diff options
Diffstat (limited to '')
| -rw-r--r-- | vorlesungen/slides/2/Makefile.inc | 2 | ||||
| -rw-r--r-- | vorlesungen/slides/2/chapter.tex | 2 | ||||
| -rw-r--r-- | vorlesungen/slides/2/images/Makefile | 32 | ||||
| -rw-r--r-- | vorlesungen/slides/2/images/quotient.inc | 186 | ||||
| -rw-r--r-- | vorlesungen/slides/2/images/quotient.ini | 7 | ||||
| -rw-r--r-- | vorlesungen/slides/2/images/quotient1.jpg | bin | 0 -> 181755 bytes | |||
| -rw-r--r-- | vorlesungen/slides/2/images/quotient1.pov | 8 | ||||
| -rw-r--r-- | vorlesungen/slides/2/images/quotient1.tex | 29 | ||||
| -rw-r--r-- | vorlesungen/slides/2/images/quotient2.jpg | bin | 0 -> 206065 bytes | |||
| -rw-r--r-- | vorlesungen/slides/2/images/quotient2.pov | 8 | ||||
| -rw-r--r-- | vorlesungen/slides/2/images/quotient2.tex | 29 | ||||
| -rw-r--r-- | vorlesungen/slides/2/quotient.tex | 110 | ||||
| -rw-r--r-- | vorlesungen/slides/2/quotientv.tex | 62 |
13 files changed, 475 insertions, 0 deletions
diff --git a/vorlesungen/slides/2/Makefile.inc b/vorlesungen/slides/2/Makefile.inc index 34dadf6..c857fec 100644 --- a/vorlesungen/slides/2/Makefile.inc +++ b/vorlesungen/slides/2/Makefile.inc @@ -15,5 +15,7 @@ chapter2 = \ ../slides/2/funktionenalgebra.tex \ ../slides/2/frobeniusnorm.tex \ ../slides/2/frobeniusanwendung.tex \ + ../slides/2/quotient.tex \ + ../slides/2/quotientv.tex \ ../slides/2/chapter.tex diff --git a/vorlesungen/slides/2/chapter.tex b/vorlesungen/slides/2/chapter.tex index 38eb21f..49e656a 100644 --- a/vorlesungen/slides/2/chapter.tex +++ b/vorlesungen/slides/2/chapter.tex @@ -13,3 +13,5 @@ \folie{2/funktionenalgebra.tex} \folie{2/frobeniusnorm.tex} \folie{2/frobeniusanwendung.tex} +\folie{2/quotient.tex} +\folie{2/quotientv.tex} diff --git a/vorlesungen/slides/2/images/Makefile b/vorlesungen/slides/2/images/Makefile new file mode 100644 index 0000000..8bce5c9 --- /dev/null +++ b/vorlesungen/slides/2/images/Makefile @@ -0,0 +1,32 @@ +# +# Makefile +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +all: quotient1.jpg quotient2.jpg quotient1.pdf quotient2.pdf + +quotient1.png: quotient1.pov quotient.inc + povray +A0.1 +W1920 +H1080 -Oquotient1.png quotient1.pov + +quotient1.jpg: quotient1.png Makefile + convert -extract 1360x1040+330+20 quotient1.png \ + -density 300 -units PixelsPerInch quotient1.jpg + +quotient2.png: quotient2.pov quotient.inc + povray +A0.1 +W1920 +H1080 -Oquotient2.png quotient2.pov + +quotient2.jpg: quotient2.png Makefile + convert -extract 1360x1040+330+20 quotient2.png \ + -density 300 -units PixelsPerInch quotient2.jpg + +quotient: quotient.ini quotient.inc quotient.pov + rm -rf quotient + mkdir quotient + povray +A0.1 -Oquotient/0.png -W1920 -H1080 quotient.ini + +quotient1.pdf: quotient1.tex quotient1.jpg + pdflatex quotient1.tex + +quotient2.pdf: quotient2.tex quotient2.jpg + pdflatex quotient2.tex + diff --git a/vorlesungen/slides/2/images/quotient.inc b/vorlesungen/slides/2/images/quotient.inc new file mode 100644 index 0000000..3fa49d1 --- /dev/null +++ b/vorlesungen/slides/2/images/quotient.inc @@ -0,0 +1,186 @@ +// +// quotient.