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Diffstat (limited to 'vorlesungen/slides/3')
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diff --git a/vorlesungen/slides/3/Makefile.inc b/vorlesungen/slides/3/Makefile.inc new file mode 100644 index 0000000..442bd15 --- /dev/null +++ b/vorlesungen/slides/3/Makefile.inc @@ -0,0 +1,37 @@ + +# +# Makefile.inc -- additional depencencies +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +chapter3 = \ + ../slides/3/motivation.tex \ + ../slides/3/inverse.tex \ + ../slides/3/polynome.tex \ + ../slides/3/division.tex \ + ../slides/3/division2.tex \ + ../slides/3/ringstruktur.tex \ + ../slides/3/teilbarkeit.tex \ + ../slides/3/ideal.tex \ + ../slides/3/nichthauptideal.tex \ + ../slides/3/nichthauptideal2.tex \ + ../slides/3/idealverband.tex \ + ../slides/3/maximalideal.tex \ + ../slides/3/quotientenring.tex \ + ../slides/3/faktorisierung.tex \ + ../slides/3/faktorzerlegung.tex \ + ../slides/3/einsetzen.tex \ + ../slides/3/maximalergrad.tex \ + ../slides/3/minimalbeispiel.tex \ + ../slides/3/fibonacci.tex \ + ../slides/3/minimalpolynom.tex \ + ../slides/3/drehmatrix.tex \ + ../slides/3/drehfaktorisierung.tex \ + ../slides/3/operatoren.tex \ + ../slides/3/adjunktion.tex \ + ../slides/3/adjalgebra.tex \ + ../slides/3/wurzel2.tex \ + ../slides/3/phi.tex \ + ../slides/3/multiplikation.tex \ + ../slides/3/chapter.tex + diff --git a/vorlesungen/slides/3/adjalgebra.tex b/vorlesungen/slides/3/adjalgebra.tex new file mode 100644 index 0000000..e65b621 --- /dev/null +++ b/vorlesungen/slides/3/adjalgebra.tex @@ -0,0 +1,43 @@ +% +% adjalgebra.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Adjunktion einer Nullstelle, abstrakt} +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +Sei $m(X)=m_0+m_1X+\dots + X^n\in \Bbbk[X]$ ein irreduzibles Polynom. + +\uncover<2->{% +\begin{block}{Existenz} +Es gibt ein ``Objekt'' $\alpha$ mit +\( +m(\alpha) = 0 +\) +\end{block}} + +\uncover<3->{% +\begin{block}{Körpererweiterung} +Der kleinste Körper, der $\Bbbk$ und $\alpha$ enthält ist +\[ +\Bbbk(\alpha) += +\left +\{ p(\alpha) +\;\left|\; +\begin{minipage}{8cm}\raggedright +$p\in\Bbbk[X]$ ein Polynom vom Grad +$\deg p<\deg m$ +\end{minipage} +\right. +\right\} +\] +\uncover<4->{Das Polynom $m$ definiert, wie mit $\alpha$ gerechnet werden +muss: +\[ +\alpha^n = -m_0-m_1\alpha-m_2\alpha^2 - \dots - m_{n-1}\alpha^{n-1} +\]} +\end{block}} + +\end{frame} diff --git a/vorlesungen/slides/3/adjunktion.tex b/vorlesungen/slides/3/adjunktion.tex new file mode 100644 index 0000000..a974a76 --- /dev/null +++ b/vorlesungen/slides/3/adjunktion.tex @@ -0,0 +1,35 @@ +% +% adjunktion.tex +% +% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\begin{frame}[t] +\frametitle{Adjunktion einer Nullstelle von $m(X)$} +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +Sei $m(X)=m_0+m_1X+\dots + X^n\in \Bbbk[X]$ ein irreduzibles Polynom. +\uncover<2->{% +\[ +X^n = -m_{n-1}X^{n-1} - \dots - m_1X - m_0 +\] +}% +\uncover<3->{% +Nullstelle $W$ als Operator betrachten: +\[ +W = \begin{pmatrix} + 0& 0& 0&\dots & 0& -m_0\\ + 1& 0& 0&\dots & 0& -m_1\\ + 0& 1& 0&\dots & 0& -m_2\\ + 0& 0& 1&\dots & 0& -m_3\\ +\vdots&\vdots&\vdots&\ddots&\vdots& \vdots\\ + 0& 0& 0&\dots & 1&-m_{n-1} +\end{pmatrix} +\]} +\uncover<4->{% +Man kann nachrechnen, dass immer $m(W)=0$. +} +\medskip + +\uncover<5->{$\Rightarrow \Bbbk(W) = \{p(W)\;|\;p\in\Bbbk[X], \deg p<\deg m\}$ +ist ein Körper, in dem $m(X)$ faktorisiert werden kann $m(X) = (X-W)q(X)$.} +\end{frame} diff --git a/vorlesungen/slides/3/chapter.tex b/vorlesungen/slides/3/chapter.tex new file mode 100644 index 0000000..3fbc3fd --- /dev/null +++ b/vorlesungen/slides/3/chapter.tex @@ -0,0 +1,33 @@ +% +% chapter.tex +% +% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi +% +\folie{3/motivation.tex} +\folie{3/inverse.tex} +\folie{3/polynome.tex} +\folie{3/division.tex} +\folie{3/division2.tex} +\folie{3/ringstruktur.tex} +\folie{3/teilbarkeit.tex} +\folie{3/ideal.tex} +\folie{3/nichthauptideal.tex} +\folie{3/nichthauptideal2.tex} +\folie{3/maximalideal.tex} +\folie{3/idealverband.tex} +\folie{3/quotientenring.tex} +\folie{3/faktorisierung.tex} +\folie{3/faktorzerlegung.tex} +\folie{3/einsetzen.tex} +\folie{3/maximalergrad.tex} +\folie{3/minimalbeispiel.tex} +\folie{3/fibonacci.tex} +\folie{3/minimalpolynom.tex} +\folie{3/drehmatrix.tex} +\folie{3/drehfaktorisierung.tex} +\folie{3/operatoren.tex} +\folie{3/adjunktion.tex} +\folie{3/adjalgebra.tex} +\folie{3/wurzel2.tex} +\folie{3/phi.tex} +\folie{3/multiplikation.tex} diff --git a/vorlesungen/slides/3/division.tex b/vorlesungen/slides/3/division.tex new file mode 100644 index 0000000..94df27b --- /dev/null +++ b/vorlesungen/slides/3/division.tex @@ -0,0 +1,32 @@ +% +% division.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Polynomdivision} +\begin{block}{Aufgabe} +Finde Quotient und Rest für +$a= X^4- X^3-7X^2+ X+6\in\mathbb{Z}[X]$ +und +$b= X^2+X+1\in\mathbb{Z}[X]$ +\end{block} +\uncover<2->{% +\begin{block}{Lösung} +\[ +\arraycolsep=1.4pt +\begin{array}{rcrcrcrcrcrcrcrcrcrcr} +\llap{$($}X^4&-& X^3&-&7X^2&+& X&+&6\rlap{$)$}&\;\mathstrut:\mathstrut&(X^2&+&X&+&1)&=&\uncover<3->{X^2}&\uncover<7->{-&2X}&\uncover<11->{-&6}=q\\ +\uncover<4->{\llap{$-($}X^4&+& X^3&+& X^2\rlap{$)$}}& & & & & & & & & & & & & & & & \\ + &\uncover<5->{-&2X^3&-&8X^2}&\uncover<6->{+& X}& & & & & & & & & & & & & & \\ + &\uncover<8->{\llap{$-($}-&2X^3&-&2X^2&-&2X\rlap{$)$}}& & & & & & & & & & & & & & \\ + & & &\uncover<9->{-&6X^2&+&3X}&\uncover<10->{+&6}& & & & & & & & & & & & \\ + & & &\uncover<12->{\llap{$-($}-&6X^2&-&6X&-&6\rlap{$)$}}& & & & & & & & & & & & \\ + & & & & & &\uncover<13->{9X&+&12\rlap{$\mathstrut=r$}}& & & & & & & & & & & & +\end{array} +\] +\uncover<14->{ +Funktioniert weil $b$ normiert ist! +} +\end{block}} +\end{frame} diff --git a/vorlesungen/slides/3/division2.tex b/vorlesungen/slides/3/division2.tex new file mode 100644 index 0000000..0602598 --- /dev/null +++ b/vorlesungen/slides/3/division2.tex @@ -0,0 +1,34 @@ +% +% division2.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Division in $\Bbbk[X]$} +\vspace{-5pt} +\begin{block}{Aufgabe} +Finde Quotienten und Rest der Polynome +$a(X) = X^4-X^3-7X^2+X+6$ +und +$b(X) = 2X^2+X+1$ +\end{block} +\uncover<2->{% +\begin{block}{Lösung} +\vspace{-15pt} +\[ +\arraycolsep=1.4pt +\renewcommand{\arraystretch}{1.2} +\begin{array}{rcrcrcrcrcrcrcrcrcrcr} +\llap{$($}X^4&-& X^3&-& 7X^2&+& X&+& 6\rlap{$)$}&\mathstrut\;:\mathstrut&(2X^2&+&X&+&1)&=&\uncover<3->{\frac12X^2}&\uncover<7->{-&\frac34X}&\uncover<11->{-\frac{27}{8}} = q\\ +\uncover<4->{\llap{$-($}X^4&+&\frac12X^3&+& \frac12X^2\rlap{$)$}}& & & & & & & & & & & & & & & \\ + &\uncover<5->{-&\frac32X^3&-&\frac{15}2X^2}&\uncover<6->{+& X}& & & & & & & & & & & & & \\ + &\uncover<8->{\llap{$-($}-&\frac32X^3&-&\frac{ 3}4X^2&-&\frac{ 3}4X\rlap{$)$}}& & & & & & & & & & & & & \\ + & & &\uncover<9->{-&\frac{27}4X^2&+&\frac{ 7}4X}&\uncover<10->{+& 6}& & & & & & & & & & & \\ + & & &\uncover<12->{\llap{$-($}-&\frac{27}4X^2&-&\frac{27}8X&-&\frac{27}{8}\rlap{$)$}}& & & & & & & & & & & \\ + & & & & & &\uncover<13->{\frac{41}8X&+&\frac{75}{8}\rlap{$\mathstrut=r$}}& & & & & & & & & & & \\ +\end{array} +\] +Funktioniert, weil man in $\Bbbk[X]$ immer normieren kann +\end{block}} + +\end{frame} diff --git a/vorlesungen/slides/3/drehfaktorisierung.tex b/vorlesungen/slides/3/drehfaktorisierung.tex new file mode 100644 index 0000000..64418d5 --- /dev/null +++ b/vorlesungen/slides/3/drehfaktorisierung.tex @@ -0,0 +1,75 @@ +% +% drehfaktorisierung.