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author | LordMcFungus <mceagle117@gmail.com> | 2021-03-22 18:05:11 +0100 |
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committer | GitHub <noreply@github.com> | 2021-03-22 18:05:11 +0100 |
commit | 76d2d77ddb2bed6b7c6b8ec56648d85da4103ab7 (patch) | |
tree | 11b2d41955ee4bfa0ae5873307c143f6b4d55d26 /buch/chapters/10-vektorenmatrizen/images | |
parent | more chapter structure (diff) | |
parent | add title image (diff) | |
download | SeminarMatrizen-76d2d77ddb2bed6b7c6b8ec56648d85da4103ab7.tar.gz SeminarMatrizen-76d2d77ddb2bed6b7c6b8ec56648d85da4103ab7.zip |
Merge pull request #1 from AndreasFMueller/master
update
Diffstat (limited to 'buch/chapters/10-vektorenmatrizen/images')
-rw-r--r-- | buch/chapters/10-vektorenmatrizen/images/Makefile | 18 | ||||
-rw-r--r-- | buch/chapters/10-vektorenmatrizen/images/gausszahlen.pdf | bin | 0 -> 19127 bytes | |||
-rw-r--r-- | buch/chapters/10-vektorenmatrizen/images/gausszahlen.tex | 48 | ||||
-rw-r--r-- | buch/chapters/10-vektorenmatrizen/images/ideale.pdf | bin | 0 -> 73185 bytes | |||
-rw-r--r-- | buch/chapters/10-vektorenmatrizen/images/ideale.tex | 72 | ||||
-rw-r--r-- | buch/chapters/10-vektorenmatrizen/images/rref.pdf | bin | 0 -> 15112 bytes | |||
-rw-r--r-- | buch/chapters/10-vektorenmatrizen/images/rref.tex | 253 | ||||
-rw-r--r-- | buch/chapters/10-vektorenmatrizen/images/strukturen.pdf | bin | 0 -> 45339 bytes | |||
-rw-r--r-- | buch/chapters/10-vektorenmatrizen/images/strukturen.tex | 122 |
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 Binary files differnew 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 Binary files differnew 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 Binary files differnew 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 Binary files differnew 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} + |