Merge pull request #1 from AndreasFMueller/master

update

9 files changed, 513 insertions, 0 deletions

diff --git a/buch/chapters/10-vektorenmatrizen/images/Makefile b/buch/chapters/10-vektorenmatrizen/images/Makefile
new file mode 100644
index 0000000..2c94e8a
--- /dev/null
+++ b/buch/chapters/10-vektorenmatrizen/images/Makefile
@@ -0,0 +1,18 @@
+#
+# Makefile -- build images
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+all:	ideale.pdf gausszahlen.pdf strukturen.pdf rref.pdf
+
+ideale.pdf:	ideale.tex
+	pdflatex ideale.tex
+
+gausszahlen.pdf:	gausszahlen.tex
+	pdflatex gausszahlen.tex
+
+strukturen.pdf:	strukturen.tex
+	pdflatex strukturen.tex
+
+rref.pdf:	rref.tex
+	pdflatex rref.tex
diff --git a/buch/chapters/10-vektorenmatrizen/images/gausszahlen.pdf b/buch/chapters/10-vektorenmatrizen/images/gausszahlen.pdf
new file mode 100644
index 0000000..181499c
--- /dev/null
+++ b/buch/chapters/10-vektorenmatrizen/images/gausszahlen.pdf
diff --git a/buch/chapters/10-vektorenmatrizen/images/gausszahlen.tex b/buch/chapters/10-vektorenmatrizen/images/gausszahlen.tex
new file mode 100644
index 0000000..6786f05
--- /dev/null
+++ b/buch/chapters/10-vektorenmatrizen/images/gausszahlen.tex
@@ -0,0 +1,48 @@
+%
+% gausszahlen.tex -- Ganze Gausssche Zahlen
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usepackage{color}
+\usetikzlibrary{arrows,intersections,math}
+\begin{document}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{tikzpicture}[>=latex,thick,scale=0.8]
+\draw[->] (-8.5,0) -- (8.5,0) coordinate[label={$\Re z$}];
+\draw[->] (0,-4.5) -- (0,4.5) coordinate[label={right:$\Im z$}];
+\foreach \x in {-8,...,8}{
+	\foreach \y in {-4,...,4}{
+		\fill (\x,\y) circle[radius=0.05];
+	}
+}
+
+
+\coordinate (O) at (0,0);
+\coordinate (A) at (2,2);
+\coordinate (B) at (-3,1);
+\coordinate (C) at (-8,-4);
+\coordinate (D) at (-1,3);
+\draw[line width=0.5pt] (A)--(D)--(B);
+\draw[->,color=red] (O) -- (A);
+\draw[->,color=red] (O) -- (B);
+\draw[->,color=blue] (O) -- (C);
+\draw[->,color=darkgreen] (O) -- (D);
+\fill[color=red] (A) circle[radius=0.08];
+\fill[color=red] (B) circle[radius=0.08];
+\fill[color=blue] (C) circle[radius=0.08];
+\fill[color=darkgreen] (D) circle[radius=0.08];
+\fill[color=black] (O) circle[radius=0.08];
+\node[color=red] at (A) [above right] {$z$};
+\node[color=red] at (B) [above left] {$w$};
+\node[color=darkgreen] at (D) [above] {$z+w$};
+\node[color=blue] at (C) [below right] {$z\cdot w$};
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/chapters/10-vektorenmatrizen/images/ideale.pdf b/buch/chapters/10-vektorenmatrizen/images/ideale.pdf
new file mode 100644