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{4pt} +\setlength{\belowdisplayskip}{4pt} +\frametitle{Faktorisierung von $X^2+X+1$} +\vspace{-3pt} +$X^2+X+1$ kann faktorisiert werden, wenn man $i\sqrt{3}$ +hinzufügt: +\uncover<2->{% +\[ +\biggl(X+\frac12+\frac{i\sqrt{3}}2\biggr) +\biggl(X+\frac12-\frac{i\sqrt{3}}2\biggr) += +X^2+X+\frac14 ++ +\frac34 +\uncover<3->{= +X^2+X+1} +\]} +\vspace{-10pt} +\uncover<4->{% +\begin{block}{Was ist $i\sqrt{3}$?} +Matrix mit Minimalpolynom $X^2+3$: +\[ +W=\begin{pmatrix}0&-3\\1&0\end{pmatrix} +\uncover<5->{% +\qquad\Rightarrow\qquad +W^2=\begin{pmatrix}3&0\\0&3\end{pmatrix} = -3I} +\uncover<6->{% +\qquad\Rightarrow\qquad +W^2+3I=0} +\] +\end{block}} +\vspace{-10pt} +\uncover<7->{% +\begin{block}{Faktorisierung von $X^2+X+1$} +\vspace{-10pt} +\begin{align*} +\uncover<8->{B_\pm +&= +-\frac12I\pm\frac12W} +& +&\uncover<10->{\Rightarrow +& +(X+B_+)(X+B_-)} +&\uncover<11->{= +(X+\frac12I+\frac12W) +(X+\frac12I-\frac12W)} +\\ +&\uncover<9->{= +\smash{ +{\textstyle\begin{pmatrix}-\frac12&-\frac32\\\frac12&-\frac12\end{pmatrix}} +}} +& +& +& +&\uncover<12->{= +X^2+X + \frac14I - \frac14W^2} +\\ +& +& +&%\Rightarrow +& +&\uncover<13->{= +X^2+X + \frac14I + \frac34I} +\uncover<14->{= +X^2+X+I} +\end{align*} +\end{block}} + +\end{frame} diff --git a/vorlesungen/slides/3/drehmatrix.tex b/vorlesungen/slides/3/drehmatrix.tex new file mode 100644 index 0000000..9e5eb65 --- /dev/null +++ b/vorlesungen/slides/3/drehmatrix.tex @@ -0,0 +1,66 @@ +% +% drehmatrix.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Analyse einer Drehung um $120^\circ$} +$D$ eine Drehung des $\mathbb{R}^3$ um $120^\circ$ +\begin{enumerate} +\item<2-> +Drehwinkel = $120^\circ\quad\Rightarrow\quad D^3 = I$ +\uncover<3->{ +$\quad\Rightarrow\quad \chi_D(X)=X^3-1$ +} +\item<4-> +$m_D(X)=X^3-1$ +\item<5-> +$m_D$ ist nicht irreduzibel, weil $m_D(1)=0$: +$ +m_D(X) = (X-1)(X^2+X+1) +$ +\item<6-> +Welche Matrix hat $X^2+X+1$ als Minimalpolynom? +\uncover<7->{% +\[ +\arraycolsep=1.4pt +W += +\biggl(\begin{array}{cc} +-\frac12 & -\frac{\sqrt{3}}2 \\ + \frac{\sqrt{3}}2 & -\frac12 +\end{array}\biggr) +\quad\Rightarrow\quad +W^2+W+I += +\biggl(\begin{array}{cc} +-\frac12 & -\frac{\sqrt{3}}2 \\ + \frac{\sqrt{3}}2 & -\frac12 +\end{array}\biggr) ++ +\biggl(\begin{array}{cc} +-\frac12 & \frac{\sqrt{3}}2 \\ + -\frac{\sqrt{3}}2 & -\frac12 +\end{array}\biggr) ++ +\biggl(\begin{array}{cc} +1&0\\0&1 +\end{array}\biggr) +=0 +\]} +\item<8-> In einer geeigneten Basis hat $D$ die Form +\[ +D=\begin{pmatrix} +1&0&0\\ +0&-\frac12 & -\frac{\sqrt{3}}2 \\ +0&\frac{\sqrt{3}}2 & -\frac12 +\end{pmatrix} +\uncover<9->{= +\begin{pmatrix} +1&0&0\\ +0&\cos 120^\circ & -\sin 120^\circ\\ +0&\sin 120^\circ & \cos 120^\circ +\end{pmatrix}} +\] +\end{enumerate} +\end{frame} diff --git a/vorlesungen/slides/3/einsetzen.tex b/vorlesungen/slides/3/einsetzen.tex new file mode 100644 index 0000000..7f54abb --- /dev/null +++ b/vorlesungen/slides/3/einsetzen.tex @@ -0,0 +1,54 @@ +% +% einsetzen.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Matrix in ein Polynom einsetzen} +\vspace{-10pt} +\[ +\begin{array}{rcrcrcrcrcrcr} +p(X)&=&a_nX^n&+&a_{n-1}X^{n-1}&+&\dots&+&a_2X^2&+&a_1X&+&a_0\phantom{I}\\ +\uncover<2->{\bigg\downarrow\hspace*{4pt}} & & +\uncover<3->{\bigg\downarrow\hspace*{4pt}} & & +\uncover<4->{\bigg\downarrow\hspace*{10pt}} & & & & +\uncover<5->{\bigg\downarrow\hspace*{4pt}} & & +\uncover<6->{\bigg\downarrow\hspace*{2pt}} & & +\uncover<7->{\bigg\downarrow\hspace*{0pt}} \\ +\uncover<2->{p(A)}&\uncover<3->{=&a_nA^n}&\uncover<4->{+&a_{n-1}A^{n-1}}&\uncover<5->{+&\dots&+&a_2A^2}&\uncover<6->{+&a_1A}&\uncover<7->{+&a_0 I} +\end{array} +\] +\vspace{-10pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\uncover<8->{% +\begin{block}{Nilpotente Matrizen} +$p(X) = (X-a)^n$ +\[ +\uncover<9->{p(A) = 0} +\uncover<10->{ +\quad\Rightarrow\quad +\text{$A-aI$ ist nilpotent}} +\] +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<11->{% +\begin{block}{Eigenwerte} +$p(X) = (X-\lambda_1)(X-\lambda_2)$,\\ +$A$ eine $2\times 2$-Matrix +\[ +\uncover<12->{p(A)=0} +\uncover<13->{\quad\Rightarrow\quad +\left\{ +\begin{aligned} +&\text{$A-\lambda_1I$ ist singulär}\\ +&\text{$A-\lambda_2I$ ist singulär} +\end{aligned} +\right.} +\] +\end{block}} +\end{column} +\end{columns} + +\end{frame} diff --git a/vorlesungen/slides/3/faktorisierung.tex b/vorlesungen/slides/3/faktorisierung.tex new file mode 100644 index 0000000..b4ea1d5 --- /dev/null +++ b/vorlesungen/slides/3/faktorisierung.tex @@ -0,0 +1,47 @@ +% +% faktorisierung.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Faktorisierung} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Primzahlen\strut} +Eine Zahl $p\in\mathbb{Z}$, $p>1$ heisst Primzahl, wenn sie nicht als Produkt +$p=ab$ mit $a,b\in\mathbb{Z},a>1, b>1$ geschrieben werden kann. +\begin{align*} +\uncover<2->{p&=7} +\\ +\uncover<3->{2021 &= 43 \cdot 47} +\\ +\uncover<4->{2048 &= 2^{11}} +\\ +\uncover<5->{4095667&=2021\cdot 2027} +\\ +\uncover<6->{p&=43, 47, 1291, 2017, 2027} +\end{align*} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<7->{% +\begin{block}{Irreduzible Polynome in $\mathbb{Q}[X]$} +Ein Polynome $p\in\mathbb{Q}[X]$, $\deg p>0$ wenn es nicht als Produkt +$p=ab$ mit $a,b\in\mathbb{Q}[X]$, $\deg a>0$, $\deg b>0$ geschrieben +werden kann. +\begin{align*} +\uncover<8->{p&=X-9} +\\ +\uncover<9->{X^2-1&= (X+1)(X-1)} +\\ +\uncover<10->{X^2-2&\text{\; irreduzibel}} +\\ +\uncover<11->{X^2-2&=(X-\sqrt{2})(X+\sqrt{2})} +\end{align*} +\uncover<12->{% +aber: $X\pm\sqrt{2}\not\in\mathbb{Q}[X]$ +} +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/3/faktorzerlegung.tex b/vorlesungen/slides/3/faktorzerlegung.tex new file mode 100644 index 0000000..eb44cf3 --- /dev/null +++ b/vorlesungen/slides/3/faktorzerlegung.tex @@ -0,0 +1,62 @@ +% +% faktorzerlegung.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Faktorzerlegung} +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{in $\mathbb{Z}$} +Jede Zahl kann eindeutig in Primfaktoren zerlegt werden: +\[ +z = p_1^{n_1}\cdot p_2^{n_2} \cdot\dots\cdot p_k^{n_k} +\] +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<2->{% +\begin{block}{in $\mathbb{Q}[X]$} +Jedes Polynom $p\in\mathbb{Q}[X]$ +kann eindeutig faktorisiert werden in irreduzible, normierte Polynome +\[ +p += +a_n +p_1^{n_1} +\cdot +p_2^{n_2} +\cdot +\dots +\cdot +p_k^{n_k} +\] +\end{block}} +\end{column} +\end{columns} +\uncover<3->{% +\begin{block}{Polynomfaktorisierung hängt vom Koeffizientenring ab} +Ist $X^2-2$ irreduzibel? +\vspace{-5pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\uncover<4->{% +\begin{block}{in $\mathbb{Q}[X]$} +\[ +X^2-2\quad\text{ist irreduzibel} +\] +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<5->{% +\begin{block}{in $\mathbb{R}[X]$} +\[ +X^2-2 = (X-\sqrt{2})(X+\sqrt{2}) +\] +\end{block}} +\end{column} +\end{columns} +\end{block}} +\end{frame} diff --git a/vorlesungen/slides/3/fibonacci.tex b/vorlesungen/slides/3/fibonacci.tex new file mode 100644 index 0000000..3d01020 --- /dev/null +++ b/vorlesungen/slides/3/fibonacci.tex @@ -0,0 +1,71 @@ +% +% fibonacci.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% + +\begin{frame}[t] +\frametitle{Fibonacci} +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\begin{block}{Fibonacci-Rekursion} +$x_i$ Fibonacci-Zahlen\uncover<2->{, d.