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Diffstat (limited to 'vorlesungen/slides/2')
23 files changed, 1339 insertions, 0 deletions
diff --git a/vorlesungen/slides/2/Makefile.inc b/vorlesungen/slides/2/Makefile.inc new file mode 100644 index 0000000..c857fec --- /dev/null +++ b/vorlesungen/slides/2/Makefile.inc @@ -0,0 +1,21 @@ + +# +# Makefile.inc -- additional depencencies +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +chapter2 = \ + ../slides/2/norm.tex \ + ../slides/2/skalarprodukt.tex \ + ../slides/2/cauchyschwarz.tex \ + ../slides/2/polarformel.tex \ + ../slides/2/funktionenraum.tex \ + ../slides/2/operatornorm.tex \ + ../slides/2/linearformnormen.tex \ + ../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/cauchyschwarz.tex b/vorlesungen/slides/2/cauchyschwarz.tex new file mode 100644 index 0000000..a24ada8 --- /dev/null +++ b/vorlesungen/slides/2/cauchyschwarz.tex @@ -0,0 +1,94 @@ +% +% cauchyschwarz.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.5,0} +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Cauchy-Schwarz-Ungleichung} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Satz (Cauchy-Schwarz)} +$\langle\;,\;\rangle$ eine positiv definite, hermitesche Sesquilinearform +\[ +{\color{darkgreen} +|\operatorname{Re}\langle u,v\rangle| +\le +|\langle u,v\rangle| +\le +\|u\|_2\cdot \|v\|_2 +} +\] +Gleichheit genau dann, wenn $u$ und $v$ linear abhängig sind +\end{block} +\begin{block}{Dreiecksungleichung} +\vspace{-12pt} +\begin{align*} +\|u+v\|_2^2 +&= +\|u\|_2^2 + 2\operatorname{Re}\langle u,v\rangle + \|v\|_2^2 +\\ +&\le +\|u\|_2^2 + 2{\color{darkgreen}|\langle u,v\rangle|} + \|v\|_2^2 +\\ +&\le +\|u\|_2^2 + 2{\color{darkgreen}\|u\|_2\cdot \|v\|_2} + \|v\|_2^2 +\\ +&=(\|u\|_2 + \|v\|_2)^2 +\end{align*} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<2->{% +\begin{proof}[Beweis] +Die quadratische Funktion +\begin{align*} +Q(t) +&= +\langle u+tv,u+tv\rangle \ge 0 +\\ +\uncover<3->{ +Q(t) +&= +\|u\|_2^2 + 2t\operatorname{Re}\langle u,v\rangle + t^2\|v\|_2^2} +\end{align*} +\uncover<4->{hat ihr Minimum bei}% +\begin{align*} +\uncover<5->{ +t&= +-\operatorname{Re}\langle u,v\rangle/\|v\|_2^2} +\intertext{\uncover<6->{mit Wert}} +\uncover<7->{ +Q(t) +&= +\|u\|_2^2 +-2\operatorname{Re}\langle u,v\rangle^2/\|v\|_2^2} +\\ +\uncover<7->{ +&\qquad + \operatorname{Re}\langle u,v\rangle^2/\|v\|_2^2} +\\ +\uncover<8->{ +0 +&\le +\|u\|_2^2-\operatorname{Re}\langle u,v\rangle^2/\|v\|_2^2} +\\ +\uncover<9->{ +\operatorname{Re}\langle u,v\rangle^2 +&\le +\|u\|_2^2\cdot\|v\|_2^2} +\\ +\uncover<10->{ +\operatorname{Re}\langle u,v\rangle +&\le +\|u\|_2\cdot\|v\|_2} +\qedhere +\end{align*} +\end{proof}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/2/chapter.tex b/vorlesungen/slides/2/chapter.tex new file mode 100644 index 0000000..49e656a --- /dev/null +++ b/vorlesungen/slides/2/chapter.tex @@ -0,0 +1,17 @@ +% +% chapter.tex +% +% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi +% +\folie{2/norm.tex} +\folie{2/skalarprodukt.tex} +\folie{2/cauchyschwarz.tex} +\folie{2/polarformel.tex} +\folie{2/funktionenraum.tex} +\folie{2/operatornorm.tex} +\folie{2/linearformnormen.tex} +\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/frobeniusanwendung.tex b/vorlesungen/slides/2/frobeniusanwendung.tex new file mode 100644 index 0000000..277d600 --- /dev/null +++ b/vorlesungen/slides/2/frobeniusanwendung.tex @@ -0,0 +1,80 @@ +% +% frobeniusanwendung.