index 0000000..439afcc
--- /dev/null
+++ b/buch/chapters/10-vektorenmatrizen/images/ideale.pdf
diff --git a/buch/chapters/10-vektorenmatrizen/images/ideale.tex b/buch/chapters/10-vektorenmatrizen/images/ideale.tex
new file mode 100644
index 0000000..9793c8e
--- /dev/null
+++ b/buch/chapters/10-vektorenmatrizen/images/ideale.tex
@@ -0,0 +1,72 @@
+%
+% ideale.tex -- Ideale in den ganzen Gaussschen Zahlen
+%
+% (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}
+\begin{tikzpicture}[>=latex,thick,scale=0.35]
+\begin{scope}[xshift=-9.5cm]
+\begin{scope}
+\clip (-8.3,-8.3) rectangle (8.3,8.3);
+	\foreach \x in {-8,...,8}{
+		\foreach \y in {-8,...,8}{
+			\fill (\x,\y) circle[radius=0.08];
+		}
+	}
+	\foreach \x in {-8,...,8}{
+		\foreach \y in {-8,...,8}{
+			\fill[color=blue]
+				({\x-2*\y},{2*\x+\y}) circle[radius=0.12];
+		}
+	}
+	\foreach \x in {-8,...,8}{
+		\draw[color=blue,line width=0.5pt]
+			({\x-2*(-8)},{2*\x+(-8)}) 
+			--
+			({\x-2*8},{2*\x+8});
+	}
+	\foreach \y in {-8,...,8}{
+		\draw[color=blue,line width=0.5pt]
+			({(-8)-2*\y},{2*(-8)+\y}) 
+			--
+			({8-2*\y},{2*8+\y});
+	}
+\end{scope}
+	\draw[->] (-8.3,0) -- (9.1,0) coordinate[label={$\Re z$}];
+	\draw[->] (0,-8.3) -- (0,8.9) coordinate[label={right:$\Im z$}];
+\end{scope}
+
+\begin{scope}[xshift=9.5cm]
+\begin{scope}
+\clip (-8.3,-8.3) rectangle (8.3,8.3);
+	\foreach \x in {-8,...,8}{
+		\foreach \y in {-8,...,8}{
+			\fill[color=red] ({\x-\y},{\x+\y}) circle[radius=0.12];
+		}
+	}
+	\foreach \x in {-8,...,8}{
+		\foreach \y in {-8,...,8}{
+			\fill (\x,\y) circle[radius=0.08];
+		}
+	}
+	\foreach \x in {-8,...,8}{
+		\draw[color=red,line width=0.5pt]
+			({\x+8},{\x-8}) -- ({\x-8},{\x+8});
+		\draw[color=red,line width=0.5pt]
+			({-8-\x},{-8+\x}) -- ({8-\x},{8+\x});
+	}
+\end{scope}
+	\draw[->] (-8.3,0) -- (9.1,0) coordinate[label={$\Re z$}];
+	\draw[->] (0,-8.3) -- (0,8.9) coordinate[label={right:$\Im z$}];
+\end{scope}
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/chapters/10-vektorenmatrizen/images/rref.pdf b/buch/chapters/10-vektorenmatrizen/images/rref.pdf
new file mode 100644
index 0000000..56fbfee
--- /dev/null
+++ b/buch/chapters/10-vektorenmatrizen/images/rref.pdf
diff --git a/buch/chapters/10-vektorenmatrizen/images/rref.tex b/buch/chapters/10-vektorenmatrizen/images/rref.tex
new file mode 100644
index 0000000..9b2bf50
--- /dev/null
+++ b/buch/chapters/10-vektorenmatrizen/images/rref.tex
@@ -0,0 +1,253 @@
+%
+% rref.tex -- Visualisierung des Gauss-Algorithmus
+%
+% (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,calc}
+\begin{document}
+\def\skala{0.21}