~h.~$x_{n+1\mathstrut}=x_{n\mathstrut}+x_{n-1\mathstrut}$} +\[ +\uncover<3->{ +v_n += +\begin{pmatrix} +x_{n+1}\\ +x_n +\end{pmatrix}} +\uncover<4->{ +\quad\Rightarrow\quad +v_n = +\underbrace{ +\begin{pmatrix} +1&1\\ +1&0 +\end{pmatrix} +}_{\displaystyle=\Phi} +v_{n-1}} +\uncover<5->{ +\quad\Rightarrow\quad +v_n += +\Phi^n +v_0}\uncover<6->{, +\; +v_0 = \begin{pmatrix} 1\\0\end{pmatrix}} +\] +\end{block} +\vspace{-5pt} +\uncover<7->{% +\begin{block}{Rekursionsformel für $\Phi$} +\vspace{-12pt} +\begin{align*} +v_{n}&=v_{n-1}+v_{n-2} +&&\uncover<8->{\Rightarrow& +\Phi^n v_0 &= \Phi^{n-1} v_0 + \Phi^{n-2}v_0} +&&\uncover<9->{\Rightarrow& +\Phi^{n-2}(\Phi^2-\Phi-I)v_0&=0} +\\ +\end{align*} +\vspace{-22pt}% + +\uncover<10->{$\Phi$ ist $\chi_\Phi(X)=m_\Phi(X) = X^2-X-1$, irreduzibel} +\end{block}} + +\uncover<11->{% +\begin{block}{Faktorisierung} +\vspace{-12pt} +\[ +(X-\Phi)(X-(I-\Phi)) +\uncover<12->{= +X^2-X +\Phi(I-\Phi)} +\uncover<13->{= +X^2-X -(\underbrace{\Phi^2-\Phi}_{\displaystyle=I}) +} +\] +\end{block}} + +\end{frame} diff --git a/vorlesungen/slides/3/ideal.tex b/vorlesungen/slides/3/ideal.tex new file mode 100644 index 0000000..f7f432e --- /dev/null +++ b/vorlesungen/slides/3/ideal.tex @@ -0,0 +1,63 @@ +% +% ideal.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Ideal} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Voraussetzungen} +$R$ ein Ring, $r\in R$ +\end{block} +\uncover<2->{% +\begin{block}{Vielfache\uncover<4->{ = Hauptideal}} +Die Menge aller Elemente, die durch $r$ teilbar sind\uncover<3->{: +\[ +(r)=rR +\]} +\uncover<4->{heisst {\em Hauptideal}} +\end{block}} +\uncover<5->{% +\begin{block}{Ideal} +$I\subset R$ mit +\(RI\subset I\), \(I+I\subset I\) +\end{block}} +\uncover<6->{% +\begin{block}{Hauptidealring} +Jedes Ideal von $R$ ist ein Hauptideal +\\ +\uncover<7->{{\usebeamercolor[fg]{title}Beispiele:} +$\mathbb{Z}$, +$\Bbbk[X]$} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<8->{% +\begin{block}{Grösster gemeinsamer Teiler} +$a,b\in R$ +\begin{align*} +\uncover<9->{(a) + (b) +&= aR + bR} +\intertext{\uncover<10->{ist eine Ideal }\uncover<11->{$\Rightarrow$ ein Hauptideal}} +&\uncover<12->{= cR}\uncover<13->{ = \operatorname{ggT}(a,b)R} +\end{align*} +\uncover<14->{Existenz des $\operatorname{ggT}(a,b)$ ist eine +gemeinsame Eigenschaft} +\end{block}} +\uncover<15->{% +\begin{block}{Allgemein} +\begin{itemize} +\item<16-> +Alle euklidischen Ringe sind Hauptidealringe +\item<17-> +Alle solchen Ringe verwenden den gleichen Algorithmus +für $\operatorname{ggT}(a,b)$ +\end{itemize} +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/3/idealverband.tex b/vorlesungen/slides/3/idealverband.tex new file mode 100644 index 0000000..3434868 --- /dev/null +++ b/vorlesungen/slides/3/idealverband.tex @@ -0,0 +1,78 @@ +% +% idealverband.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Idealverband} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\node at (0,0) {$\mathbb{Z}$}; + +\uncover<2->{ +\node at (-6,-2) {$2\mathbb{Z}$}; +\node at (-2,-2) {$3\mathbb{Z}$}; +\node at (2,-2) {$5\mathbb{Z}$}; +\node at (6,-2) {$7\mathbb{Z}$}; +\node at (7,-2) {$\dots$}; +} + +\uncover<3->{ +\node at (-4,-4) {$6\mathbb{Z}$}; +\node at (-2,-4) {$10\mathbb{Z}$}; +\node at (0,-4) {$15\mathbb{Z}$}; +\node at (2,-4) {$21\mathbb{Z}$}; +\node at (4,-4) {$35\mathbb{Z}$}; +\node at (6,-4) {$\dots$}; +} + +\uncover<4->{ +\node at (-2,-6) {$30\mathbb{Z}$}; +\node at (0,-6) {$70\mathbb{Z}$}; +\node at (2,-6) {$105\mathbb{Z}$}; +} + +\uncover<5->{ + \node at (-5,-6) {$\dots$}; + \node at (5,-6) {$\dots$}; +} + +\uncover<2->{ +\draw[shorten >= 0.4cm, shorten <=0.4cm] (0,0) -- (-6,-2); +\draw[shorten >= 0.4cm, shorten <=0.4cm] (0,0) -- (-2,-2); +\draw[shorten >= 0.4cm, shorten <=0.4cm] (0,0) -- (2,-2); +\draw[shorten >= 0.4cm, shorten <=0.4cm] (0,0) -- (6,-2); +} + +\uncover<3->{ +\draw[shorten >= 0.4cm, shorten <=0.4cm] (-6,-2) -- (-4,-4); +\draw[shorten >= 0.4cm, shorten <=0.4cm] (-6,-2) -- (-2,-4); + +\draw[shorten >= 0.4cm, shorten <=0.4cm] (-2,-2) -- (-4,-4); +\draw[shorten >= 0.4cm, shorten <=0.4cm] (-2,-2) -- (0,-4); +\draw[shorten >= 0.4cm, shorten <=0.4cm] (-2,-2) -- (2,-4); + +\draw[shorten >= 0.4cm, shorten <=0.4cm] (2,-2) -- (-2,-4); +\draw[shorten >= 0.4cm, shorten <=0.4cm] (2,-2) -- (0,-4); +\draw[shorten >= 0.4cm, shorten <=0.4cm] (2,-2) -- (4,-4); + +\draw[shorten >= 0.4cm, shorten <=0.4cm] (6,-2) -- (2,-4); +\draw[shorten >= 0.4cm, shorten <=0.4cm] (6,-2) -- (4,-4); +} + +\uncover<4->{ +\draw[shorten >= 0.4cm, shorten <=0.4cm] (-2,-6) -- (-4,-4); +\draw[shorten >= 0.4cm, shorten <=0.4cm] (-2,-6) -- (-2,-4); +\draw[shorten >= 0.4cm, shorten <=0.4cm] (-2,-6) -- (0,-4); + +\draw[shorten >= 0.4cm, shorten <=0.4cm] (0,-6) -- (-2,-4); +\draw[shorten >= 0.4cm, shorten <=0.4cm] (0,-6) -- (4,-4); + +\draw[shorten >= 0.4cm, shorten <=0.4cm] (2,-6) -- (0,-4); +\draw[shorten >= 0.4cm, shorten <=0.4cm] (2,-6) -- (2,-4); +\draw[shorten >= 0.4cm, shorten <=0.4cm] (2,-6) -- (4,-4); +} + +\end{tikzpicture} +\end{center} +\end{frame} diff --git a/vorlesungen/slides/3/images/Makefile b/vorlesungen/slides/3/images/Makefile new file mode 100644 index 0000000..e338fcf --- /dev/null +++ b/vorlesungen/slides/3/images/Makefile @@ -0,0 +1,55 @@ +# +# Makefile -- build images +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +all: hauptideal.jpg nichthauptideal.jpg hauptideal2.jpg hauptidealX.jpg \ + hauptidealR.jpg hauptidealXR.jpg ring.jpg + +ring.png: ring.pov common.inc + povray +A0.1 +W1920 +H1080 -Oring.png ring.pov +ring.jpg: ring.png + convert ring.png -density 300 -units PixelsPerInch ring.jpg + +hauptideal.png: hauptideal.pov common.inc + povray +A0.1 +W1920 +H1080 -Ohauptideal.png hauptideal.pov +hauptideal.jpg: hauptideal.png + convert hauptideal.png -density 300 -units PixelsPerInch \ + hauptideal.jpg + +hauptidealR.png: hauptidealR.pov common.inc + povray +A0.1 +W1920 +H1080 -OhauptidealR.png hauptidealR.pov +hauptidealR.jpg: hauptidealR.png + convert hauptidealR.png -density 300 -units PixelsPerInch \ + hauptidealR.jpg + +hauptideal2.png: hauptideal2.pov common.inc + povray +A0.1 +W1920 +H1080 -Ohauptideal2.png hauptideal2.pov +hauptideal2.jpg: hauptideal2.png + convert hauptideal2.png -density 300 -units PixelsPerInch \ + hauptideal2.jpg + +hauptidealX.png: hauptidealX.pov common.inc + povray +A0.1 +W1920 +H1080 -OhauptidealX.png hauptidealX.pov +hauptidealX.jpg: hauptidealX.png + convert hauptidealX.png -density 300 -units PixelsPerInch \ + hauptidealX.jpg + +hauptidealXR.png: hauptidealXR.pov common.inc + povray +A0.1 +W1920 +H1080 -OhauptidealXR.png hauptidealXR.pov +hauptidealXR.jpg: hauptidealXR.png + convert hauptidealXR.png -density 300 -units PixelsPerInch \ + hauptidealXR.jpg + +nichthauptideal.png: nichthauptideal.pov common.inc + povray +A0.1 +W1920 +H1080 -Onichthauptideal.png nichthauptideal.pov +nichthauptideal.jpg: nichthauptideal.png + convert nichthauptideal.png -density 300 -units PixelsPerInch \ + nichthauptideal.jpg + +ideal: ideal.pov ideal.ini common.inc + rm -rf ideal + mkdir ideal + povray +A0.1 +W1920 +H1080 -Oideal/i.png ideal.ini + + diff --git a/vorlesungen/slides/3/images/common.inc b/vorlesungen/slides/3/images/common.inc new file mode 100644 index 0000000..36c4e6b --- /dev/null +++ b/vorlesungen/slides/3/images/common.inc @@ -0,0 +1,277 @@ +// +// common.inc +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#version 