diff options
Diffstat (limited to 'vorlesungen/slides/5')
20 files changed, 356 insertions, 3 deletions
diff --git a/vorlesungen/slides/5/Makefile.inc b/vorlesungen/slides/5/Makefile.inc index 0858369..4ca3de4 100644 --- a/vorlesungen/slides/5/Makefile.inc +++ b/vorlesungen/slides/5/Makefile.inc @@ -5,8 +5,11 @@ # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule # chapter5 = \ + ../slides/5/verzerrung.tex \ ../slides/5/motivation.tex \ ../slides/5/charpoly.tex \ + ../slides/5/kernbildintro.tex \ + ../slides/5/kernbilder.tex \ ../slides/5/kernbild.tex \ ../slides/5/ketten.tex \ ../slides/5/dimension.tex \ diff --git a/vorlesungen/slides/5/beispiele/Makefile b/vorlesungen/slides/5/beispiele/Makefile index c03ea62..05bd5b5 100644 --- a/vorlesungen/slides/5/beispiele/Makefile +++ b/vorlesungen/slides/5/beispiele/Makefile @@ -3,7 +3,7 @@ # # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule # -all: kern bild kb kombiniert.jpg +all: kern bild kb kombiniert.jpg leer.jpg drei.jpg kern: kern1.jpg kern2.jpg bild: bild1.jpg bild2.jpg @@ -19,6 +19,8 @@ bild2.png: JK1.inc common.inc bild2.pov kern1.png: JK1.inc common.inc kern1.pov kern2.png: JK1.inc common.inc kern2.pov kombiniert.png: JK1.inc common.inc kombiniert.pov +leer.png: JK1.inc common.inc leer.pov +drei.png: JK1.inc common.inc drei.pov %.png: %.pov povray +A0.1 -W1920 -H1080 -O$@ $< diff --git a/vorlesungen/slides/5/beispiele/bild1.jpg b/vorlesungen/slides/5/beispiele/bild1.jpg Binary files differindex e8bdc1c..879fae8 100644 --- a/vorlesungen/slides/5/beispiele/bild1.jpg +++ b/vorlesungen/slides/5/beispiele/bild1.jpg diff --git a/vorlesungen/slides/5/beispiele/bild2.jpg b/vorlesungen/slides/5/beispiele/bild2.jpg Binary files differindex 8e5dde5..2597c95 100644 --- a/vorlesungen/slides/5/beispiele/bild2.jpg +++ b/vorlesungen/slides/5/beispiele/bild2.jpg diff --git a/vorlesungen/slides/5/beispiele/drei.jpg b/vorlesungen/slides/5/beispiele/drei.jpg Binary files differnew file mode 100644 index 0000000..35f9034 --- /dev/null +++ b/vorlesungen/slides/5/beispiele/drei.jpg diff --git a/vorlesungen/slides/5/beispiele/drei.pov b/vorlesungen/slides/5/beispiele/drei.pov new file mode 100644 index 0000000..bdc9630 --- /dev/null +++ b/vorlesungen/slides/5/beispiele/drei.pov @@ -0,0 +1,22 @@ +// +// drei.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// + +#include "common.inc" +#include "JK.inc" + +arrow(O, j21, at, orange2) +//arrow(O, k21, at, gruen2) +//arrow(O, k22, at, gruen2) +gerade(j21, orange2) +//ebene(k21, k22, gruen2) + +#declare at = 0.7 * at; + +arrow(O, j11, at, orange1) +arrow(O, j12, at, orange1) +arrow(O, k11, at, gruen1) +ebene(j11, j12, orange1) + diff --git a/vorlesungen/slides/5/beispiele/kern1.jpg b/vorlesungen/slides/5/beispiele/kern1.jpg Binary files differindex f5fcf7d..5c99664 100644 --- a/vorlesungen/slides/5/beispiele/kern1.jpg +++ b/vorlesungen/slides/5/beispiele/kern1.jpg diff --git a/vorlesungen/slides/5/beispiele/kern2.jpg