From 84a06435159346d7768b0a1a6164caf8e0ef0da0 Mon Sep 17 00:00:00 2001 From: henrydatei Date: Fri, 12 Oct 2018 10:19:54 +0200 Subject: [PATCH] Fixed Issues #16 #17 #18 #19 #21 #22 #23 #24 #25 #26 --- .../TeX_files/Das_Minimalpolynom.tex | 4 ++-- .../TeX_files/Das_Standardskalarprodukt.tex | 2 +- .../TeX_files/Die_Jordan_Normalform.tex | 2 +- .../TeX_files/Die_adjungierte_Abbildung.tex | 2 +- .../TeX_files/Euklidische_und_unitaere_VR.tex | 14 +++++++------- .../TeX_files/Nilpotente_Endomorphismen.tex | 4 ++-- ...rthogonale_und_unitaere_Endomorphismen.tex | 4 ++-- .../TeX_files/Orthogonalitaet.tex | 3 ++- .../TeX_files/Trigonalisierbarkeit.tex | 1 - .../LAAG 1+2 Semester_Fehm/Vorlesung LAAG.pdf | Bin 1190360 -> 1270001 bytes 10 files changed, 18 insertions(+), 18 deletions(-) diff --git a/2. Semester/LAAG 1+2 Semester_Fehm/TeX_files/Das_Minimalpolynom.tex b/2. Semester/LAAG 1+2 Semester_Fehm/TeX_files/Das_Minimalpolynom.tex index 905eba6..442199b 100644 --- a/2. Semester/LAAG 1+2 Semester_Fehm/TeX_files/Das_Minimalpolynom.tex +++ b/2. Semester/LAAG 1+2 Semester_Fehm/TeX_files/Das_Minimalpolynom.tex @@ -47,9 +47,9 @@ \end{definition} \begin{mathematica}[Minimalpolynom] - Die Funktion für das Minimalpolynom $p$ mit der Variable $t$ in Mathematica bzw. WolframAlpha lautet: + Die Funktion für das Minimalpolynom $p$ mit der Variable $t$ von einer Matrix $A$ in Mathematica bzw. WolframAlpha lautet: \begin{align} - \texttt{MinimalPolynomial[p,x]}\notag + \texttt{MinimalPolynomial[A,x]}\notag \end{align} \end{mathematica} diff --git a/2. Semester/LAAG 1+2 Semester_Fehm/TeX_files/Das_Standardskalarprodukt.tex b/2. Semester/LAAG 1+2 Semester_Fehm/TeX_files/Das_Standardskalarprodukt.tex index 328049d..7ee2420 100644 --- a/2. Semester/LAAG 1+2 Semester_Fehm/TeX_files/Das_Standardskalarprodukt.tex +++ b/2. Semester/LAAG 1+2 Semester_Fehm/TeX_files/Das_Standardskalarprodukt.tex @@ -23,7 +23,7 @@ Sei zunächst $K=\real$. \item Für $x,x',y,y'\in\real^n$ und $\lambda\in\real$ ist: \begin{align} \langle x+x',y\rangle &= \langle x,y\rangle + \langle x',y\rangle\notag\\ - \langle \lambda x,y\rangle =&= \lambda \langle x,y \rangle\notag \\ + \langle \lambda x,y\rangle &= \lambda \langle x,y \rangle\notag \\ \langle x,y+y' \rangle &= \langle x,y \rangle + \langle x,y'\rangle\notag \\ \langle x,\lambda y \rangle &= \lambda \langle x,y \rangle\notag \end{align} diff --git a/2. Semester/LAAG 1+2 Semester_Fehm/TeX_files/Die_Jordan_Normalform.tex b/2. Semester/LAAG 1+2 Semester_Fehm/TeX_files/Die_Jordan_Normalform.tex index 415eaf6..f03e223 100644 --- a/2. Semester/LAAG 1+2 Semester_Fehm/TeX_files/Die_Jordan_Normalform.tex +++ b/2. Semester/LAAG 1+2 Semester_Fehm/TeX_files/Die_Jordan_Normalform.tex @@ -37,7 +37,7 @@ A=\begin{pmatrix}1&3&\; \\ \;&1&4 \\ \;&\; & 2\end{pmatrix}\in\Mat_3(\real)\notag \end{align} $\chi_A(t)=(t-1)^2(t-2)$ - $\Rightarrow \real^3=\underbrace{\Hau(f,1)}_{\dim = 2}\oplus\underbrace{\Hau(f,2)}_{\dim 1}$ \\ + $\Rightarrow \real^3=\underbrace{\Hau(f,1)}_{\dim = 2}\oplus\underbrace{\Hau(f,2)}_{\dim = 1}$ \\ $\Hau(f,1)=\Ker((f-\id)^2)=L((A-\mathbbm{1})^2,0)$ \\ $\Hau(f,2)=\Ker(f-2\id)=\Eig(f,2)=L(A-2\mathbbm{1},0)$ \\ \begin{align} diff --git a/2. Semester/LAAG 1+2 Semester_Fehm/TeX_files/Die_adjungierte_Abbildung.tex b/2. Semester/LAAG 