3.7; +#include "colors.inc" + +global_settings { + assumed_gamma 1 +} + +#declare imagescale = 0.40; +#declare O = <0, 0, 0>; +#declare at = 0.10; + +#declare Xunten = -10; +#declare Xoben = 10; +#declare Yunten = -8; +#declare Yoben = 8; +#declare Zunten = 0; +#declare Zoben = 20; + +#declare phi0 = 2 * pi * 290 / 360; + +camera { + location <60 * cos(2*pi*T+phi0), 15, 60 * sin(2*pi*T+phi0) + 10> + look_at <0, -2, 10> + right 16/9 * x * imagescale + up y * imagescale +} + +light_source { + <-14, 20, -50> color White + area_light <1,0,0> <0,0,1>, 10, 10 + adaptive 1 + jitter +} + +light_source { + <41, 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 + +arrow(< -12.0, 0.0, 0 >, < 12.0, 0.0, 0.0 >, at, Gray) +arrow(< 0.0, 0.0, -2.0>, < 0.0, 0.0, 22.0 >, at, Gray) +arrow(< 0.0, -10.0, 0 >, < 0.0, 10.0, 0.0 >, at, Gray) + +#macro kasten() + box { <-10.5,-8.5,-0.5>, <10.5,8.5,20.5> } +#end + +#declare gruen = rgb<0.2,0.4,0.2>; +#declare blau = rgb<0.0,0.4,0.8>; +#declare rot = rgb<1.0,0.4,0.0>; + +#declare r = 0.4; + +#macro Zring() + union { + #declare X = Xunten; + #while (X <= Xoben + 0.5) + #declare Y = Yunten; + #while (Y <= Yoben + 0.5) + #declare Z = Zunten; + #while (Z <= Zoben + 0.5) + sphere { <X, Y, Z>, r } + + #declare Z = Z + 1; + #end + #declare Y = Y + 1; + #end + #declare X = X + 1; + #end + pigment { + color rot + } + finish { + specular 0.9 + metallic + } + } +#end + +#macro Hauptideal() + union { + #declare A = Xunten; + #while (A <= Xoben + 0.5) + #declare B = Zunten; + #while (B <= Zoben + 0.5) + #declare Y = A + B; + #if ((Y >= Yunten - 0.5) & (Y <= Yoben + 0.5)) + sphere { <A, Y, B>, r } + #end + #declare B = B + 1; + #end + #declare A = A + 1; + #end + pigment { + color blau + } + finish { + specular 0.9 + metallic + } + } +#end + +#macro HauptidealR() + intersection { + kasten() + #declare n = vnormalize(< 1, -2, 1 >); + plane { n, 0.1 } + plane { -n, 0.1 } + pigment { + color blau + } + finish { + specular 0.9 + metallic + } + } +#end + +#macro Ideal2() + union { + #declare X = Xunten; + #while (X <= Xoben + 0.5) + #declare Y = Yunten; + #while (Y <= Yoben + 0.5) + #declare Z = Zunten; + #while (Z <= Zoben + 0.5) + sphere { <X, Y, Z>, r } + #declare Z = Z + 2; + #end + #declare Y = Y + 2; + #end + #declare X = X + 2; + #end + pigment { + color gruen + } + finish { + specular 0.9 + metallic + } + } +#end + +#macro IdealX() + union { + #declare Y = Yunten; + #while (Y <= Yoben + 0.5) + #declare Z = Zunten; + #while (Z <= Zoben + 0.5) + sphere { <0, Y, Z>, r } + #declare Z = Z + 1; + #end + #declare Y = Y + 1; + #end + pigment { + color gruen + } + finish { + specular 0.9 + metallic + } + } +#end + +#macro IdealXR() + intersection { + kasten() + plane { <1,0,0>, 0.1 } + plane { <-1,0,0>, 0.1 } + pigment { + color gruen + } + finish { + specular 0.9 + metallic + } + } +#end + +#macro Nichthauptideal() + union { + #declare X = Xunten/2; + #while (X <= Xoben/2 + 0.5) + #declare Y = Yunten; + #while (Y <= Yoben + 0.5) + #declare Z = 0; + #while (Z <= Zoben + 0.5) + sphere { <2*X,Y,Z>, r } + #declare Z = Z + 1; + #end + #declare Y = Y + 1; + #end + #declare X = X + 1; + #end + pigment { + color gruen + } + finish { + specular 0.9 + metallic + } + } +#end + +#macro NichthauptidealKomplement() + union { + #declare X = Xunten + 1; + #while (X <= Xoben + 0.5) + #declare Y = Yunten; + #while (Y <= Yoben + 0.5) + #declare Z = Zunten; + #while (Z <= Zoben + 0.5) + sphere { <X,Y,Z>, r } + #declare Z = Z + 1; + #end + #declare Y = Y + 1; + #end + #declare X = X + 2; + #end + pigment { + color rot + } + finish { + specular 0.9 + metallic + } + } +#end + + + + + + + diff --git a/vorlesungen/slides/3/images/hauptideal.jpg b/vorlesungen/slides/3/images/hauptideal.jpg Binary files differnew file mode 100644 index 0000000..769f53c --- /dev/null +++ b/vorlesungen/slides/3/images/hauptideal.jpg diff --git a/vorlesungen/slides/3/images/hauptideal.pov b/vorlesungen/slides/3/images/hauptideal.pov new file mode 100644 index 0000000..a934e57 --- /dev/null +++ b/vorlesungen/slides/3/images/hauptideal.pov @@ -0,0 +1,10 @@ +// +// hauptideal.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#declare T = 0; +#include "common.inc" + +Hauptideal() + diff --git a/vorlesungen/slides/3/images/hauptideal2.jpg b/vorlesungen/slides/3/images/hauptideal2.jpg Binary files differnew file mode 100644 index 0000000..51823f3 --- /dev/null +++ b/vorlesungen/slides/3/images/hauptideal2.jpg diff --git a/vorlesungen/slides/3/images/hauptideal2.pov b/vorlesungen/slides/3/images/hauptideal2.pov new file mode 100644 index 0000000..9da5a1a --- /dev/null +++ b/vorlesungen/slides/3/images/hauptideal2.pov @@ -0,0 +1,10 @@ +// +// hauptideal2.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#declare T = 0; +#include "common.inc" + +Ideal2() + diff --git a/vorlesungen/slides/3/images/hauptidealR.jpg b/vorlesungen/slides/3/images/hauptidealR.jpg Binary files differnew file mode 100644 index 0000000..fae5840 --- /dev/null +++ b/vorlesungen/slides/3/images/hauptidealR.jpg diff --git a/vorlesungen/slides/3/images/hauptidealR.pov b/vorlesungen/slides/3/images/hauptidealR.pov new file mode 100644 index 0000000..330e523 --- /dev/null +++ b/vorlesungen/slides/3/images/hauptidealR.pov @@ -0,0 +1,10 @@ +// +// hauptidealR.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#declare T = 0; +#include "common.inc" + +HauptidealR() + diff --git a/vorlesungen/slides/3/images/hauptidealX.jpg b/vorlesungen/slides/3/images/hauptidealX.jpg Binary files differnew file mode 100644 index 0000000..f9b4540 --- /dev/null +++ b/vorlesungen/slides/3/images/hauptidealX.jpg diff --git a/vorlesungen/slides/3/images/hauptidealX.pov b/vorlesungen/slides/3/images/hauptidealX.pov new file mode 100644 index 0000000..d0045f9 --- /dev/null +++ b/vorlesungen/slides/3/images/hauptidealX.pov @@ -0,0 +1,10 @@ +// +// hauptidealX.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#declare T = 0; +#include "common.inc" + +IdealX() + diff --git a/vorlesungen/slides/3/images/hauptidealXR.jpg b/vorlesungen/slides/3/images/hauptidealXR.jpg Binary files differnew file mode 100644 index 0000000..d8906c8 --- /dev/null +++ b/vorlesungen/slides/3/images/hauptidealXR.jpg diff --git a/vorlesungen/slides/3/images/hauptidealXR.pov b/vorlesungen/slides/3/images/hauptidealXR.pov new file mode 100644 index 0000000..5daa3e6 --- /dev/null +++ b/vorlesungen/slides/3/images/hauptidealXR.pov @@ -0,0 +1,10 @@ +// +// hauptidealXR.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#declare T = 0; +#include "common.inc" + +IdealXR() + diff --git a/vorlesungen/slides/3/images/ideal.ini b/vorlesungen/slides/3/images/ideal.ini new file mode 100644 index 0000000..66aa191 --- /dev/null +++ b/vorlesungen/slides/3/images/ideal.ini @@ -0,0 +1,7 @@ +Input_File_Name=ideal.pov +Initial_Frame=0 +Final_Frame=2500 +Initial_Clock=0 +Final_Clock=5 +Cyclic_Animation=off +Pause_when_Done=off diff --git a/vorlesungen/slides/3/images/ideal.pov b/vorlesungen/slides/3/images/ideal.pov new file mode 100644 index 0000000..88afaf7 --- /dev/null +++ b/vorlesungen/slides/3/images/ideal.pov @@ -0,0 +1,26 @@ +// +// ideal.