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#version 3.7; +#include "colors.inc" + +global_settings { + assumed_gamma 1 +} + +#declare imagescale = 0.035; +#declare O = <0, 0, 0>; +#declare at = 0.015; + +camera { + location <8, 15, -50> + look_at <0.4, 0.2, 0.4> + right 16/9 * x * imagescale + up y * imagescale +} + +light_source { + <-4, 20, -50> color White + area_light <1,0,0> <0,0,1>, 10, 10 + adaptive 1 + jitter +} + +sky_sphere { + pigment { + color rgb<1,1,1> + } +} + +#macro arrow(from, to, arrowthickness, c) +#declare arrowdirection = vnormalize(to - from); +#declare arrowlength = vlength(to - from); +union { + sphere { + from, 1.1 * arrowthickness + } + cylinder { + from, + from + (arrowlength - 5 * arrowthickness) * arrowdirection, + arrowthickness + } + cone { + from + (arrowlength - 5 * arrowthickness) * arrowdirection, + 2 * arrowthickness, + to, + 0 + } + pigment { + color c + } + finish { + specular 0.9 + metallic + } +} +#end + +#macro kasten() + box { <-0.5,-0.5,-0.5>, <1.5,1,1.5> } +#end + + +arrow(<-0.6,0,0>, <1.6,0,0>, at, White) +arrow(<0,0,-0.6>, <0,0,1.6>, at, White) +arrow(<0,-0.6,0>, <0,1.2,0>, at, White) + +#declare U = <-1,3,-0.5>; +#declare V1 = <1,0.2,0>; +#declare V2 = <0,0.2,1>; + +#macro gerade(richtung, farbe) + intersection { + kasten() + cylinder { -U + richtung, U + richtung, at } + pigment { + color farbe + } + finish { + specular 0.9 + metallic + } + } +#end + +#declare A = <0.8, -0.2, 0>; +#declare B = <0.2, 0.8, 0>; + +#macro ebene(vektor1, vektor2) +#declare n = vcross(vektor1,vektor2); + + +intersection { + kasten() + plane { n, 0.005 } + plane { -n, 0.005 } + pigment { + color rgbf<0.8,0.8,1,0.7> + } + finish { + specular 0.9 + metallic + } +} + +intersection { + kasten() + union { + #declare Xstep = 0.45; + #declare X = -5 * Xstep; + #while (X < 5.5 * Xstep) + cylinder { X*vektor1 - 5*vektor2, X*vektor1 + 5*vektor2, at/2 } + #declare X = X + Xstep; + #end + #declare Ystep = 0.45; + #declare Y = -5 * Ystep; + #while (Y < 5.5 * Ystep) + cylinder { -5*vektor1 + Y*vektor2, 5*vektor1 + Y*vektor2, at/2 } + #declare Y = Y + Ystep; + #end + } + pigment { + color rgb<0.9,0.9,1> + } + finish { + specular 0.9 + metallic + } +} +#end + + +gerade(O, Red) + +#declare gruen = rgb<0.2,0.4,0.2>; +#declare blau = rgb<0,0.4,0.8>; +#declare rot = rgb<1,0.4,0.0>; + +#macro repraesentanten(vektor1, vektor2) + +#declare d1 = A.x*vektor1 + A.y*vektor2; +#declare d2 = B.x*vektor1 + B.y*vektor2; + +arrow(0, d1 + d2, at, rot) +gerade(d1 + d2, rot) + +gerade(d1, blau) +arrow(O, d1, at, blau) +cylinder { d1, d1 + d2, 0.6 * at + pigment { + color gruen + } + finish { + specular 0.9 + metallic + } +} + +gerade(d2, gruen) +arrow(O, d2, at, gruen) +cylinder { d2, d1 + d2, 0.6 * at + pigment { + color blau + } + finish { + specular 0.9 + metallic + } +} + +#end + +#macro vektorraum(s) +#declare b1 = V1 + s * 0.03 * U; +#declare b2 = V2 + s * 0.03 * U; + +ebene(b1, b2) +repraesentanten(b1, b2) +#end + diff --git a/vorlesungen/slides/2/images/quotient.ini b/vorlesungen/slides/2/images/quotient.ini new file mode 100644 index 0000000..f62b21a --- /dev/null +++ b/vorlesungen/slides/2/images/quotient.ini @@ -0,0 +1,7 @@ +Input_File_Name="quotient.pov" +Initial_Frame=0 +Final_Frame=100 +Initial_Clock=-1 +Final_Clock=1 +Cyclic_Animation=off +Pause_when_Done=off diff --git a/vorlesungen/slides/2/images/quotient1.jpg b/vorlesungen/slides/2/images/quotient1.jpg Binary files differnew file mode 100644 index 0000000..aeb713e --- /dev/null +++ b/vorlesungen/slides/2/images/quotient1.jpg diff --git a/vorlesungen/slides/2/images/quotient1.pov b/vorlesungen/slides/2/images/quotient1.pov new file mode 100644 index 0000000..60bab7f --- /dev/null +++ b/vorlesungen/slides/2/images/quotient1.pov @@ -0,0 +1,8 @@ +// +// quotient1.