b/vorlesungen/slides/5/beispiele/kern2.jpg Binary files differindex c3b4b79..87d18ac 100644 --- a/vorlesungen/slides/5/beispiele/kern2.jpg +++ b/vorlesungen/slides/5/beispiele/kern2.jpg diff --git a/vorlesungen/slides/5/beispiele/kernbild.m b/vorlesungen/slides/5/beispiele/kernbild.m index ca6bd6a..28cd552 100644 --- a/vorlesungen/slides/5/beispiele/kernbild.m +++ b/vorlesungen/slides/5/beispiele/kernbild.m @@ -47,7 +47,8 @@ fprintf(f, "#declare j12 = < %.5f, %.5f, %.5f>;\n", E1(1,2), E1(2,2), E1(3,2)); fprintf(f, "\n"); # k = 2 -D = rref(E1 * E1); +E12 = E1 * E1 +D = rref(E12); K2 = [ -D(1,2), -D(1,3); 1, 0; @@ -60,7 +61,7 @@ K2 fprintf(f, "// Kern und Bild von (C - %.3f I)^2\n", lambda2); fprintf(f, "#declare k21 = < %.5f, %.5f, %.5f>;\n", K2(1,1), K2(2,1), K2(3,1)); fprintf(f, "#declare k22 = < %.5f, %.5f, %.5f>;\n", K2(1,2), K2(2,2), K2(3,2)); -fprintf(f, "#declare j21 = < %.5f, %.5f, %.5f>;\n", E2(1,1), E2(2,1), E2(3,1)); +fprintf(f, "#declare j21 = < %.5f, %.5f, %.5f>;\n", E12(1,1), E12(2,1), E12(3,1)); fprintf(f, "\n"); fclose(f); diff --git a/vorlesungen/slides/5/beispiele/kernbild1.jpg b/vorlesungen/slides/5/beispiele/kernbild1.jpg Binary files differindex bda5ce4..87e874e 100644 --- a/vorlesungen/slides/5/beispiele/kernbild1.jpg +++ b/vorlesungen/slides/5/beispiele/kernbild1.jpg diff --git a/vorlesungen/slides/5/beispiele/kernbild2.jpg b/vorlesungen/slides/5/beispiele/kernbild2.jpg Binary files differindex 4139918..1160b31 100644 --- a/vorlesungen/slides/5/beispiele/kernbild2.jpg +++ b/vorlesungen/slides/5/beispiele/kernbild2.jpg diff --git a/vorlesungen/slides/5/beispiele/kombiniert.jpg b/vorlesungen/slides/5/beispiele/kombiniert.jpg Binary files differindex c214596..9cb789c 100644 --- a/vorlesungen/slides/5/beispiele/kombiniert.jpg +++ b/vorlesungen/slides/5/beispiele/kombiniert.jpg diff --git a/vorlesungen/slides/5/beispiele/kombiniert.pov b/vorlesungen/slides/5/beispiele/kombiniert.pov new file mode 100644 index 0000000..c187d08 --- /dev/null +++ b/vorlesungen/slides/5/beispiele/kombiniert.pov @@ -0,0 +1,22 @@ +// +// kombiniert.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// + +#include "common.inc" +#include "JK.inc" + +arrow(O, j21, at, orange2) +arrow(O, k21, at, gruen2) +arrow(O, k22, at, gruen2) +gerade(j21, orange2) +ebene(k21, k22, gruen2) + +#declare at = 0.7 * at; + +arrow(O, j11, at, orange1) +arrow(O, j12, at, orange1) +arrow(O, k11, at, gruen1) +ebene(j11, j12, orange1) + diff --git a/vorlesungen/slides/5/beispiele/leer.jpg b/vorlesungen/slides/5/beispiele/leer.jpg Binary files differnew file mode 100644 index 0000000..9789887 --- /dev/null +++ b/vorlesungen/slides/5/beispiele/leer.jpg diff --git a/vorlesungen/slides/5/beispiele/leer.pov b/vorlesungen/slides/5/beispiele/leer.pov new file mode 100644 index 0000000..f4653d9 --- /dev/null +++ b/vorlesungen/slides/5/beispiele/leer.pov @@ -0,0 +1,9 @@ +// +// leer.