+\def\r{0.4}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\def\pivot#1#2{
+	\fill[color=red!20] ({#1-0.5},{-#2+0.5}) circle[radius=\r];
+	\draw[color=red] ({#1-0.5},{-#2+0.5}) circle[radius=\r];
+}
+
+\def\spalteoben#1#2#3{
+	\fill[color=blue!20] ({(#1)-0.5+\r},{-(#3)})
+		-- ({(#1)-0.5+\r},{-(#2)+0.5}) arc (0:180:\r)
+		-- ({(#1)-0.5-\r},{-(#3)}) -- cycle;
+	\draw[color=blue] ({(#1)-0.5+\r},{-(#3)})
+		-- ({(#1)-0.5+\r},{-(#2)+0.5}) arc (0:180:\r)
+		-- ({(#1)-0.5-\r},{-(#3)});
+}
+
+\def\spalteunten#1#2#3{
+	\fill[color=blue!20] ({(#1)-0.5-\r},{-(#2)+1})
+		-- ({(#1)-0.5-\r},{-(#3)+0.5}) arc (-180:0:\r)
+		-- ({(#1)-0.5+\r},{-(#2)+1});
+	\draw[color=blue] ({(#1)-0.5-\r},{-(#2)+1})
+		-- ({(#1)-0.5-\r},{-(#3)+0.5}) arc (-180:0:\r)
+		-- ({(#1)-0.5+\r},{-(#2)+1});
+}
+
+\def\fuellung{
+	\fill[color=gray!50] (0,0) rectangle (8,-6);
+}
+\def\rahmen{
+	\draw (0,0) rectangle (8,-6);
+	\draw (7,0) -- (7,-6);
+}
+
+\def\eins#1#2{
+	\fill[color=gray] ({#1-1},{-#2}) rectangle ({#1},{-#2+1});
+}
+
+\def\null#1#2#3{
+	\fill[color=white] ({#1-1-0.01},{-#3-0.01})
+		rectangle ({#1+0.01},{-#2+1+0.01});
+}
+
+\fill[color=darkgreen!20] (-1.0,-10.81) rectangle (67.0,5);
+\fill[color=orange!20] (-1.0,-27) rectangle (67.0,-11.94);
+
+\node at (33,2) [above] {Vorwärtsreduktion};
+\node at (33,-24) [below] {Rückwärtseinsetzen};
+
+\draw[->] (9,-3.375)--(11,-3.375);
+\draw[->] (21,-3.375)--(23,-3.375);
+\draw[->] (33,-3.375)--(35,-3.375);
+\draw[->] (45,-3.375)--(47,-3.375);
+
+\draw[->] (57,-3.375) .. controls (62,-3.375) .. (62,-7.5);
+\draw[->] (62,-15.375) .. controls (62,-19.375) .. (57,-19.375);
+
+\draw[<-] (9,-19.375)--(11,-19.375);
+\draw[<-] (21,-19.375)--(23,-19.375);
+\draw[<-] (33,-19.375)--(35,-19.375);
+\draw[<-] (45,-19.375)--(47,-19.375);
+
+\begin{scope}[xshift=-0.5cm,scale=1.125]
+\fuellung
+\pivot{1}{1} 
+\spalteoben{1}{2}{6}
+\rahmen
+\end{scope}
+
+\begin{scope}[xshift=11.5cm,scale=1.125]
+\fuellung
+\eins{1}{1}
+\null{1}{2}{6}
+\pivot{2}{2}
+\spalteoben{2}{3}{6}
+\rahmen
+\end{scope}
+
+\begin{scope}[xshift=23.54cm,scale=1.125]
+\fuellung
+\eins{1}{1}
+\null{1}{2}{6}
+\eins{2}{2}
+\null{2}{3}{6}
+\pivot{3}{3}
+\spalteoben{3}{4}{6}
+\rahmen
+\end{scope}
+
+\begin{scope}[xshift=35.5cm,scale=1.125]
+\fuellung
+\eins{1}{1}
+\null{1}{2}{6}
+\eins{2}{2}
+\null{2}{3}{6}
+\eins{3}{3}
+\null{3}{4}{6}
+\null{4}{4}{6}
+\pivot{5}{4}
+\spalteoben{5}{5}{6}
+\rahmen
+\end{scope}
+
+\begin{scope}[xshift=47.5cm,scale=1.125]
+\fuellung
+\eins{1}{1}
+\null{1}{2}{6}
+\eins{2}{2}
+\null{2}{3}{6}
+\eins{3}{3}
+\null{3}{4}{6}
+\eins{5}{4}
+\null{5}{5}{6}
+\null{6}{5}{6}
+\pivot{7}{5}
+\spalteoben{7}{6}{6}
+\rahmen