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Anwendung der Frobenius-Norm} +\vspace{-18pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Ableitung nach $X\in M_{m\times n}(\mathbb{R})$} +Die Ableitung $Df=\partial f/\partial X$ der Funktion +$f\colon M_{m\times n}(\mathbb{R})\to \mathbb{R}$ ist die Matrix +mit Einträgen +\begin{align*} +\biggl( +\frac{\partial f}{\partial X} +\biggr)_{ij} +&= +\frac{\partial f}{\partial x_{ij}} += +D_{ij}f +\end{align*} +\end{block} +\uncover<2->{% +\begin{block}{Richtungsableitung} +\uncover<5->{Die Matrix $Df$ ist ein Gradient:} +\begin{align*} +\frac{\partial}{\partial t}f(X+tY)\bigg|_{t=0} +&=\uncover<3->{ +\sum_{i,j} +D_{ij} f(X) \cdot y_{ij}} +\\ +&\uncover<4->{= +\langle D_{ij}f(X), Y\rangle_F} +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<6->{% +\begin{block}{Quadratische Minimalprobleme} +$A=A^t,B,X\in M_n(\mathbb{R})$, Minimum von +\begin{align*} +f(X)&=\langle X,AX\rangle_F + \langle B,X\rangle_F +\intertext{\uncover<7->{Folgerungen:}} +\uncover<8->{ +\langle X,AY\rangle_F&=\langle AX,Y\rangle_F +} +\\ +\uncover<9->{ +D\langle B,\mathstrut\cdot\mathstrut\rangle_F +&= +B +} +\\ +\uncover<10->{ +D_X\langle X, AY\rangle_F +&=AY +} +\\ +\uncover<11->{ +D_Y\langle X, AY\rangle_F +&=AX +} +\\ +\uncover<12->{ +Df &= 2AX + B +} +\intertext{\uncover<13->{Minimum:}} +\uncover<14->{ +X&=-\frac12 A^{-1}B +} +\end{align*} +\uncover<15->{(Kalman-Filter)} +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/2/frobeniusnorm.tex b/vorlesungen/slides/2/frobeniusnorm.tex new file mode 100644 index 0000000..461005a --- /dev/null +++ b/vorlesungen/slides/2/frobeniusnorm.tex @@ -0,0 +1,96 @@ +% +% frobeniusnorm.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Frobenius-Norm} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Skalarprodukt} +$A,B\in M_{m\times n}(\mathbb{C})$ +\begin{align*} +\langle A,B\rangle_F +&\uncover<2->{= +\sum_{i,j} \overline{a}_{ik}b_{ik}} +\uncover<3->{= +\operatorname{Spur} A^*B} +\\ +\uncover<4->{ +\|A\|_F^2 +&= +\langle A,A\rangle} +\uncover<5->{= +\sum_{i,k} |a_{ik}|^2} +\end{align*} +\uncover<6->{% +$\Rightarrow M_{m\times n}(\mathbb{C})$ ist ein normierter Raum} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<12->{% +\begin{block}{Singulärwertzerlegung} +\vspace{-12pt} +\begin{align*} +\uncover<13->{ +A +&= +U\Sigma V^*} +\\ +\uncover<14->{ +A^*A +&= +V\Sigma^*U^*U\Sigma V^*} +\uncover<15->{= +V\Sigma^*\Sigma V^*} +\\ +\uncover<16->{% +\operatorname{Spur}{A^*A} +&= +\operatorname{Spur}V\Sigma^*\Sigma V^*} +\\ +\uncover<17->{% +&= +\operatorname{Spur}V^*V\Sigma^*\Sigma} +\\ +\uncover<18->{% +&= +\operatorname{Spur}\Sigma^*\Sigma} +\uncover<19->{= +\sum_{i} |\sigma_i|^2} +\end{align*} +\end{block}} +\end{column} +\end{columns} +\uncover<7->{% +\begin{block}{Produkt} +\vspace{-10pt} +\begin{align*} +\|AB\|_F +\uncover<8->{= +\sum_{i,j} +\biggl| +\sum_{k} +a_{ik}b_{kj} +\biggr|^2} +&\uncover<9->{\le +\sum_{i,j} +\biggl( +\sum_k |a_{ik}|^2 +\biggr) +\biggl( +\sum_l |b_{lj}|^2 +\biggr)} +\\ +\uncover<10->{ +&= +\sum_{i,k} |a_{ik}|^2 +\sum_{l,j} |b_{lj}|^2} +\uncover<11->{= +\|A\|_F\cdot \|B\|_F} +\end{align*} +\end{block}} +\end{frame} diff --git a/vorlesungen/slides/2/funktionenalgebra.tex b/vorlesungen/slides/2/funktionenalgebra.tex new file mode 100644 index 0000000..9116be4 --- /dev/null +++ b/vorlesungen/slides/2/funktionenalgebra.tex @@ -0,0 +1,88 @@ +% +% funktionenalgebra.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Funktionenalgebra} +\vspace{-17pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Algebra $C([0,1])$} +Funktionenraum +\[ +C([0,1]) += +\{f\colon[0,1]\to\mathbb{C}\;|\;\text{$f$ stetig}\} +\] +mit Supremum-Norm\uncover<2->{ und punktweisem Produkt +\[ +(f\cdot g)(x) += +f(x)\cdot g(x) +\]} +\end{block} +\vspace{-8pt} +\uncover<3->{% +\begin{block}{Algebranorm} +\vspace{-12pt} +\begin{align*} +\|f\cdot g\|_\infty +&= +\sup_{x\in[0,1]} |f(x)g(x)| +\\ +\uncover<4->{ +&\le +\sup_{x\in[0,1]}|f(x)| +\sup_{y\in[0,1]}|g(y)| +} +\\ +\uncover<5->{ +&= +\|f\|_\infty \cdot \|g\|_\infty +} +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<6->{% +\begin{block}{Faltungs-Algebra $L^2([0,1])$} +Funktionenraum +\[ +L^2=\{f\colon \mathbb{R}\to\mathbb{C}\;|\;\text{$f$ $1$-periodisch}\} +\] +mit $L^2$-Skalarprodukt\uncover<7->{ und Faltungsprodukt +\[ +f*g(x) += +\int_0^1 +\underbrace{f(x-t)}_{(=\gamma_x\check{f})(t)} g(t)\,dx +\]} +\end{block}} +\vspace{-21pt} +\uncover<8->{% +\begin{block}{Norm} +\vspace{-12pt} +\begin{align*} +\|f*g\|_2^2 +&\uncover<9->{=\int_0^1 | +\langle \gamma_x\check{f},g\rangle +|^2\,dx} +\\ +\uncover<10->{ +&\le +\int_0^1 +\|\gamma_t\check{f}\|_2^2 +\|g\|_2^2 +\,dx} +\\ +\uncover<11->{ +&=\|f\|_2^2\cdot \|g\|_2^2 +} +\end{align*} +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/2/funktionenraum.tex b/vorlesungen/slides/2/funktionenraum.tex new file mode 100644 index 0000000..f7733cc --- /dev/null +++ b/vorlesungen/slides/2/funktionenraum.tex @@ -0,0 +1,70 @@ +% +% funktionenraum.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Funktionenraum} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Supremum-Norm} +Vektorraum +\[ +C([a,b]) += +\{f\colon[a,b]\to\mathbb{R}\;|\; \text{$f$ stetig}\} +\] +\only<2->{wird Banachraum }% +mit der Norm +\(\displaystyle +\|f\| += +\|f\|_{\infty} += +\sup_{x\in[a,b]} |f(x)| +\) +\end{block} +\uncover<3->{% +\begin{block}{$L^1$-Norm} +Vektorraum +\[ +L^1([a,b]) += +\{f\colon[a,b]\;|\;\text{$f$ integrierbar}\} +\] +\only<4->{wird Banachraum }% +mit der Norm +\[ +\|f\|_1 += +\int_a^b |f(x)|\,dx +\] +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<5->{% +\begin{block}{$L^2$-Norm} +Vektorraum +\[ +L^2([a,b]) += +\{f\colon[a,b]\to\mathbb{R}\;|\; \|f\|_2^2<\infty\} +\] +mit Skalarprodukt +\begin{align*} +\langle f,g\rangle +&= +\int_a^b \overline{f}(x)g(x)\,dx +\\ +\|f\|_2^2 +&= +\int_a^b |f(x)|^2\,dx +\end{align*} +\uncover<6->{ist ein Banachraum} +\end{block}} +\end{column} +\end{columns} +\end{frame} 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/linearformnormen.tex b/vorlesungen/slides/2/linearformnormen.tex new file mode 100644 index 0000000..8993f66 --- /dev/null +++ b/vorlesungen/slides/2/linearformnormen.tex @@ -0,0 +1,76 @@ +% +% linearformnormen.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Linearformen} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Linearformen $\varphi\colon L^1\to\mathbb{R}$} +Beispiel: $g\in C([a,b])$ +\[ +\varphi(f) += +\int_a^b g(x)f(x)\,dx +\] +\uncover<2->{% +erfüllt +\begin{align*} +|\varphi(f)| +&= +\biggl|\int_a^b g(x)f(x)\,dx\biggr| +\\ +\uncover<3->{ +&\le \|g\|_\infty\cdot \|f\|_1 +} +\end{align*}} +\uncover<4->{% +und hat daher die Operatornorm +\[ +\|\varphi\|_{C([a,b])^*} += +\|g\|_\infty +\]} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Linearformen $\varphi\colon L^2\to\mathbb{R}$} +\uncover<5->{% +Darstellungssatz von Riesz: $\exists g\in L^2$ +\[ +\varphi(f) = \langle g,f\rangle +\]} +\uncover<6->{% +erfüllt Cauchy-Schwarz} +\begin{align*} +\uncover<7->{ +|\varphi(f)| +&= +|\langle g,f\rangle|} +\\ +\uncover<8->{ +&\le +\|g\|_2 \cdot \|f\|_2 +} +\end{align*} +\uncover<9->{% +und hat daher die Operatornorm +\[ +\|\varphi\|_{L^2([a,b])^*} += \|g\|_2 +\]} +\end{block} +\end{column} +\end{columns} + +\vspace{8pt} +{\usebeamercolor[fg]{title} +\uncover<10->{% +$\Rightarrow$ +Operatornorm hängt von den Vektorraumnormen ab} +} +\end{frame} diff --git a/vorlesungen/slides/2/norm.tex b/vorlesungen/slides/2/norm.tex new file mode 100644 index 0000000..35d2513 --- /dev/null +++ b/vorlesungen/slides/2/norm.tex @@ -0,0 +1,58 @@ +% +% norm.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Norm} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Wozu} +Ziel: Konvergenz von Folgen, Grenzwert in einem Vektorraum +\end{block} +\uncover<7->{% +\begin{block}{Cauchy-Folge} +Eine Folge $(x_n)_{n\in\mathbb{N}}$ von Vektoren in $V$ heisst +{\em Cauchy-Folge}, +wenn es für alle $\varepsilon >0$ ein $N$ gibt mit +\[ +\|x_n-x_m\| < \varepsilon\; \forall n,m>N +\] +\end{block}} +\vspace{-8pt} +\uncover<8->{% +\begin{block}{Grenzwert} +$x\in V$ heisst Grenzwert der Folge $x_n$, wenn es für alle $\varepsilon>0$ +ein $N$ gibt mit +\[ +\| x-x_n\| < \varepsilon \;\forall n>N +\] +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<2->{% +\begin{block}{Definition} +$V$ ein $\mathbb{R}$-Vektorraum. +Eine Funktion +\[ +\|\cdot\| \colon V \to \mathbb{R}_{\ge 0} : v \mapsto \|v\| +\] +heisst eine {\em Norm}, wenn +\begin{itemize} +\item<3-> $\| v \|>0$ für $v\ne 0$ +\item<4-> $\|\lambda v\| = |\lambda|\cdot\|v\|$ +\item<5-> $\| u + v \| \le \|u\| + \|v\|$ (Dreiecksungleichung) +\end{itemize} +\uncover<6->{% +Ein Vektorraum mit einer Norm heisst {\em normierter Raum}} +\end{block}} +\uncover<9->{% +\begin{block}{Banach-Raum} +Normierter Raum, in dem jede Cauchy-Folge einen Grenwzert hat +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/2/operatornorm.tex b/vorlesungen/slides/2/operatornorm.tex new file mode 100644 index 0000000..d20461a --- /dev/null +++ b/vorlesungen/slides/2/operatornorm.tex @@ -0,0 +1,59 @@ +% +% operatorname.