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// + +#include "common.inc" +#include "JK.inc" + diff --git a/vorlesungen/slides/5/chapter.tex b/vorlesungen/slides/5/chapter.tex index c19222c..96eea29 100644 --- a/vorlesungen/slides/5/chapter.tex +++ b/vorlesungen/slides/5/chapter.tex @@ -3,8 +3,11 @@ % % (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi % +\folie{5/verzerrung.tex} \folie{5/motivation.tex} \folie{5/charpoly.tex} +\folie{5/kernbildintro.tex} +\folie{5/kernbilder.tex} \folie{5/kernbild.tex} \folie{5/ketten.tex} \folie{5/dimension.tex} diff --git a/vorlesungen/slides/5/kernbilder.tex b/vorlesungen/slides/5/kernbilder.tex new file mode 100644 index 0000000..08581ff --- /dev/null +++ b/vorlesungen/slides/5/kernbilder.tex @@ -0,0 +1,68 @@ +% +% kernbilder.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup + +\definecolor{grueneins}{rgb}{0.0,0.4,0.0} +\definecolor{gruenzwei}{rgb}{0.0,0.4,0.8} +\definecolor{orangeeins}{rgb}{1.0,0.6,0.0} +\definecolor{orangezwei}{rgb}{0.8,0.0,0.4} + +\begin{frame}[t] +\frametitle{Kerne und Bilder} +\vspace{-15pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\uncover<2->{ +\begin{scope}[xshift=-4cm,yshift=1.9cm] +\node at (0,0) {\includegraphics[width=3.6cm]{../slides/5/beispiele/bild1.jpg}}; +\node[color=orangeeins] at (1.6,1.3) [right] {$\mathcal{J}^1(A)$}; +\end{scope} +} + +\uncover<3->{ +\begin{scope}[xshift=-4cm,yshift=-1.9cm] +\node at (0,0) {\includegraphics[width=3.6cm]{../slides/5/beispiele/bild2.jpg}}; +\node[color=orangezwei] at (0.9,0.5) {$\mathcal{J}^2(A)$}; +\end{scope} +} + +\begin{scope}[xshift=0cm,yshift=0cm] +\uncover<1>{ +\node at (0,0) {\includegraphics[width=3.6cm]{../slides/5/beispiele/leer.jpg}}; +} +\uncover<2>{ +\node at (0,0) {\includegraphics[width=3.6cm]{../slides/5/beispiele/bild1.jpg}}; +} +\uncover<3>{ +\node at (0,0) {\includegraphics[width=3.6cm]{../slides/5/beispiele/bild2.jpg}}; +} +\uncover<4>{ +\node at (0,0) {\includegraphics[width=3.6cm]{../slides/5/beispiele/drei.jpg}}; +} +\uncover<5->{ +\node at (0,0) {\includegraphics[width=3.6cm]{../slides/5/beispiele/kombiniert.jpg}}; +} +\end{scope} + +\uncover<4->{ +\begin{scope}[xshift=4cm,yshift=1.9cm] +\node at (0,0) {\includegraphics[width=3.6cm]{../slides/5/beispiele/kern1.jpg}}; +\node[color=grueneins] at (1.0,1.3) [right] {$\mathcal{K}^1(A)$}; +\end{scope} +} + +\uncover<5->{ +\begin{scope}[xshift=4cm,yshift=-1.9cm] +\node at (0,0) {\includegraphics[width=3.6cm]{../slides/5/beispiele/kern2.jpg}}; +\node[color=gruenzwei] at (0.7,-0.6) {$\mathcal{K}^2(A)$}; +\end{scope} +} + +\end{tikzpicture} +\end{center} +\end{frame} +\egroup diff --git a/vorlesungen/slides/5/kernbildintro.tex b/vorlesungen/slides/5/kernbildintro.tex new file mode 100644 index 0000000..9fd7849 --- /dev/null +++ b/vorlesungen/slides/5/kernbildintro.tex @@ -0,0 +1,89 @@ +% +% kernbildintro.