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#declare T = clock; +#include "common.inc" + +#if (T < 1) +Zring() +#else + #if (T < 2) + Hauptideal() + #else + #if (T < 3) + Ideal2() + #else + #if (T < 4) + IdealX() + #else + Nichthauptideal() + NichthauptidealKomplement() + #end + #end + #end +#end diff --git a/vorlesungen/slides/3/images/nichthauptideal.jpg b/vorlesungen/slides/3/images/nichthauptideal.jpg Binary files differnew file mode 100644 index 0000000..55858d0 --- /dev/null +++ b/vorlesungen/slides/3/images/nichthauptideal.jpg diff --git a/vorlesungen/slides/3/images/nichthauptideal.pov b/vorlesungen/slides/3/images/nichthauptideal.pov new file mode 100644 index 0000000..72a6330 --- /dev/null +++ b/vorlesungen/slides/3/images/nichthauptideal.pov @@ -0,0 +1,10 @@ +// +// hauptideal.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#declare T = 0; +#include "common.inc" + +Nichthauptideal() +NichthauptidealKomplement() diff --git a/vorlesungen/slides/3/images/ring.jpg b/vorlesungen/slides/3/images/ring.jpg Binary files differnew file mode 100644 index 0000000..27721b1 --- /dev/null +++ b/vorlesungen/slides/3/images/ring.jpg diff --git a/vorlesungen/slides/3/images/ring.pov b/vorlesungen/slides/3/images/ring.pov new file mode 100644 index 0000000..f854335 --- /dev/null +++ b/vorlesungen/slides/3/images/ring.pov @@ -0,0 +1,10 @@ +// +// ring.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#declare T = 0; +#include "common.inc" + +Zring() + diff --git a/vorlesungen/slides/3/inverse.tex b/vorlesungen/slides/3/inverse.tex new file mode 100644 index 0000000..4ad22d2 --- /dev/null +++ b/vorlesungen/slides/3/inverse.tex @@ -0,0 +1,89 @@ +% +% inverse.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Inverse Matrix} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.24\textwidth} +\begin{block}{Imaginäre Einheit} +\vspace{-15pt} +\begin{align*} +J &= \begin{pmatrix} 0&-1\\1&0\end{pmatrix} +\\ +0&= +J^2 + I +\\ +0&= +J+J^{-1} +\\ +J^{-1}&=-J +\end{align*} +\end{block} +\end{column} +\begin{column}{0.25\textwidth} +\uncover<2->{% +\begin{block}{Wurzel $\sqrt{2}$} +\vspace{-15pt} +\begin{align*} +W&=\begin{pmatrix}0&2\\1&0\end{pmatrix} +\\ +0 &= X^2-2 +\\ +0 &= W-2W^{-1} +\\ +W^{-1}&=\frac12 W +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.41\textwidth} +\uncover<3->{% +\begin{block}{Drehmatrix} +\vspace{-15pt} +\begin{align*} +D&=\begin{pmatrix} +\cos \frac{\pi}{1291} & -\sin\frac{\pi}{1291}\\ +\sin \frac{\pi}{1291} & \cos\frac{\pi}{1291} +\end{pmatrix} +\\ +0 &= \ifthenelse{\boolean{presentation}}{\only<-3>{D^{1291}+I\phantom{+\frac{\mathstrut}{\mathstrut}}}}{} +\only<4->{D^2-2D\cos\frac{\pi\mathstrut}{1291\mathstrut}+I} +\\ +0 &= \ifthenelse{\boolean{presentation}}{\only<-3>{D^{1290}+D^{-1}\phantom{+\frac{\mathstrut}{\mathstrut}}}}{} +\only<4->{D-2\cos\frac{\pi\mathstrut}{1291\mathstrut}+D^{-1}} +\\ +D^{-1} +&= \only<-3>{-D^{1290}\phantom{+\frac{\mathstrut}{\mathstrut}}}% +\only<4->{-D+2I\cos\frac{\pi\mathstrut}{1291\mathstrut}} +\end{align*} +\end{block}} +\end{column} +\end{columns} +\vspace{-25pt} +\uncover<5->{ +\begin{block}{3D-Beispiel} +$p(x) = -x^3-5x^2+5x+1$ +\[ +A= +\begin{pmatrix*}[r] +-5&-1&1\\ +-5&-2&3\\ +-1&-1&2 +\end{pmatrix*} +\quad\Rightarrow\quad +A^{-1} += +A^2+5A-5I += +\begin{pmatrix*}[r] +-1& 1&-1\\ + 7&-9&10\\ + 3&-4& 5 +\end{pmatrix*} +\] +\end{block}} +\vspace{-10pt} + +\end{frame} diff --git a/vorlesungen/slides/3/maximalergrad.tex b/vorlesungen/slides/3/maximalergrad.tex new file mode 100644 index 0000000..d33ddc0 --- /dev/null +++ b/vorlesungen/slides/3/maximalergrad.tex @@ -0,0 +1,72 @@ +% +% maximalergrad.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Jede Matrix hat eine Polynomrelation} +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\vspace{-5pt} +\begin{block}{Dimension des Matrizenrings} +Der Ring $M_{n}(\Bbbk)$ ist ein $n^2$-dimensionaler Vektorraum mit +Basis +{\tiny +\begin{align*} +&\uncover<2->{\begin{pmatrix} +1&0&\dots&0\\ +0&0&\dots&0\\ +\vdots&\vdots&\ddots&\vdots\\ +\end{pmatrix}} +& +&\uncover<3->{\begin{pmatrix} +0&1&\dots&0\\ +0&0&\dots&0\\ +\vdots&\vdots&\ddots&\vdots\\ +\end{pmatrix}} +& +&\uncover<4->{\dots} +& +&\uncover<5->{\begin{pmatrix} +0&0&\dots&1\\ +0&0&\dots&0\\ +\vdots&\vdots&\ddots&\vdots\\ +\end{pmatrix}} +\\ +&\uncover<6->{\begin{pmatrix} +0&0&\dots&0\\ +1&0&\dots&0\\ +\vdots&\vdots&\ddots&\vdots\\ +\end{pmatrix}} +& +&\uncover<7->{\begin{pmatrix} +0&0&\dots&0\\ +0&1&\dots&0\\ +\vdots&\vdots&\ddots&\vdots\\ +\end{pmatrix}} +& +&\uncover<8->{\dots} +& +&\uncover<9->{\begin{pmatrix} +0&0&\dots&0\\ +0&0&\dots&1\\ +\vdots&\vdots&\ddots&\vdots\\ +\end{pmatrix}} +\end{align*}} +\end{block} +\vspace{-10pt} +\uncover<10->{% +\begin{block}{Potenzen von $A$} +Die $n^2+1$ Matrizen $I,A,A^2,\dots,A^{n^2-1},A^{n^2}$ müssen linear abhängig +sein: +\[ +\uncover<11->{ +a_0I+a_1A+a_2A^2+\dots+a_{n^2-1}A^{n^2-1}+a_{n^2}A^{n^2} = 0 +} +\] +\uncover<12->{d.~h.~$p(X) = a_0+a_1X+a_2X^2+\dots +a_{n^2-1}X^{n^2-1}+a_{n^2}A^{n^2}\in\Bbbk[X]$ ist ein Polynom mit $p(A)=0$.} +\end{block}} +\uncover<13->{% +$\Rightarrow$ $A$ über die Eigenschaften (Faktorisierung) von $p$ studieren +} +\end{frame} diff --git a/vorlesungen/slides/3/maximalideal.tex b/vorlesungen/slides/3/maximalideal.tex new file mode 100644 index 0000000..21a945a --- /dev/null +++ b/vorlesungen/slides/3/maximalideal.tex @@ -0,0 +1,64 @@ +% +% maximalideal.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Maximale Ideale} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Teilbarkeit} +$a|b$ +\uncover<2->{$\Rightarrow$ +$b\in aR$} +\uncover<3->{$\Rightarrow$ +$bR\subset aR$} +\end{block} +\uncover<4->{% +\begin{block}{Nicht mehr teilbar} +$a\in R$ nicht faktorisierbar +\\ +\uncover<5->{$\Rightarrow$ +\\ +es gibt kein Ideal zwischen $aR$ und $R$} +\\ +\uncover<6->{$\Leftrightarrow$ +\\ +$J$ ein Ideal +$aR \subset J \subset R$, dann ist +$J=aR$ oder $J=R$} +\end{block}} +\uncover<7->{ +\begin{block}{maximales Ideal} +$I\subset R$ heisst maximal, wenn für jedes Ideal $J$ +mit $I\subset J\subset R$ gilt +$I=J$ oder $J=R$ +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<8->{ +\begin{block}{Beispiele} +\begin{itemize} +\item Primzahlen $p$ erzeugen maximale Ideale in $\mathbb{Z}$ +\item<9-> Irreduzible Polynome erzeugen maximale Ideale in $\Bbbk[X]$ +\end{itemize} +\end{block}} +\uncover<10->{% +\begin{block}{Körper} +$M\subset R$ ein maximales Ideal, dann ist +$R/M$ ein Körper +\end{block}} +\uncover<11->{% +\begin{block}{Beispiel} +\begin{itemize} +\item +$\mathbb{F}_p = \mathbb{Z}/p\mathbb{Z}$ +\item<12-> +$m$ ein irreduzibles Polynom: +$\Bbbk[X]/ (m)$ ist ein Körper +\end{itemize} +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/3/minimalbeispiel.tex b/vorlesungen/slides/3/minimalbeispiel.tex new file mode 100644 index 0000000..f94cf8d --- /dev/null +++ b/vorlesungen/slides/3/minimalbeispiel.tex @@ -0,0 +1,36 @@ +% +% minimalbeispiel.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Beispiel für $p(A)=0$} +\begin{block}{Potenzen einer $2\times 2$-Matrix $A$} +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\vspace{-10pt} +\[ +I ={\tiny\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}},\quad +A ={\tiny\begin{pmatrix} 3 & 2 \\ -1 & -2 \end{pmatrix}},\quad +\uncover<2->{A^2={\tiny\begin{pmatrix} 7 & 2 \\ -1 & 2 \end{pmatrix}}} +\uncover<3->{,\quad A^3={\tiny\begin{pmatrix} 19 & 10 \\ -5 & -6 \end{pmatrix}}} +\uncover<4->{,\quad A^4={\tiny\begin{pmatrix} 47 & 18 \\ -9 & 2 \end{pmatrix}}} +\] +\end{block} +\vspace{-5pt} +\uncover<5->{% +\begin{block}{linear abhängig} +Bereits die ersten $3$ sind linear abhängig: +\[ +-4I - A + A^2 += +-4\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} +-\begin{pmatrix} 3 & 2 \\ -1 & -2 \end{pmatrix} ++\begin{pmatrix} 7 & 2 \\ -1 & 2 \end{pmatrix} += +\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} +\] +\uncover<6->{$p(X) = X^2 - X - 4 \in \mathbb{Q}[X]$ hat die Eigenschaft +$p(A)=0$} +\end{block}} +\end{frame} diff --git a/vorlesungen/slides/3/minimalpolynom.tex b/vorlesungen/slides/3/minimalpolynom.tex new file mode 100644 index 0000000..2b36a65 --- /dev/null +++ b/vorlesungen/slides/3/minimalpolynom.tex @@ -0,0 +1,30 @@ +% +% minimalpolynom.