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Operatornorm} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\uncover<2->{% +\begin{block}{Lineare Operatoren} +$A\colon U\to V$ lineare Abbildung mit $U$, $V$ normiert +\end{block}} +\uncover<3->{% +\begin{block}{Operatornorm} +eines linearen Operators $A$: +\[ +\|A\| += +\sup_{\|x\|_U\le 1} \|Ax\|_V +\] +\uncover<4->{$\Rightarrow \|Ax\| \le \| A \|\cdot \|x\|$} +\end{block}} +\uncover<5->{% +\begin{block}{Stetigkeit} +Wenn $\|A\|<\infty$, dann ist $A$ stetig, d.~h. +\[ +\lim_{n\to\infty} Ax_n += +A\lim_{n\to\infty} x_n +\] +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<6->{% +\begin{block}{Algebranorm} +$A$ ein normierter Raum, der auch ein Algebra ist. +Dann heisst $A$ eine normierte Algebra, wenn +\[ +\| ab\| \le \| a\|\cdot \|b\| +\quad\forall a,b\in A +\] +\end{block}} +\vspace{-10pt} +\uncover<7->{% +\begin{block}{Operatoralgebra} +$U$ ein normierter Raum, dann ist die Algebra der linearen Operatoren +$A\colon U\to U$ mit der Operatornorm eine normierte Algebra +\end{block}} +\uncover<8->{% +\begin{block}{Banach-Algebra} +Ein Banach-Raum, der auch eine normierte Algebra ist +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/2/polarformel.tex b/vorlesungen/slides/2/polarformel.tex new file mode 100644 index 0000000..ebdbf81 --- /dev/null +++ b/vorlesungen/slides/2/polarformel.tex @@ -0,0 +1,113 @@ +% +% polarformel.tex +% +% (c) 2021 Prod Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkcolor}{rgb}{0,0.6,0} +\def\yone{-2.1} +\def\ytwo{-3.55} +\def\ythree{-5.0} +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Polarformel} +\vspace{-5pt} +\begin{block}{Aufgabe} +$\langle x,y\rangle$ aus Werten von $\|\cdot\|_2$ rekonstruieren: + +\end{block} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\node at (0,0) {$ +\begin{aligned} +\uncover<2->{ +\|x+ty\|_2^2 +&= +\|x\|_2^2 ++t\langle x,y\rangle ++\overline{t}\langle y,x\rangle ++ \|y\|_2^2} +\\ +\uncover<3->{ +&= +\|x\|_2^2 ++t\langle x,y\rangle ++\overline{t\langle x,y\rangle} ++ \|y\|_2^2} +\\ +\uncover<4->{ +&= +\|x\|_2^2 ++2\operatorname{Re}(t\langle x,y\rangle) ++ \|y\|_2^2} +\end{aligned}$}; + +\uncover<5->{ + \draw[->] (-1,-0.9) -- (-3.3,{\yone+0.25}); + \node at (-3.5,\yone) {$ + \|x\pm y\|_2^2 + = + \|x\|_2^2 + \pm2\operatorname{Re}\langle x,y\rangle + + + \|y\|_2^2 + $}; +} + +\uncover<8->{ + \draw[->] (1,-0.9) -- (3.3,{\yone+0.25}); + \node at (3.5,\yone) {$ + \|x\pm iy\|_2^2 + = + \|x\|_2^2 + \pm2i\operatorname{Im}\langle x,y\rangle + + + \|y\|_2^2 + $}; +} + +\uncover<6->{ + \draw[->] (-3.5,{\yone-0.2}) -- (-3.5,{\ytwo+0.2}); + \node at (-3.5,\ytwo) {$\operatorname{Re}\langle