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "quotient.inc" + +vektorraum(-1) diff --git a/vorlesungen/slides/2/images/quotient1.tex b/vorlesungen/slides/2/images/quotient1.tex new file mode 100644 index 0000000..30d82d2 --- /dev/null +++ b/vorlesungen/slides/2/images/quotient1.tex @@ -0,0 +1,29 @@ +% +% quotient1.tex -- Vektorraumquotient +% +% (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} +\definecolor{darkgreen}{rgb}{0,0.6,0} +\definecolor{darkred}{rgb}{0.7,0,0} +\def\skala{1} +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\node at (0,0) {\includegraphics[width=8cm]{quotient1.jpg}}; + +\node[color=blue] at (0.7,-1.3) {$v$}; +\node[color=darkgreen] at (-1.0,0.1) {$w$}; +\node[color=orange] at (2.5,0.1) {$v+w$}; +\node[color=darkred] at (-2.1,-0.9) {$0$}; +\node[color=darkred] at (-3.1,2.4) {$U$}; + +\end{tikzpicture} +\end{document} + diff --git a/vorlesungen/slides/2/images/quotient2.jpg b/vorlesungen/slides/2/images/quotient2.jpg Binary files differnew file mode 100644 index 0000000..345cf22 --- /dev/null +++ b/vorlesungen/slides/2/images/quotient2.jpg diff --git a/vorlesungen/slides/2/images/quotient2.pov b/vorlesungen/slides/2/images/quotient2.pov new file mode 100644 index 0000000..771425d --- /dev/null +++ b/vorlesungen/slides/2/images/quotient2.pov @@ -0,0 +1,8 @@ +// +// quotient2.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "quotient.inc" + +vektorraum(1) diff --git a/vorlesungen/slides/2/images/quotient2.tex b/vorlesungen/slides/2/images/quotient2.tex new file mode 100644 index 0000000..607fd03 --- /dev/null +++ b/vorlesungen/slides/2/images/quotient2.tex @@ -0,0 +1,29 @@ +% +% quotient2.tex -- Vektorraumquotient +% +% (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} +\definecolor{darkgreen}{rgb}{0,0.6,0} +\definecolor{darkred}{rgb}{0.7,0,0} +\def\skala{1} +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\node at (0,0) {\includegraphics[width=8cm]{quotient2.jpg}}; + +\node[color=blue] at (0.57,-0.94) {$v$}; +\node[color=darkgreen] at (-1.15,0.65) {$w$}; +\node[color=orange] at (2.15,1) {$v+w$}; +\node[color=darkred] at (-2.1,-0.9) {$0$}; +\node[color=darkred] at (-3.1,2.4) {$U$}; + +\end{tikzpicture} +\end{document} + diff --git a/vorlesungen/slides/2/quotient.tex b/vorlesungen/slides/2/quotient.tex new file mode 100644 index 0000000..24b0523 --- /dev/null +++ b/vorlesungen/slides/2/quotient.tex @@ -0,0 +1,110 @@ +% +% quotient.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkred}{rgb}{0.7,0,0} +\definecolor{darkgreen}{rgb}{0,0.6,0} +\def\s{0.3} +\def\punkt#1#2{({#1-3*#2},{8*#2})} +\def\gerade#1{ +\draw[darkgreen,line width=1.4pt] + \punkt{#1}{1} + -- + \punkt{#1}{-1}; +} +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Quotientenraum} +\vspace{-18pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Einen Unterraum ``ignorieren''} +{\usebeamercolor[fg]{title}Gegeben:} $U\subset V$ ein Unterraum +\\ +{\usebeamercolor[fg]{title}Gesucht:} Eine Projektion auf einen Vektorraum, +in dem die Richtungen in $U$ zu $0$ gemacht werden +\end{block} +\uncover<2->{% +\begin{block}{Projektion} +In $V$ Klassen bilden: +\[ +\pi +\colon +v\mapsto +\llbracket v\rrbracket += +v+U +\] +\end{block}} +\vspace{-12pt} +\uncover<3->{% +\begin{block}{Quotientenraum} +\vspace{-12pt} +\begin{align*} +V/U +&= +\{ v+U\;|\; v\in V \} +\\ +\uncover<4->{\pi(\lambda v)&=\lambda v+U= \lambda \pi(v)} +\\ +\uncover<5->{\pi(v+w) +&= +v+w+U} +\ifthenelse{\boolean{presentation}}{ +\only<6>{= +v+U+w+U}}{} +\uncover<7->{= +\pi(v) + \pi(w)} +\phantom{blubb} +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\coordinate (U) at (-3,8); +\def\t{0.03} +\begin{scope} +\clip (-2,-2) rectangle (4,4.8); +\draw[color=darkred,line width=2pt] (-3,8) -- (1.5,-4); +\node[color=darkred] at (-1.45,4.6) {$U$}; +\node[color=darkred] at (-0.05,-0.05) [above left] {$0$}; + +\gerade{2.5} + +\ifthenelse{\boolean{presentation}}{ + \foreach \n in {8,...,25}{ + \pgfmathparse{(\n-12)*0.04} + \xdef\s{\pgfmathresult} + \only<\n>{ + \draw[color=blue,line width=1.2pt] + \punkt{-5}{-2*\s} -- \punkt{5}{2*\s}; + \draw[->,color=blue,line width=2pt] + (0,0) -- \punkt{2.5}{\s}; + \node[color=blue] at \punkt{2.5}{\s} + [above right] {$v'$}; + } + } +}{ + \xdef\s{0.35} + \draw[color=blue,line width=1.2pt] + \punkt{-5}{-2*\s} -- \punkt{5}{2*\s}; + \draw[->,color=blue,line width=2pt] (0,0) -- \punkt{2.5}{\s}; + \node[color=blue] at \punkt{2.5}{\s} [above right] {$v'$}; +} + +\draw[->,color=darkgreen,line width=1.4pt] (0,0) -- \punkt{2.5}{0.1}; + +\node[color=darkgreen] at \punkt{2.5}{0.1} [above right] {$v$}; + +\end{scope} +\draw[->] (-2,0) -- (4,0) coordinate[label={$x$}]; +\draw[->] (0,-2) -- (0,5) coordinate[label={right:$x$}]; +\end{tikzpicture} +\end{center} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/2/quotientv.tex b/vorlesungen/slides/2/quotientv.tex new file mode 100644 index 0000000..dc01f21 --- /dev/null +++ b/vorlesungen/slides/2/quotientv.tex @@ -0,0 +1,62 @@ +% +% quotientv.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkred}{rgb}{0.7,0,0} +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame}[t] +\frametitle{Quotient} +\vspace{-18pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.33\textwidth} +\begin{block}{Repräsentanten} +Jeder Unterraum $W\subset V$ mit +$W\cap U = \{0\}$ +kann als Menge von Repräsentanten +für +\begin{align*} +V/U +&= +\{v+U\;|\;v \in V\} +\\ +&\simeq W +\end{align*} +dienen. +\end{block} +\uncover<3->{% +\begin{block}{Orthogonalraum} +Mit Skalarprodukt ist +$W=U^\perp$ eine bevorzugte Wahl +\end{block}} +\end{column} +\begin{column}{0.66\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\only<1>{ + \node at (0,0) + {\includegraphics[width=8.5cm]{../slides/2/images/quotient1.jpg}}; + \node[color=darkgreen] at (-0.5,0.3) {$v$}; + \node[color=blue] at (0.7,-1.4) {$w$}; + \node[color=orange] at (2.7,0.1) {$v+w$}; + \fill[color=white,opacity=0.5] (3.7,1.0) circle[radius=0.25]; + \node at (3.7,1.0) {$W$}; +} +\only<2->{ + \node at (0,0) + {\includegraphics[width=8.5cm]{../slides/2/images/quotient2.jpg}}; + \node[color=darkgreen] at (-0.75,0.95) {$v$}; + \node[color=blue] at (0.6,-1.05) {$w$}; + \node[color=orange] at (2.36,1.05) {$v+w$}; + \fill[color=white,opacity=0.5] (3.7,2.9) circle[radius=0.25]; + \node at (3.7,2.9) {$W$}; +} +\node[color=darkred] at (-3.3,2.6) {$U$}; +\node[color=darkred] at (-2.25,-1.0) {$0$}; +\end{tikzpicture} +\end{center} +\end{column} +\end{columns} +\end{frame} +\egroup |
