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Diffstat (limited to 'vorlesungen/slides/1/algebrastruktur.tex')
| -rw-r--r-- | vorlesungen/slides/1/algebrastruktur.tex | 73 |
1 files changed, 73 insertions, 0 deletions
diff --git a/vorlesungen/slides/1/algebrastruktur.tex b/vorlesungen/slides/1/algebrastruktur.tex new file mode 100644 index 0000000..9647c04 --- /dev/null +++ b/vorlesungen/slides/1/algebrastruktur.tex @@ -0,0 +1,73 @@ +% +% algebrastruktur.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup + +\definecolor{darkgreen}{rgb}{0,0.6,0} + +\begin{frame}[t] +\frametitle{Algebra über $\Bbbk$} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\pgfmathparse{atan(7/4)} +\xdef\a{\pgfmathresult} +\fill[color=red!40,opacity=0.5] + ({-4-2.5},{2+1.0}) + -- + ({-2.5},{-3-1.0}) + -- + ({2.5},{-3-1.0}) + -- + ({-4+2.5},{2+1.0}) + -- cycle; +\fill[color=blue!40,opacity=0.5] + ({4-2.5},{2+1.0}) + -- + ({-2.5},{-3-1.0}) + -- + ({2.5},{-3-1.0}) + -- + ({4+2.5},{2+1.0}) + -- cycle; +\fill[color=darkgreen!40,opacity=0.5] + ({-4-2.5},{2+1.0}) + -- + ({-4-2.5+2*(4/7)},{2-1}) + -- + ({+4+2.5-2*(4/7)},{2-1}) + -- + ({+4+2.5},{2+1}) + -- + cycle; +\node at ({-3-0.5},2) {Skalarmultiplikation}; + +\node at (3.5,2.2) {Multiplikation}; +\node at (3.5,1.8) {\tiny Halbgruppe}; + +\node at (0,-2.8) {Addition}; +\node at (0,-3.2) {\tiny Gruppe}; + +\node[color=blue] at (4.8,-0.5) [rotate=\a] {Ring\strut}; +\node[color=red] at (-4.8,-0.5) [rotate=-\a] {Vektorraum\strut}; + +\node[color=darkgreen] at (0,2.6) {$(\lambda a)b=\lambda(ab)$}; + +\node[color=red] at (-2.5,-0.5) {$\displaystyle +\begin{aligned} +\lambda(a+b)&=\lambda a + \lambda b\\ +(\lambda+\mu)a&=\lambda a +\mu a +\end{aligned}$}; + +\node[color=blue] at (2.5,-0.5) {$\displaystyle +\begin{aligned} +a(b+c)&=ab+ac\\ +(a+b)c&=ac+bc +\end{aligned}$}; + +\end{tikzpicture} +\end{center} +\end{frame} + +\egroup |