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Minimalpolynom} +\begin{block}{Definition} +Zu jeder $n\times n$-Matrix $A$ +gibt es ein Polynom $m_A(X)\in\Bbbk[X]$ minimalen Grades $\deg m_A\le n^2$ +derart, dass $m_A(A)=0$. +\end{block} +\uncover<2->{% +\begin{block}{Strategie} +Das Minimalpolynom ist eine ``Invariante'' der Matrix $A$ +\end{block}} +\uncover<3->{% +\begin{block}{Satz von Cayley-Hamilton} +Für jede $n\times n$-Matrix $A\in M_n(\Bbbk)$ gilt $\chi_A(A)=0$ +\uncover<4->{% +\[ +\Downarrow +\] +Das Minimalpolynom $m_A\in \Bbbk[X]$ ist ein Teiler +des charakteristischen Polynoms $\chi_A\in \Bbbk[X]$} +\\ +\uncover<5->{$\Rightarrow $ +Faktorzerlegung on $\chi_A(X)$ ermitteln!} +\end{block}} +\end{frame} diff --git a/vorlesungen/slides/3/motivation.tex b/vorlesungen/slides/3/motivation.tex new file mode 100644 index 0000000..048e6a2 --- /dev/null +++ b/vorlesungen/slides/3/motivation.tex @@ -0,0 +1,108 @@ +% +% motivation.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Motivation} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.24\textwidth} +\begin{block}{Imaginäre Einheit} +\vspace{-15pt} +\begin{align*} +J &= \begin{pmatrix} 0&-1\\1&0\end{pmatrix} +\\ +p(X) &= X^2 + 1 +\\ +p(J) &= J^2 + I = 0 +\end{align*} +\end{block} +\end{column} +\begin{column}{0.25\textwidth} +\uncover<2->{% +\begin{block}{Wurzel $\sqrt{2}$} +\vspace{-15pt} +\begin{align*} +W&=\begin{pmatrix}0&2\\1&0\end{pmatrix} +\\ +p(X) &= X^2-2 +\\ +p(W) &= W^2-2I=0 +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.41\textwidth} +\uncover<3->{% +\begin{block}{Drehmatrix} +\vspace{-15pt} +\begin{align*} +D&=\begin{pmatrix} +\cos \frac{\pi}{1291} & -\sin\frac{\pi}{1291}\\ +\sin \frac{\pi}{1291} & \cos\frac{\pi}{1291} +\end{pmatrix} +\\ +p(X)&= +\ifthenelse{\boolean{presentation}}{\only<-3>{X^{1291}+1\phantom{+\frac{\mathstrut}{\mathstrut}}}}{} +\only<4->{X^2-2X\cos\frac{\pi\mathstrut}{1291\mathstrut}+I} +\\ +p(D) &= \ifthenelse{\boolean{presentation}}{\only<-3>{D^{1291}+I\phantom{+\frac{\mathstrut}{\mathstrut}}}}{} +\only<4->{D^2-2D\cos\frac{\pi\mathstrut}{1291\mathstrut}+I} +\end{align*} +\end{block}} +\end{column} +\end{columns} +\vspace{-20pt} +\uncover<5->{ +\begin{block}{3D-Beispiel} +$p(x) = -x^3-5x^2+5x+1$ +\[ +\ifthenelse{\boolean{presentation}}{ +\only<5-8>{ +A= +\begin{pmatrix*}[r] +-5&-1&1\\ +-5&-2&3\\ +-1&-1&2 +\end{pmatrix*}} +\only<6-8>{ +\quad\Rightarrow\quad}}{} +\uncover<6->{ +- +\only<-9>{A^3}\only<10->{ +\begin{pmatrix*}[r] +-169&-35&35\\ +-185&-39&40\\ + -45&-10&11 +\end{pmatrix*}} +-5 +\only<-8>{A^2}\only<9->{ +\begin{pmatrix*}[r] +29&6&-6\\ +32&6&-5\\ + 8&1& 0 +\end{pmatrix*}} ++5 +\only<-7>{A}\only<8->{ +\begin{pmatrix*}[r] +-5&-1&1\\ +-5&-2&3\\ +-1&-1&2 +\end{pmatrix*}} ++ +\only<-6>{I}\only<7->{ +\begin{pmatrix*}[r] +1&0&0\\ +0&1&0\\ +0&0&1 +\end{pmatrix*}} +} +\uncover<11->{=0} +\] +\end{block}} +\vspace{-10pt} +\uncover<12->{% +{\usebeamercolor[fg]{title}$\Rightarrow$ +Rechenregeln von Matrizen können durch Polynome ausgedrückt werden} +} +\end{frame} diff --git a/vorlesungen/slides/3/multiplikation.tex b/vorlesungen/slides/3/multiplikation.tex new file mode 100644 index 0000000..13f4e03 --- /dev/null +++ b/vorlesungen/slides/3/multiplikation.tex @@ -0,0 +1,180 @@ +% +% multiplikation.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\def\N{21} +\begin{frame}[t,fragile] +\frametitle{Multiplikation mit $\alpha$ in $\mathbb{Z}(\alpha)$} +\vspace{-18pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=0.92] + +\node[color=red] at (-3.2,3.2) [above right] {$\mathbb{Z}(\sqrt{2})$}; +\node[color=blue] at (3.5,3.2) [above left] {$\sqrt{2}\mathbb{Z}(\sqrt{2})$}; + +\pgfmathparse{sqrt(2)} +\xdef\a{\pgfmathresult} +\pgfmathparse{-int(3.2/\a)} +\xdef\ymin{\pgfmathresult} +\pgfmathparse{int(3.2/\a)} +\xdef\ymax{\pgfmathresult} + +\draw[->] (-3.2,0) -- (3.5,0) coordinate[label={$\mathbb{Z}$}]; +\draw[->] (0,-3.2) -- (0,3.6) coordinate[label={right:$\mathbb{Z}\sqrt{2}$}]; + +\def\punkt#1#2#3{ + ({(1-(#3))*(#1)+2*(#3)*(#2)},{((1-(#3))*(#2)+(#3)*(#1))*\a}) +} + +\foreach \x in {-3,...,3}{ + \draw[color=red,line width=0.5pt] + \punkt{\x}{\ymin}{0} -- \punkt{\x}{\ymax}{0}; + \foreach \y in {\ymin,...,\ymax}{ + \fill[color=red] \punkt{\x}{\y}{0} circle[radius=0.08]; + } +} +\foreach \y in {\ymin,...,\ymax}{ + \draw[color=red,line width=0.5pt] + \punkt{-3}{\y}{0} -- \punkt{3}{\y}{0}; +} + + +\def\bildnetz#1{ + \pgfmathparse{(#1-1)/(\N-1)} + \xdef\t{\pgfmathresult} + \only<#1>{ + \uncover<2->{ + \draw[->,color=blue,line width=1.4pt] + (0,\a) -- \punkt{0}{1}{\t}; + \draw[->,color=blue,line width=1.4pt] + (1,0) -- \punkt{1}{0}{\t}; + } + \foreach \x in {-3,...,3}{ + \draw[color=blue,line width=0.5pt] + \punkt{\x}{\ymin}{\t} -- \punkt{\x}{\ymax}{\t}; + \foreach \y in {\ymin,...,\ymax}{ + \fill[color=blue] + \punkt{\x}{\y}{\t} + circle[radius=0.06]; + } + } + \foreach \y in {\ymin,...,\ymax}{ + \draw[color=blue,line width=0.5pt] + \punkt{-3}{\y}{\t} -- \punkt{3}{\y}{\t}; + } + } +} + +\begin{scope} +\clip (-3.2,-3.2) rectangle (3.2,3.2); +\ifthenelse{\boolean{presentation}}{ + \foreach \T in {1,...,\N}{ + \bildnetz{\T} + } +}{ + \bildnetz{\N} +} +\end{scope} + +\uncover<\N->{ +\begin{scope}[yshift=-2.5cm] +\fill[color=white,opacity=0.8] (-1.5,-0.8) rectangle (1.5,0.8); +\draw[line width=0.2pt] (-1.5,-0.8) rectangle (1.5,0.8); +\node at (0,0) {$\displaystyle W=\begin{pmatrix}0&2\\1&0\end{pmatrix}$}; +\end{scope} +} + +\node at (0,-3.7) {$\alpha^2 = 2$}; + +\end{tikzpicture} +\end{center} +\end{column} +\begin{column}{0.48\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=0.92] + +\node[color=red] at (-3.2,3.2) [above right] {$\mathbb{Z}(\varphi)$}; +\node[color=blue] at (3.5,3.2) [above left] {$\varphi\mathbb{Z}(\varphi)$}; + +\pgfmathparse{(sqrt(5)+1)/2} +\xdef\a{\pgfmathresult} +\pgfmathparse{-int(3.3/\a)} +\xdef\ymin{\pgfmathresult} +\pgfmathparse{int(3.3/\a)} +\xdef\ymax{\pgfmathresult} +\def\punkt#1#2#3{ + ({(1-(#3))*(#1)+(#3)*(#2)},{((1-(#3))*(#2)+(#3)*(#1+#2))*\a}) +} + +\draw[->] (-3.2,0) -- (3.5,0) coordinate[label={$\mathbb{Z}$}]; +\draw[->] (0,-3.2) -- (0,3.6) coordinate[label={right:$\mathbb{Z}\varphi$}]; + +\foreach \x in {-3,...,3}{ + \draw[color=red,line width=0.5pt] + \punkt{\x}{\ymin}{0} -- \punkt{\x}{\ymax}{0}; + \foreach \y in {\ymin,...,\ymax}{ + \fill[color=red] \punkt{\x}{\y}{0} circle[radius=0.08]; + } +} +\foreach \y in {\ymin,...,\ymax}{ + \draw[color=red,line width=0.5pt] + \punkt{-3}{\y}{0} -- \punkt{3}{\y}{0}; +} + +\def\bildnetz#1{ + \pgfmathparse{(#1-1)/(\N-1)} + \xdef\t{\pgfmathresult} + \only<#1>{ + \uncover<2->{ + \draw[->,color=blue,line width=1.4pt] + (0,\a) -- \punkt{0}{1}{\t}; + \draw[->,color=blue,line width=1.4pt] + (1,0) -- \punkt{1}{0}{\t}; + } + \foreach \x in {-3,...,3}{ + \draw[color=blue,line width=0.5pt] + \punkt{\x}{\ymin}{\t} -- \punkt{\x}{\ymax}{\t}; + \foreach \y in {\ymin,...,\ymax}{ + \fill[color=blue] \punkt{\x}{\y}{\t} + circle[radius=0.06]; + } + } + \foreach \y in {\ymin,...,\ymax}{ + \draw[color=blue,line width=0.5pt] + \punkt{-3}{\y}{\t} -- \punkt{3}{\y}{\t}; + } + } +} + +\begin{scope} + +\clip (-3.2,-3.2) rectangle (3.2,3.2); +\ifthenelse{\boolean{presentation}}{ + \foreach \T in {1,...,\N}{ + \bildnetz{\T} + } +}{ + \bildnetz{\N} +} +\end{scope} + +\uncover<\N->{ +\begin{scope}[yshift=-2.5cm] +\fill[color=white,opacity=0.8] (-1.5,-0.8) rectangle (1.5,0.8); +\draw[line width=0.2pt] (-1.5,-0.8) rectangle (1.5,0.8); +\node at (0,0) {$\displaystyle \Phi=\begin{pmatrix}0&1\\1&1\end{pmatrix}$}; +\end{scope} +} + +\node at (0,-3.7) {$\alpha^2 = \alpha + 1$}; + +\end{tikzpicture} +\end{center} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/3/nichthauptideal.tex b/vorlesungen/slides/3/nichthauptideal.tex new file mode 100644 index 0000000..46074b9 --- /dev/null +++ b/vorlesungen/slides/3/nichthauptideal.tex @@ -0,0 +1,78 @@ +% +% nichthauptideal.