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diff --git a/buch/chapters/10-vektorenmatrizen/images/strukturen.tex b/buch/chapters/10-vektorenmatrizen/images/strukturen.tex new file mode 100644 index 0000000..0006699 --- /dev/null +++ b/buch/chapters/10-vektorenmatrizen/images/strukturen.tex @@ -0,0 +1,122 @@ +% +% strukturen.tex -- Bezug der verschiedenen algebraischen Strukturen +% +% (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} +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\definecolor{darkgreen}{rgb}{0,0.6,0} + +% assoziative Verknüpfung +\draw[rounded corners=1cm] (-7,-11.5) rectangle (7,7); + +\begin{scope}[yshift=6cm] +\node at (0,0.5) [left] {{\bf assoziative Verknüpfung}:\strut}; +\node at (0,0.5) [right] {$a(bc)=(ab)c\;\forall a,b,c$\strut}; +\node at (0,-0.3) {\small $\mathbb{N}$, $\Sigma^*$}; +\end{scope} + +% Gruppe +\fill[rounded corners=1cm,color=gray!40] (-6.5,-11.0) rectangle (6.5,5.3); +\draw[rounded corners=1cm] (-6.5,-11.0) rectangle (6.5,5.3); + +\begin{scope}[xshift=-3cm,yshift=4.3cm] +\node at (0,0.5) [left] {{\bf Gruppe}:}; +\node at (0,0.5) [right] {neutrales Element $e$:\strut}; +\node at (3.3,0.5) [right] {$eg=ge=g$\strut}; +\node at (5.7,0.5) [right] {$\forall g\in G$\strut}; +\node at (0,0.0) [right] {inverses Element $g^{-1}$:\strut}; +\node at (3.3,0.0) [right] {$gg^{-1}=g^{-1}g=e$\strut}; +\node at (5.7,0.0) [right] {$\forall g\in G$\strut}; +\node at (3,-1) {\small $\mathbb{Z}$, $\operatorname{GL}_n(\mathbb R)$, $S_n$, $A_n$}; +\end{scope} + +% abelsche Gruppe +\fill[rounded corners=0.7cm,color=gray!20] (-6.2,-10.7) rectangle (6.2,2.7); +\draw[rounded corners=0.7cm] (-6.2,-10.7) rectangle (6.2,2.7); +\begin{scope}[yshift=1.5cm] +\node at (0,0.5) [left] {{\bf abelsche Gruppe}:\strut}; +\node at (0,0.5) [right] {$a+b=b+a\;\forall a,b$\strut}; +\node at (0,0.0) {Addition\strut}; + +\node at (0,-1) {\small $\mathbb{Q}^*$, $\operatorname{SO}(2)$, $C_n$ }; +\end{scope} + +\fill[rounded corners=0.5cm,color=white] (-2,-10.5) rectangle (6,-0.5); +\fill[rounded corners=0.5cm,color=blue!20] (-6,-10.0) rectangle (2,0); +%\draw[rounded corners=0.5cm] (-6,-10.0) rectangle (2,0); + +% Vektorraum +\begin{scope}[yshift=-1cm] +\node at (-5.8,0.5) [right] {{\bf Vektorraum}:\strut}; +\node at (-5.8,0.0) [right] {Skalarmultiplikation\strut}; + +\node at (-5.8,-0.5) [right] {$\lambda(a+b)=\lambda a+\lambda b$\strut}; +\node at (-5.8,-1.0) [right] {$(\lambda+\mu)a=\lambda a+\mu a$\strut}; +\node at (-5.8,-1.5) [right] {$\forall\lambda,\mu\in \Bbbk\;\forall a,b\in V$}; + +\node at (-5.8,-2.5) [right] {\small $\mathbb{R}^n$, $\mathbb{C}^n$, $l^2$}; +\end{scope} + +\fill[rounded corners=0.5cm,color=red!40,opacity=0.5] + (-2,-10.5) rectangle (6,-0.5); +\draw[rounded corners=0.5cm] (-2,-10.5) rectangle (6,-0.5); + +\begin{scope}[yshift=-1cm] +\node at (0,0.0) {{\bf Algebra}:\strut}; +\node at (0,-1.0) {$a(\lambda b) = \lambda ab$\strut}; +\node at (0,-1.5) {$\forall a,b\in A, \lambda\in \Bbbk$\strut}; +\node at (0,-3.0) {\small $c_0(\mathbb{R})$}; +\end{scope} + +\begin{scope}[yshift=-1cm] +\node at (5.8,0) [left] {{\bf Ring}:}; +\node at (5.8,-0.5) [left] {Multiplikation}; + +\node at (5.8,-1.0) [left] {$a(b+c)=ab+ac$\strut}; +\node at (5.8,-1.5) [left] {$(a+b)c=ac+bc$\strut}; +\node at (5.8,-2.0) [left] {$\forall a,b,c\in R$\strut}; + +\node at (5.8,-3) [left] {\small $c_0(\mathbb{Z})$, $L^2(\mathbb R)$}; +\end{scope} + +\fill[rounded corners=0.3cm,color=yellow!20,opacity=0.5] + (-1.8,-10.3) rectangle (5.8,-4.5); +\draw[rounded corners=0.3cm] (-1.8,-10.3) rectangle (5.8,-4.5); + +% boundary of blue area +\draw[rounded corners=0.5cm] (-6,-10.0) rectangle (2,0); + +\begin{scope}[yshift=-5cm] +\node at (5.6,0) [left] {{\bf Ring mit Eins}:}; +\node at (5.6,-1) [left] {$1\cdot a= a\cdot 1 = a\forall a\in R$\strut}; +\node at (5.6,-3) [left] {\small $\mathbb{Z}[X]$, $M_n(\mathbb{Z})$}; +\end{scope} + +\begin{scope}[yshift=-5cm] +\node at (0,0) {{\bf Algebra mit Eins}}; +\node at (0,-1.2) {\small $M_n(\mathbb R)$, $C([a,b])$}; +\end{scope} + +\fill[rounded corners=0.1cm,color=darkgreen!20] + (-1.6,-9.8) rectangle (1.6,-6.9); +\draw[rounded corners=0.1cm] (-1.6,-9.8) rectangle (1.6,-6.9); + +\begin{scope}[yshift=-7cm] +\node at (0,-0.3) {{\bf Körper}:\strut}; +\node at (0,-1) {$a\in K\setminus\{0\}\Rightarrow \exists a^{-1}$\strut}; +\node at (0,-2.2) {\small $\mathbb{F}_p$, $\mathbb{R}$, $\mathbb{C}$, $\mathbb{Q}(X)$}; +\end{scope} + +\end{tikzpicture} +\end{document} + |