usepackage{pgfplots}
\usepackage{csvsimple}
\usetikzlibrary{arrows,intersections,math}
\begin{document}
\def\skala{1}
\definecolor{darkgreen}{rgb}{0,0.6,0}
\begin{tikzpicture}[>=latex,thick,scale=\skala]
\def\s{0.5}
\def\inset{0.05}
\def\w{8}
\def\zeile#1#2{
\fill[color=#2] ({0+\inset},{(12-#1)*\s+\inset})
rectangle ({\w*\s-\inset},{(13-#1)*\s-\inset});
}
\def\marke#1#2{
\node at ({0.5*\w*\s},{12.5-#1)*\s}) {$#2\mathstrut$};
}
\def\gauss{
\draw (0,0) rectangle ({\w*\s},{12*\s});
\draw (0,{7*\s}) -- ({\w*\s},{7*\s});
}
\draw[->,color=red,line width=1pt] ({0.1*\s},{(12.5-1)*\s})
to[out=180,in=90] (-3.6,-2);
\draw[->,color=red,line width=1pt] ({0.1*\s},{(12.5-2)*\s})
to[out=180,in=90] (-2.2,-2);
\draw[->,color=red,line width=1pt] ({0.1*\s},{(12.5-4)*\s})
to[out=180,in=90] (-0.7,-2);
\draw[->,color=darkgreen,line width=1pt] ({0.1*\s},{(12.5-7)*\s})
to[out=180,in=90] (0.9,-2);
\draw[->,color=darkgreen,line width=1pt] ({0.1*\s},{(12.5-8)*\s})
to[out=180,in=90] (1.6,-2);
\draw[->,color=darkgreen,line width=1pt] ({(\w-0.1)*\s},{(12.5-12)*\s})
to[out=0,in=90] (2.6,-2);
\draw[->,line width=2pt] ({\w*\s+0.1},{6*\s}) -- (5.4,{6*\s});
\node at ({0.5*(\w*\s+5.5)},{6*\s}) [above] {Gauss};
\begin{scope}
\zeile{1}{red!30}
\zeile{2}{red!30}
\zeile{4}{red!30}
\zeile{7}{darkgreen!30}
\zeile{8}{darkgreen!30}
%\zeile{10}{darkgreen!30}
\zeile{12}{darkgreen!30}
\marke{1}{\scriptstyle\partial_{k+1}e_1^{(k+1)}}
\marke{2}{\scriptstyle\partial_{k+1}e_2^{(k+1)}}
\marke{3}{\scriptstyle\partial_{k+1}e_3^{(k+1)}}
\marke{4}{\vdots}
\marke{5}{\scriptstyle\partial_{k+1}e_{n_{k+1}}^{(k+1)}}
\marke{6}{\scriptstyle z_1^{(k)}}
\marke{7}{\scriptstyle z_2^{(k)}}
\marke{8}{\scriptstyle z_3^{(k)}}
\marke{9}{\scriptstyle z_4^{(k)}}
\marke{10}{\vdots}
\marke{11}{\scriptstyle z_{l-1}^{(k)}}
\marke{12}{\scriptstyle z_{l}^{(k)}}
\gauss
\end{scope}
\begin{scope}[xshift=5.5cm]
\zeile{1}{black!20}
\zeile{2}{black!20}
\zeile{3}{black}
\marke{3}{\color{white}0}
\zeile{4}{black!20}
\zeile{5}{black}
\marke{5}{\color{white}0}
\zeile{6}{black}
\marke{6}{\color{white}0}
\zeile{7}{black!20}
\zeile{8}{black!20}
\zeile{9}{black}
\marke{9}{\color{white}0}
\zeile{10}{black}
\marke{10}{\color{white}0}
\zeile{11}{black}
\marke{11}{\color{white}0}
\zeile{12}{black!20}
\gauss
\end{scope}
\node at (-4.4,-2) [below right] {$\{
{\color{red}\partial_{k+1}e_1^{(k+1)}},
{\color{red}\partial_{k+1}e_2^{(k+1)}},
{\color{red}\partial_{k+1}e_{i_3}^{(k+1)}},\dots,
{\color{darkgreen}z_2^{(k)}},
{\color{darkgreen}z_3^{(k)}},
\dots
{\color{darkgreen}z_l^{(k)}}
\} = {\color{red}\mathcal{B}_k} \cup {\color{darkgreen}\mathcal{Z}_k'}$};
\end{tikzpicture}
\end{document}
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