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Nicht-Hauptideal in $\mathbb{Z}[X]$} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Hauptideal\uncover<2->{ = ``Gerade''}} +\vspace{-10pt} +\begin{align*} +\langle X+1\rangle&=(X+1) = {\color{red}(X+1)\cdot\mathbb{Z}[X]} +\end{align*} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=0.4] +\draw[->] (-6.3,0) -- (6.8,0) coordinate[label={$\mathbb{Z}$}]; +\draw[->] (0,-6.2) -- (0,6.6) coordinate[label={right:$\mathbb{Z}X$}]; +\foreach \x in {-6,...,6}{ + \fill[color=red] (\x,\x) circle[radius=0.12]; +} +\end{tikzpicture} +\end{center} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<3->{% +\begin{block}{Ideal mit zwei Erzeugenden} +\vspace{-10pt} +\begin{align*} +\uncover<6->{ +{\color{darkgreen} +\langle 2,X\rangle +} +&=} +\uncover<5->{ +{\color{red}2\cdot\mathbb{Z}[X]} +} +\uncover<6->{+} +\uncover<4->{ +{\color{blue}X\cdot\mathbb{Z}[X]} +} +\end{align*} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=0.4] +\draw[->] (-6.3,0) -- (6.9,0) coordinate[label={$\mathbb{Z}$}]; +\draw[->] (0,-6.2) -- (0,7.0) coordinate[label={right:$\mathbb{Z}X$}]; +\uncover<6->{ + \foreach \x in {-6,-4,...,6}{ + \foreach \y in {-6,...,6}{ + \fill[color=darkgreen] (\x,\y) circle[radius=0.20]; + } + } +} +\uncover<5->{ + \foreach \x in {-6,-4,...,6}{ + \foreach \y in {-6,-4,...,6}{ + \fill[color=red] (\x,\y) circle[radius=0.16]; + } + } +} +\uncover<4->{ + \foreach \y in {-6,...,6}{ + \fill[color=blue] (0,\y) circle[radius=0.12]; + } +} +\end{tikzpicture} +\end{center} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/3/nichthauptideal2.tex b/vorlesungen/slides/3/nichthauptideal2.tex new file mode 100644 index 0000000..e1424ff --- /dev/null +++ b/vorlesungen/slides/3/nichthauptideal2.tex @@ -0,0 +1,95 @@ +% +% nichthauptideal2.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame}[t] +\frametitle{Das Ideal $\langle 2,X\rangle \subset \mathbb{Z}[X]$} +\vspace{-12pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\def\c{\clip (-2.8,-2.0) rectangle (2.8,2.0);} + +\def\labels{ + \fill[color=white,opacity=0.5] (1.5,-0.1) circle[radius=0.2]; + \node at (1.5,-0.1) {$1$}; + \fill[color=white,opacity=0.5] (-0.9,1.7) circle[radius=0.2]; + \node at (-0.9,1.7) {$X$}; + \fill[color=white,opacity=0.5] (0.8,0.7) circle[radius=0.2]; + \node at (0.8,0.7) {$X^2$}; +} + +\only<-3>{ +\begin{scope}[xshift=3.0cm,yshift=1.9cm] + \begin{scope} + \c + \node at (0,0) + {\includegraphics[width=7cm]{../slides/3/images/ring.jpg}}; + \end{scope} + \node[color=orange] at (1.9,0.1) [right] {$\mathbb{Z}[X]$}; +\end{scope} +} + +\uncover<2->{ +\begin{scope}[xshift=-3.0cm,yshift=1.9cm] + \begin{scope} + \c + \node at (0,0) + {\includegraphics[width=7cm]{../slides/3/images/hauptideal.jpg}}; + \end{scope} + \node[color=blue] at (-0.2,-1.2) {$(X+1)\cdot\mathbb{Z}[X]$}; + \labels +\end{scope} +} + +\uncover<3->{ +\begin{scope}[xshift=-3.0cm,yshift=-1.9cm] + \begin{scope} + \c + \node at (0,0) + {\includegraphics[width=7cm]{../slides/3/images/hauptideal2.jpg}}; + \end{scope} + \node[color=darkgreen] at (-3.0,-0.8) {$2\cdot\mathbb{Z}[X]$}; +\end{scope} + +\begin{scope}[xshift=3.0cm,yshift=-1.9cm] + \begin{scope} + \c + \node at (0,0) + {\includegraphics[width=7cm]{../slides/3/images/hauptidealX.jpg}}; + \end{scope} + \node[color=darkgreen] at (2.5,-0.8) {$X\cdot\mathbb{Z}[X]$}; + \labels +\end{scope} +} + +\uncover<4->{ +\begin{scope}[xshift=3.0cm,yshift=1.9cm] + \begin{scope} + \c + \node at (0,0) + {\includegraphics[width=7cm]{../slides/3/images/nichthauptideal.jpg}}; + \end{scope} + \node[color=orange] at (1.9,0.1) [right] {$\mathbb{Z}[X]$}; + \node[color=darkgreen] at (1.9,-0.4) [right] {$\langle 2,X\rangle$}; +\end{scope} +} + +\draw[color=gray!50] (-6.6,0) -- (6.4,0); +\draw[color=gray!50] (0,-3.8) -- (0,3.8); + +\begin{scope}[xshift=3.0cm,yshift=1.9cm] + \fill[color=white,opacity=0.5] (1.5,-0.6) circle[radius=0.2]; + \node at (1.5,-0.6) {$1$}; + \fill[color=white,opacity=0.5] (-0.4,1.7) circle[radius=0.2]; + \node at (-0.4,1.7) {$X$}; +\end{scope} + +\end{tikzpicture} +\end{center} + +\end{frame} +\egroup diff --git a/vorlesungen/slides/3/operatoren.tex b/vorlesungen/slides/3/operatoren.tex new file mode 100644 index 0000000..d646353 --- /dev/null +++ b/vorlesungen/slides/3/operatoren.tex @@ -0,0 +1,51 @@ +% +% operatoren.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{$X$ als Operator} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.38\textwidth} +\begin{block}{Polynome} +$a(X)=a_0+a_1X+\dots+a_nX^n$ +\uncover<2->{% +\[ +a(X) += +\begin{pmatrix} +a_0\\a_1\\a_2\\a_3\\\vdots\\a_n +\end{pmatrix} +\]} +\end{block} +\end{column} +\begin{column}{0.58\textwidth} +\uncover<3->{% +\begin{block}{Multiplikation mit $X$} +\strut +\[ +\begin{pmatrix} +1\\0\\0\\0\\\vdots\\0 +\end{pmatrix} +\uncover<4->{\overset{\cdot X}{\mapsto} +\begin{pmatrix} +0\\1\\0\\0\\\vdots\\0 +\end{pmatrix}} +\uncover<5->{\overset{\cdot X}{\mapsto} +\begin{pmatrix} +0\\0\\1\\0\\\vdots\\0 +\end{pmatrix}} +\uncover<6->{\overset{\cdot X}{\mapsto} +\begin{pmatrix} +0\\0\\0\\1\\\vdots\\0 +\end{pmatrix}} +\uncover<7->{\overset{\cdot X}{\mapsto}\dots} +\uncover<8->{\overset{\cdot X}{\mapsto} +\begin{pmatrix} +0\\0\\0\\0\\\vdots\\1 +\end{pmatrix}} +\] +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/3/phi.tex b/vorlesungen/slides/3/phi.tex new file mode 100644 index 0000000..ee0814c --- /dev/null +++ b/vorlesungen/slides/3/phi.tex @@ -0,0 +1,85 @@ +% +% phi.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{$\mathbb{Q}(\varphi)=\mathbb{Q}[X]/(X^2-X-1)$} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Der Ring $\mathbb{Z}(\varphi)$} +$\mathbb{Z}(\varphi)$ als Teilrung: +{\color{blue} +\[ +R=\{a+b\varphi\;|\; a,b\in\mathbb{Z}\} +\]}% +\uncover<2->{$\varphi\not\in\mathbb{Q}$}\uncover<3->{ +$\Rightarrow$ +$1$ und $\varphi$ sind inkommensurabel}\uncover<4->{ +$\Rightarrow$ +$R$ dicht in $\mathbb{R}$} +\end{block} +\uncover<5->{% +\begin{block}{Algebraische Konstruktion} +\uncover<8->{% +Das Polynom $X^2-X-1$ ist irreduzibel als Polynom in $\mathbb{Q}[X]$} +\[ +\uncover<8->{\mathbb{Q}[X]/(X^2-X-1) +=} +{\color{red}\{a+b\varphi\;|\;a,b\in\mathbb{Z}\}} +\]\uncover<7->{% +mit der Rechenregel: $X^2=X+1$} +\end{block}} +\uncover<9->{% +\begin{block}{Körper} +$\mathbb{Q}(\varphi) = \mathbb{Q}[X]/(X^2+X+1)$ +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=0.92] +\begin{scope} +\pgfmathparse{(sqrt(5)-1))/2} +\xdef\gphi{\pgfmathresult} +\clip (-3.2,-3.2) rectangle (3.2,3.2); +\foreach \x in {-10,...,10}{ + \pgfmathparse{int(\x/\gphi)-10} + \xdef\s{\pgfmathresult} + \pgfmathparse{int(\x/\gphi)+10} + \xdef\t{\pgfmathresult} + \foreach \y in {\s,...,\t}{ + \uncover<4->{ + \fill[color=blue] ({\x-\y*\gphi},0) + circle[radius=0.05]; + } + \uncover<6->{ + \draw[color=blue,line width=0.1pt] + ({\x-\y*\gphi-3.2},3.2) + -- + ({\x-\y*\gphi+3.2},-3.2); + } + } +} +\end{scope} + +\draw[->] (-3.2,0) -- (3.5,0) coordinate[label={$\mathbb{Z}$}]; + +\uncover<5->{ + \draw[->] (0,-3.2) -- (0,3.5) coordinate[label={right:$\mathbb{Z}X$}]; + + \foreach \x in {-3,...,3}{ + \foreach \y in {-5,...,5}{ + \fill[color=red] + ({\x},{\y*\gphi}) circle[radius=0.08]; + } + } +} + +\end{tikzpicture} +\end{center} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/3/polynome.tex b/vorlesungen/slides/3/polynome.tex new file mode 100644 index 0000000..d7179a0 --- /dev/null +++ b/vorlesungen/slides/3/polynome.tex @@ -0,0 +1,29 @@ +% +% polynome.