x,y\rangle + = + \frac12\bigl( + \|x+y\|_2^2-\|x-y\|_2^2 + \bigr) + $}; +} + +\uncover<9->{ + \draw[->] (3.5,{\yone-0.2}) -- (3.5,{\ytwo+0.2}); + \node at (3.5,\ytwo) {$ + \operatorname{Im}\langle x,y\rangle + = + \frac1{2i}\bigl( + \|x+iy\|_2^2-\|x-iy\|_2^2 + \bigr) + $}; +} + +\uncover<7->{ + \draw[->] (-3.3,{\ytwo-0.25}) -- (-1.5,{\ythree+0.25}); + \node at (0,\ythree) {$ + \langle x,y\rangle + = + \frac12\bigl( + \|x+y\|_2^2-\|x-y\|_2^2 + \uncover<10->{ + + + \|x+iy\|_2^2-\|x-iy\|_2^2 + } + \bigr)$}; +} + +\uncover<10->{ + \draw[->] (3.3,{\ytwo-0.25}) -- (1.5,{\ythree+0.25}); +} + +\end{tikzpicture} +\end{center} +\end{frame} +\egroup 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 diff --git a/vorlesungen/slides/2/skalarprodukt.tex b/vorlesungen/slides/2/skalarprodukt.tex new file mode 100644 index 0000000..99d8a73 --- /dev/null +++ b/vorlesungen/slides/2/skalarprodukt.tex @@ -0,0 +1,96 @@ +% +% skalarprodukt.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Skalarprodukt} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Positiv definite, symmetrische Bilinearform} +$\langle \;\,,\;\rangle\colon V\times V\to \mathbb{R}$ +\begin{itemize} +\item<2-> +Bilinear: +\begin{align*} +\langle \alpha u+\beta v,w\rangle +&= +\alpha\langle u,w\rangle ++ +\beta\langle v,w\rangle +\\ +\langle u,\alpha v+\beta w\rangle +&= +\alpha\langle u,v\rangle ++ +\beta\langle u,w\rangle +\end{align*} +\item<3-> +Symmetrisch: $\langle u,v\rangle = \langle v,u\rangle$ +\item<4-> +$\langle x,x\rangle >0 \quad\forall x\ne 0$ +\end{itemize} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<5->{% +\begin{block}{Positive definite, hermitesche Sesquilinearform} +$\langle \;\,,\;\rangle\colon V\times V\to \mathbb{C}$ +\begin{itemize} +\item<6-> +Sesquilinear: +\begin{align*} +\langle \alpha u+\beta v,w\rangle +&= +\overline{\alpha}\langle u,w\rangle ++ +\overline{\beta}\langle v,w\rangle +\\ +\langle u,\alpha v+\beta w\rangle +&= +\alpha\langle u,v\rangle ++ +\beta\langle u,w\rangle +\end{align*} +\item<7-> +Hermitesch: $\langle u,v\rangle = \overline{\langle v,u\rangle}$ +\item<8-> +$\langle x,x\rangle >0 \quad\forall x\ne 0$ +\end{itemize} +\end{block}} +\end{column} +\end{columns} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.28\textwidth} +\uncover<9->{% +\begin{block}{$2$-Norm} +$\|v\|_2^2 = \langle v,v\rangle$ +\\ +$\|v\|_2 = \sqrt{\langle v,v\rangle}$ +\end{block}} +\end{column} +\begin{column}{0.78\textwidth} +\uncover<10->{% +\begin{itemize} +\item<11-> $\|v\|_2 = \sqrt{\langle v,v\rangle} > 0\quad\forall v\ne 0$ +\item<12-> $\| \lambda v \|_2 += +\sqrt{\langle \lambda v,\lambda v\rangle\mathstrut} += +\sqrt{\overline{\lambda}\lambda\langle v,v\rangle} += +|\lambda|\cdot \|v\|_2$ +\item<13-> +\raisebox{-8pt}{ +$\begin{aligned} +\|u+v\|_2^2 &= \|u\|_2^2 + 2{\color{red}\operatorname{Re}\langle u,v\rangle} + \|v\|_2^2 +\\ +(\|u\|_2+\|v\|_2)^2 &= \|u\|_2^2 + 2{\color{red}\|u\|_2\|v\|_2} + \|v\|_2^2 +\end{aligned}$} +\end{itemize}} +\end{column} +\end{columns} +\end{frame} |
