Merge pull request #1 from AndreasFMueller/master

update

1 files changed, 110 insertions, 0 deletions

diff --git a/vorlesungen/slides/2/quotient.tex b/vorlesungen/slides/2/quotient.tex
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+++ 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}