+\end{scope}
+
+\begin{scope}[xshift=57.5cm,yshift=-8cm,scale=1.125]
+\fuellung
+\eins{1}{1}
+\null{1}{2}{6}
+\eins{2}{2}
+\null{2}{3}{6}
+\eins{3}{3}
+\null{3}{4}{6}
+\null{4}{4}{6}
+\eins{5}{4}
+\null{5}{5}{6}
+\null{6}{5}{6}
+\eins{7}{5}
+\null{7}{6}{6}
+\rahmen
+\end{scope}
+
+\begin{scope}[xshift=47.5cm,yshift=-16cm,scale=1.125]
+\fuellung
+\eins{1}{1}
+\null{1}{2}{6}
+\eins{2}{2}
+\null{2}{3}{6}
+\eins{3}{3}
+\null{3}{4}{6}
+\null{4}{4}{6}
+\eins{5}{4}
+\null{5}{5}{6}
+\null{6}{5}{6}
+\eins{7}{5}
+\null{7}{6}{6}
+\spalteunten{7}{1}{4}
+\rahmen
+\end{scope}
+
+\begin{scope}[xshift=35.5cm,yshift=-16cm,scale=1.125]
+\fuellung
+\eins{1}{1}
+\null{1}{2}{6}
+\eins{2}{2}
+\null{2}{3}{6}
+\eins{3}{3}
+\null{3}{4}{6}
+\null{4}{4}{6}
+\eins{5}{4}
+\null{5}{5}{6}
+\null{6}{5}{6}
+\eins{7}{5}
+\null{7}{6}{6}
+\null{7}{1}{4}
+\spalteunten{5}{1}{3}
+\rahmen
+\end{scope}
+
+\begin{scope}[xshift=23.5cm,yshift=-16cm,scale=1.125]
+\fuellung
+\eins{1}{1}
+\null{1}{2}{6}
+\eins{2}{2}
+\null{2}{3}{6}
+\eins{3}{3}
+\null{3}{4}{6}
+\null{4}{4}{6}
+\eins{5}{4}
+\null{5}{5}{6}
+\null{6}{5}{6}
+\eins{7}{5}
+\null{7}{6}{6}
+\null{7}{1}{4}
+\null{5}{1}{3}
+\spalteunten{3}{1}{2}
+\rahmen
+\end{scope}
+
+\begin{scope}[xshift=11.5cm,yshift=-16cm,scale=1.125]
+\fuellung
+\eins{1}{1}
+\null{1}{2}{6}
+\eins{2}{2}
+\null{2}{3}{6}
+\eins{3}{3}
+\null{3}{4}{6}
+\null{4}{4}{6}
+\eins{5}{4}
+\null{5}{5}{6}
+\null{6}{5}{6}
+\eins{7}{5}
+\null{7}{6}{6}
+\null{7}{1}{4}
+\null{5}{1}{3}
+\null{3}{1}{2}
+\spalteunten{2}{1}{1}
+\rahmen
+\end{scope}
+
+\begin{scope}[xshift=-0.5cm,yshift=-16cm,scale=1.125]
+\fuellung
+\eins{1}{1}
+\null{1}{2}{6}
+\eins{2}{2}
+\null{2}{3}{6}
+\eins{3}{3}
+\null{3}{4}{6}
+\null{4}{4}{6}
+\eins{5}{4}
+\null{5}{5}{6}
+\null{6}{5}{6}
+\eins{7}{5}
+\null{7}{6}{6}
+\null{7}{1}{4}
+\null{5}{1}{3}
+\null{3}{1}{2}
+\null{2}{1}{1}
+\rahmen
+\end{scope}
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/chapters/10-vektorenmatrizen/images/strukturen.pdf b/buch/chapters/10-vektorenmatrizen/images/strukturen.pdf
new file mode 100644
index 0000000..14f7e59
--- /dev/null
+++ b/buch/chapters/10-vektorenmatrizen/images/strukturen.pdf
diff --git a/buch/chapters/10-vektorenmatrizen/images/strukturen.tex b/buch/chapters/10-vektorenmatrizen/images/strukturen.tex
new file mode 100644
index 0000000..02ca71d
--- /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,-10.1) 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,-10.1) 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,-9.9) rectangle (1.6,-6.9);
+\draw[rounded corners=0.1cm] (-1.6,-9.9) 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}
+