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup + +\definecolor{grueneins}{rgb}{0.0,0.4,0.0} +\definecolor{gruenzwei}{rgb}{0.0,0.4,0.8} +\definecolor{orangeeins}{rgb}{1.0,0.6,0.0} +\definecolor{orangezwei}{rgb}{0.8,0.0,0.4} + +\begin{frame}[t] +\frametitle{Bilder und Kerne} +\vspace{-15pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\begin{scope}[xshift=-3.4cm] + +\only<1>{ +\node at (0,0) {\includegraphics[width=6.6cm]{../slides/5/beispiele/leer.jpg}}; +} +\only<2-3>{ +\node at (0,0) {\includegraphics[width=6.6cm]{../slides/5/beispiele/bild1.jpg}}; +} +\uncover<4->{ +\node at (0,0) {\includegraphics[width=6.6cm]{../slides/5/beispiele/bild2.jpg}}; +} +\uncover<2->{ + \fill[color=white,opacity=0.7] (0.1,2.18) rectangle (4,2.64); + \node[color=orangeeins] at (0,2.4) [right] + {$\operatorname{im} A = \{Av\;|v\in\mathbb{R}^n\}$}; +} +\uncover<4->{ + \node[color=orangezwei] at (4,0.7) [left] + {$\operatorname{im} A^2 = \{A^2v\;|v\in\mathbb{R}^n\}$}; +} +\end{scope} + +\begin{scope}[xshift=3.4cm] + +\uncover<2->{ +\fill[color=orangeeins!40] (-1,0.5) rectangle (1.8,2); +} +\uncover<4->{ +\fill[color=orangezwei!40] (-1.1,-1.7) rectangle (-0.,-0.3); +} + +\node at (0,0) {\begin{minipage}{6cm} +\begin{align*} +A&={\scriptstyle\begin{pmatrix*}[r] + -0.979& -0.142& 0.917\\ + -0.260& -0.643& 1.069\\ + -0.285& -0.449& 0.823 +\end{pmatrix*}} +\\ +\operatorname{Rang}A&=2 +\\ +\uncover<3->{ +A^2&={\scriptstyle\begin{pmatrix*}[r] + 0.734& -0.181& -0.295\\ + 0.118& -0.029& -0.047\\ + 0.161& -0.039& -0.065 +\end{pmatrix*}}}\\ +\uncover<3->{ +\operatorname{Rang}A^2&=1} +\end{align*} +\end{minipage}}; + +\only<5>{ +\node at (0,0) {\includegraphics[width=6.6cm]{../slides/5/beispiele/kern1.jpg}}; +} + +\uncover<6->{ +\node at (0,0) {\includegraphics[width=6.6cm]{../slides/5/beispiele/kern2.jpg}}; +\node[color=gruenzwei] at (-1.35,-3.0) [right] {$\ker A^2 = \{v\;|\; A^2v=0\}$}; +} + +\uncover<5->{ +\node[color=grueneins] at (-0.9,3.1) [right] {$\ker A = \{v\;|\; Av=0\}$}; +} + +\end{scope} + +\end{tikzpicture} +\end{center} +\end{frame} +\egroup diff --git a/vorlesungen/slides/5/verzerrung.tex b/vorlesungen/slides/5/verzerrung.tex new file mode 100644 index 0000000..8d6514c --- /dev/null +++ b/vorlesungen/slides/5/verzerrung.tex @@ -0,0 +1,121 @@ +% +% verzerrung.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\def\r{1.10} +\def\s{1.12} +\def\q{1.23} +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame}[t] +\frametitle{Verzerrung} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.49\textwidth} +\begin{block}{Abbildung $A\colon v\mapsto Av$} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=2.5] +\draw[color=blue,line width=1.2pt] (0,0) circle[radius=1]; + +\coordinate (a1) at (0.974,0.171); +\coordinate (a2) at (0.037,1.018); + +\coordinate (v1) at (-0.5216,0.8532); +\coordinate (v2) at (-0.3343,-0.9425); + +\foreach \a in {0,5,...,355}{ + \draw[color=red,line width=1.2pt] + ($cos(\a)*(a1)+sin(\a)*(a2)$) -- + ($cos(\a+5)*(a1)+sin(\a+5)*(a2)$); +} +\foreach \a in {1,...,144}{ + \only<\a>{ + \fill[color=red,line width=1.4pt] + ($cos(\a*5)*(a1)+sin(\a*5)*(a2)$) circle[radius=0.03]; + \draw[->,color=red,line