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Polynome} +$R$ ein Ring, z.~B.~$\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$ + +\begin{definition} +Polynome in $X$ mit Koeffizienten in $R$: +\[ +R[X] += +\{ +a(X)\;|\; +a(X) = a_nX^n+a_{n-1}X^{n-1} + \dots a_2X^2+a_1X + a_0, a_k\in R +\} +\] +\end{definition} + +\begin{itemize} +\item<2-> {\em Grad} des Polynoms: $\deg a(X) = \deg a = n$ +\item<3-> $\deg 0 = -\infty$ +\item<4-> {\em normiertes Polynom}: $a_n=1$ +\end{itemize} + + +\end{frame} diff --git a/vorlesungen/slides/3/quotientenring.tex b/vorlesungen/slides/3/quotientenring.tex new file mode 100644 index 0000000..4aa9e43 --- /dev/null +++ b/vorlesungen/slides/3/quotientenring.tex @@ -0,0 +1,59 @@ +% +% Quotientenring.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Quotientenring} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Quotientenring} +$I\subset R$ ein Ideal +\\ +\uncover<2->{ +$R/I$ hat eine Ringstruktur: +\begin{align*} +\uncover<3->{\pi(s)&=s+I} +\\ +\uncover<4->{\pi(s)\pi(r)&= (s+I)(r+I)}\\ + &\uncover<5->{= sr +\underbrace{sI + rI}_{\subset RI\subset I} + II = sr+I} +\\ +\uncover<6->{\pi(s)+\pi(r)&= (s+I)+(r+I)}\\ + &\uncover<7->{=s+r+I=\pi(s+r)} +\end{align*}} +\end{block} +\vspace{-15pt} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<7->{% +\begin{block}{Beispiele} +\begin{itemize} +\item +$\mathbb{Z}/(n)=\mathbb{Z}/n\mathbb{Z}$, +$\mathbb{F}_p=\mathbb{Z}/(p)=\mathbb{Z}/p\mathbb{Z}$ +\item<8-> +$p\in\Bbbk[X]$ +$\Rightarrow$ +$\Bbbk[X]/(p)$ ist ein Ring +\end{itemize} +\end{block}} +\uncover<9->{% +\begin{block}{Algebraideal} +$I\subset A$ +\begin{itemize} +\item<10-> +$I$ ein Unterraum von $A$ als Vektorraum +\item<11-> +$I$ ein Ideal von $A$ als Ring +\end{itemize} +\end{block}} +\uncover<12->{% +\begin{block}{Quotientenalgebra} +$A/I$ ist eine Algebra +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/3/ringstruktur.tex b/vorlesungen/slides/3/ringstruktur.tex new file mode 100644 index 0000000..d653300 --- /dev/null +++ b/vorlesungen/slides/3/ringstruktur.tex @@ -0,0 +1,50 @@ +% +% ringstruktur.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Ringstruktur} +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.46\textwidth} +\begin{block}{Ring} +Menge $R$ mit zwei zweistelligen Verknüfpungen $+$ und $\cdot$ +mit +\begin{enumerate} +\item<3-> +$R$ ist abelsche Gruppe bezüglich $+$ +\item<5-> +$R\setminus\{0\}$ ist ein Monoid bezüglich $\cdot$ +\item<7-> +Für alle $a,b,c\in R$ +\begin{align*} +a(b+c) &= ab+ac +\\ +(a+b)c &= ac+bc +\end{align*} +\end{enumerate} +\end{block} +\end{column} +\begin{column}{0.50\textwidth} +\uncover<2->{% +\begin{block}{Polynomring} +$R$ ein Ring, $R[X]$ ``erbt'' Addition und Multiplikation mit +\begin{enumerate} +\item<4-> +$R[X]$ ist abelsche Gruppe bezüglich $+$ +\item<6-> +$R[X]\setminus\{0\}$ ist ein Monoid bezüglich $\cdot$ +\item<8-> +Für alle $a,b,c\in R[X]$ +\begin{align*} +a(b+c) &= ab+ac +\\ +(a+b)c &= ac+bc +\end{align*} +\end{enumerate} +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/3/teilbarkeit.tex b/vorlesungen/slides/3/teilbarkeit.tex new file mode 100644 index 0000000..a5ea9b9 --- /dev/null +++ b/vorlesungen/slides/3/teilbarkeit.tex @@ -0,0 +1,47 @@ +% +% teilbarkeit.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Teilen} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Teilen in $\mathbb{Z}$} +Zu zwei Zahlen $a,b\in \mathbb{Z}$, \only<3->{{\color<3-4>{red}$a>b$}, }gibt es +immer \only<3->{{\color<3-4>{red}genau}} ein Paar $q,r\in\mathbb{Z}$ derart, dass +\begin{align*} +a&=bq+r +\\ +\uncover<3->{{\color<3-4>{red}r}&{\color<3-4>{red}< b}} +\end{align*} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<2->{% +\begin{block}{Teilen in $\mathbb{Q}[X]$} +Zu zwei Polynomen $a,b\in\mathbb{Q}[X]$, \only<4->{{\color<4>{red}$\deg a > \deg b$},} +gibt es +immer \only<4->{{\color<4>{red}bis auf eine Einheit genau }}% +ein Paar $q,r\in\mathbb{Q}[X]$ derart, dass +\begin{align*} +a&=bq+r +\\ +\uncover<4->{{\color<4>{red}\deg r}&{\color<4>{red}< \deg b}} +\end{align*} +\end{block}} +\end{column} +\end{columns} +\uncover<5->{% +\begin{block}{Allgemein: euklidischer Ring} +Nullteilerfreier Ring $R$ mit einer Funktion +$d\colon R\setminus{0}\to\mathbb{N}$ mit +\begin{itemize} +\item Für $x,y\in R$ gilt $d(xy) \ge d(x)$. +\item Für $x,y\in R$ gibt es $q,r\in R$ derart +$x=qy +r$ mit $d(y)>d(r)$ +\end{itemize} +Euklidische Ringe haben ähnliche Eigenschaften wie Polynomringe +\end{block}} +\end{frame} diff --git a/vorlesungen/slides/3/wurzel2.tex b/vorlesungen/slides/3/wurzel2.tex new file mode 100644 index 0000000..d20bfc4 --- /dev/null +++ b/vorlesungen/slides/3/wurzel2.tex @@ -0,0 +1,83 @@ +% +% wurzel2.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{$\mathbb{Z}(\sqrt{2})\only<7->{ = \mathbb{Z}[X]/(X^2-2)}$} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Der Ring $\mathbb{Z}(\sqrt{2})$} +$\mathbb{Z}(\sqrt{2})$ als Teilring: +{\color{blue} +\[ +R=\{ a+b\sqrt{2}\;|\; a,b\in\mathbb{Z} \} \subset \mathbb{R} +\]}% +\uncover<2->{$\sqrt{2}\not\in\mathbb{Q}$}\uncover<3->{ +$\Rightarrow$ +$1$ und $\sqrt{2}$ sind inkommensurabel}\uncover<4->{ +$\Rightarrow$ +$R$ dicht in $\mathbb{R}$} +\end{block} +\uncover<5->{% +\begin{block}{Algebraische Konstruktion} +\uncover<8->{% +Das Polynom $X^2-2$ ist irreduzibel als Polynom in $\mathbb{Q}[X]$} +\[ +\uncover<8->{\mathbb{Z}[X]/(X^2-2) +=} +{\color{red}\{a+bX\;|\;a,b\in\mathbb{Z}\}} +\]\uncover<7->{% +mit Rechenregel: $X^2=2$} +\end{block}} +\uncover<9->{% +\begin{block}{Körper} +$\mathbb{Q}(\sqrt{2}) = \mathbb{Q}[X]/(X^2-2)$ +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=0.92] +\begin{scope} +\clip (-3.2,-3.2) rectangle (3.2,3.2); +\foreach \x in {-10,...,10}{ + \pgfmathparse{int(\x/sqrt(2))-5} + \xdef\s{\pgfmathresult} + \pgfmathparse{int(\x/sqrt(2))+5} + \xdef\t{\pgfmathresult} + \foreach \y in {\s,...,\t}{ + \uncover<4->{ + \fill[color=blue] ({\x-\y*sqrt(2)},0) + circle[radius=0.05]; + } + \uncover<6->{ + \draw[color=blue,line width=0.1pt] + ({\x-\y*sqrt(2)-3.2},3.2) + -- + ({\x-\y*sqrt(2)+3.2},-3.2); + } + } +} +\end{scope} + +\draw[->] (-3.2,0) -- (3.5,0) coordinate[label={$\mathbb{Z}$}]; + +\uncover<5->{ + \draw[->] (0,-3.2) -- (0,3.5) coordinate[label={right:$\mathbb{Z}X$}]; + + \foreach \x in {-3,...,3}{ + \foreach \y in {-2,...,2}{ + \fill[color=red] + ({\x},{\y*sqrt(2)}) circle[radius=0.08]; + } + } +} + +\end{tikzpicture} +\end{center} +\end{column} +\end{columns} +\end{frame} |