width=1.4pt] (0,0) -- + ($cos(\a*5)*(a1)+sin(\a*5)*(a2)$); + \draw[->,color=blue,line width=1.4pt] (0,0) -- ({5*\a}:1); + \fill[color=blue] ({5*\a}:1) circle[radius=0.03]; + \node[color=blue] at ({5*\a}:\r) {$v$}; + \node[color=red] at ($\s*cos(\a*5)*(a1)+\s*sin(\a*5)*(a2)$) + {$Av$}; + } +} + +\begin{scope} +\clip (-1.2,-1.1) rectangle (1.2,1.1); +\draw[color=darkgreen,line width=0.7pt] ($-2*(v1)$) -- ($2*(v1)$); +\draw[color=darkgreen,line width=0.7pt] ($-2*(v2)$) -- ($2*(v2)$); +\draw[->,color=darkgreen,line width=1.5pt] (0,0) -- (v1); +\draw[->,color=darkgreen,line width=1.5pt] (0,0) -- (v2); +\end{scope} + +\draw[->] (-\q,0) -- (1.2,0) coordinate[label={$x$}]; +\draw[->] (0,-1.2) -- (0,1.2) coordinate[label={right:$y$}]; + +\node[color=darkgreen] at (v1) [above left] {$v_1$}; +\node[color=darkgreen] at (v2) [below left] {$v_2$}; + +\end{tikzpicture} +\end{center} +\end{block} +\end{column} +\begin{column}{0.49\textwidth} +\uncover<73->{% +\begin{block}{Abbildung $A\colon v\mapsto (A-\lambda)v$} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=2.5] +\draw[color=blue,line width=1.2pt] (0,0) circle[radius=1]; + +\coordinate (a1) at (0.121,0.343); +\coordinate (a2) at (0.074,0.209); + +\coordinate (v1) at (-0.5216,0.8532); +\coordinate (v2) at (-0.3343,-0.9425); + +\begin{scope} +\clip (-1.2,-1.2) rectangle (1.2,1.2); +\draw[color=darkgreen,line width=0.7pt] ($-2*(v1)$) -- ($2*(v1)$); +\draw[color=darkgreen,line width=0.7pt] ($-2*(v2)$) -- ($2*(v2)$); +\end{scope} + +\foreach \a in {0,5,...,355}{ + \draw[color=red!60,line width=4pt] + ($cos(\a)*(a1)+sin(\a)*(a2)$) -- + ($cos(\a+5)*(a1)+sin(\a+5)*(a2)$); +} +\foreach \a in {73,...,144}{ + \only<\a>{ + \fill[color=red,line width=1.4pt] + ($cos(\a*5)*(a1)+sin(\a*5)*(a2)$) circle[radius=0.03]; + \draw[->,color=red,line width=1.4pt] (0,0) -- + ($cos(\a*5)*(a1)+sin(\a*5)*(a2)$); + \draw[->,color=blue,line width=1.4pt] (0,0) -- ({5*\a}:1); + \fill[color=blue] ({5*\a}:1) circle[radius=0.03]; + \node[color=blue] at ({5*\a}:\r) {$v$}; + \node[color=red] at ($\s*cos(\a*5)*(a1)+\s*sin(\a*5)*(a2)$) + {$(A-\lambda)v$}; + } +} + +\begin{scope} +\clip (-1.2,-1.1) rectangle (1.2,1.1); +\draw[->,color=darkgreen,line width=1.5pt] (0,0) -- (v1); +\draw[->,color=darkgreen,line width=1.5pt] (0,0) -- (v2); +\end{scope} + +\draw[->] (-\q,0) -- (1.2,0) coordinate[label={$x$}]; +\draw[->] (0,-1.2) -- (0,1.2) coordinate[label={right:$y$}]; + +\node[color=darkgreen] at (v1) [above left] {$v_1$}; +\node[color=darkgreen] at (v2) [below left] {$v_2$}; + +\end{tikzpicture} +\end{center} + +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/5/verzerrung/verzerrung.m b/vorlesungen/slides/5/verzerrung/verzerrung.m new file mode 100644 index 0000000..028e7f9 --- /dev/null +++ b/vorlesungen/slides/5/verzerrung/verzerrung.m @@ -0,0 +1,13 @@ +# +# verzerrung.m +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# + +rand("seed", 4712); + +A = eye(2) + 1.0 * (rand(2,2) - 0.5 * ones(2,2)) + +[V, lambda] = eig(A) + +B = A - lambda(1,1) * eye(2) |