1+2 Semester_Fehm/TeX_files/Die_adjungierte_Abbildung.tex index 02f3255..942f0ea 100644 --- a/2. Semester/LAAG 1+2 Semester_Fehm/TeX_files/Die_adjungierte_Abbildung.tex +++ b/2. Semester/LAAG 1+2 Semester_Fehm/TeX_files/Die_adjungierte_Abbildung.tex @@ -88,7 +88,7 @@ Sei $K=\real$ oder $K=\comp$ und $V$ ein endlichdimensionaler unitärer $K$-Vekt \end{align} so ist \begin{align} - 0=\skalar{x}{f_1(y)}-\skalar{x}{f_2(y)}=\skalar{x}{f_1(y),f_2(y)}\quad\forall x,y\in V\notag + 0=\skalar{x}{f_1(y)}-\skalar{x}{f_2(y)}=\skalar{x}{f_1(y)-f_2(y)}\quad\forall x,y\in V\notag \end{align} da $\skalar{\cdot}{\cdot}$ nicht ausgeartet ist, folgt daraus, dass $f_1=f_2$. \end{itemize} diff --git a/2. Semester/LAAG 1+2 Semester_Fehm/TeX_files/Euklidische_und_unitaere_VR.tex b/2. Semester/LAAG 1+2 Semester_Fehm/TeX_files/Euklidische_und_unitaere_VR.tex index 32afe5a..80cde39 100644 --- a/2. Semester/LAAG 1+2 Semester_Fehm/TeX_files/Euklidische_und_unitaere_VR.tex +++ b/2. Semester/LAAG 1+2 Semester_Fehm/TeX_files/Euklidische_und_unitaere_VR.tex @@ -38,16 +38,16 @@ Im Fall $K=\comp$: ÜA \end{proof} -\begin{definition}[(semi)definit, euklidischer VR, unitärer VR] +\begin{definition}[(semi)definit, euklidischer Vektorraum, unitärer Vektorraum] Sei $s$ eine hermitesche Sesquilinearform auf $V$. Ist $s(x,x)\ge 0$ für alle $x\in V$, so heißt $s$ \emph{positiv} \begriff{semidefinit}. Ist $s(x,x)>0$ für alle $0\neq x\in V$, so heißt $s$ \emph{positiv} \begriff{definit} (oder ein \emph{Skalarprodukt}). Eine hermitesche Matrix $A\in\Mat_n(K)$ heißt \emph{positiv (semi)definit}, wenn $s_A$ dies ist. - Einen endlichdimensionalen $K$-VR zusammen mit positiv definiten hermiteschen Sesquilinearformen nennt man einen \begriff{euklidischen} bzw. \begriff{unitären} VR (oder auch \emph{Prähilbertraum}). Wenn nicht anderes angegeben, notieren wir die Sesquilinearform mit $\skalar{\cdot}{\cdot}$. + Einen endlichdimensionalen $K$-Vektorraum zusammen mit positiv definiten hermiteschen Sesquilinearformen nennt man einen \begriff{euklidischen} bzw. \begriff{unitären} Vektorraum (oder auch \emph{Prähilbertraum}). Wenn nicht anderes angegeben, notieren wir die Sesquilinearform mit $\skalar{\cdot}{\cdot}$. \end{definition} \begin{example} - Der Standardraum $V=K^n$ zusammen mit dem Standardskalarprodukt ist ein euklidischer bzw. unitärer VR. + Der Standardraum $V=K^n$ zusammen mit dem Standardskalarprodukt ist ein euklidischer bzw. unitärer Vektorraum. \end{example} \begin{example} @@ -55,21 +55,21 @@ \end{example} \begin{proposition} - Ist $V$ ein unitärer VR und $U\subseteq V$ ein UVR, so ist $U$ mit der Einschränkung des Skalarprodukts wieder ein unitärer VR. + Ist $V$ ein unitärer Vektorraum und $U\subseteq V$ ein Untervektorraum, so ist $U$ mit der Einschränkung des Skalarprodukts wieder ein unitärer Vektorraum. \end{proposition} \begin{proof} klar, die Einschränkung ist wieder positiv definit. \end{proof} \begin{definition} - Ist $V$ ein unitärer VR, so definiert man die Norm von $x\in V$ als + Ist $V$ ein unitärer Vektorraum, so definiert man die Norm von $x\in V$ als \begin{align} \Vert x\Vert = \sqrt{\skalar{x}{x}}\in\real_{\ge 0}\notag \end{align} \end{definition} \begin{proposition} - Die Norm eines unitären VR erfüllt die folgenden Eigenschaften: + Die Norm eines unitären Vektorraums erfüllt die folgenden Eigenschaften: \begin{itemize} \item Für $x\in V$ ist $\Vert x\Vert =0\iff x=0$ \item Für $x\in V$ und $\lambda\in K$ ist $\Vert \lambda x\Vert=\vert \lambda\vert \cdot \Vert x\Vert$ @@ -85,7 +85,7 @@ \end{proof} \begin{proposition} - Ist $V$ ein unitärer VR, so gilt für $x,y\in V$: + Ist $V$ ein unitärer Vektorraum, so gilt für $x,y\in V$: \begin{align} \vert \skalar{x}{y}\vert \le \Vert x\Vert\cdot \Vert y\Vert\notag \end{align} diff --git a/2. Semester/LAAG 1+2 Semester_Fehm/TeX_files/Nilpotente_Endomorphismen.tex b/2. Semester/LAAG 1+2 Semester_Fehm/TeX_files/Nilpotente_Endomorphismen.tex index 3960cc0..51da90a 100644 --- a/2. Semester/LAAG 1+2 Semester_Fehm/TeX_files/Nilpotente_Endomorphismen.tex +++ b/2. Semester/LAAG 1+2 Semester_Fehm/TeX_files/Nilpotente_Endomorphismen.tex @@ -33,7 +33,7 @@ Da $\dim_K(V)=n<\infty$ gibt es ein $d$ mit $\Ker(f^d)=\Ker(f^{d+i})$ und $\Imag \begin{proposition}[Lemma von \person{Fitting}] \proplbl{satz_6_4} - Seien $V_i=\Ker(f^i)$, $W_i=\Image(f^i)$, $d=\min\{i:V_i=V_{i+1}\}$. Dann sind + Seien $V_i=\Ker(f^i)$, $W_i=\Image(f^i)$, $d=\min\{i\mid V_i=V_{i+1}\}$. Dann sind \begin{align} \{0\}&=V_0\subsetneq V_1\subsetneq ...\subsetneq V_d=V_{d+1}=...\notag \\ V&= W_0\supsetneq W_1\supsetneq ... \supsetneq W_d=W_{d+1}=...\notag @@ -41,7 +41,7 @@ Da $\dim_K(V)=n<\infty$ gibt es ein $d$ mit $\Ker(f^d)=\Ker(f^{d+i})$ und $\Imag Folgen $f$-invarianter UVR und $V=V_d\oplus W_d$. \end{proposition} \begin{proof} - Da $f^i$ und $f^j$ für beliebige $i,j$ kommutieren, sind $V_i$ und $V_j$ nach \propref{lemma_6_3} $f$-invariant für jedes $i$. Aus $\dim_K(V_i)+\dim_K(W_i)=n$ folgt $d=\min\{i:W_i=W_{i+1}\}$, insbesondere ist $\Image(f^d)=\Image(f^{d+1})=f(\Image(f^d))$, somit $W_{d+i}=\Image(f^{d+i})=W_d$ für $i\ge 0$, also auch $V_d=V_{d+i}$ für alle $i\ge 0$. \\ + Da $f^i$ und $f^j$ für beliebige $i,j$ kommutieren, sind $V_i$ und $V_j$ nach \propref{lemma_6_3} $f$-invariant für jedes $i$. Aus $\dim_K(V_i)+\dim_K(W_i)=n$ folgt $d=\min\{i\mid W_i=W_{i+1}\}$, insbesondere ist $\Image(f^d)=\Image(f^{d+1})=f(\Image(f^d))$, somit $W_{d+i}=\Image(f^{d+i})=W_d$ für $i\ge 0$, also auch $V_d=V_{d+i}$ für alle $i\ge 0$. \\ Insbesondere ist $f^d\vert_{W_d}:W_d\to W_{2d}=W_d$ surjektiv, also auch injektiv, also $V_d\cap W_d=\{0\}$. Aus der Dimensionsformel II.4.12 folgt dann $\dim_K(V_d+W_d)=\dim_K(V_d)+\dim_K(W_d)=\dim_K(V)$. Folglich ist $V_d+W_d=V$ und $V_d\cap W_d=\{0\}$, also $V=V_d\oplus W_d$. \end{proof} diff --git a/2. Semester/LAAG 1+2 Semester_Fehm/TeX_files/Orthogonale_und_unitaere_Endomorphismen.tex b/2. Semester/LAAG 1+2 Semester_Fehm/TeX_files/Orthogonale_und_unitaere_Endomorphismen.tex index e313694..e37ca99 100644 --- a/2. Semester/LAAG 1+2 Semester_Fehm/TeX_files/Orthogonale_und_unitaere_Endomorphismen.tex +++ b/2. Semester/LAAG 1+2 Semester_Fehm/TeX_files/Orthogonale_und_unitaere_Endomorphismen.tex @@ -100,10 +100,10 @@ Sei $V$ ein euklidischer bzw. unitärer Vektorraum und $f\in\End_K(V)$. \begin{theorem} \proplbl{6_5_9} - Sei $K=\comp$ und $f\in\End_K(V)$. Ist $f$ unitär, so besitzt $V$ eine Orthonormalbasis aus Eigenvektoren von f. + Sei $K=\comp$ und $f\in\End_K(V)$. Ist $f$ unitär, so besitzt $V$ eine Orthonormalbasis aus Eigenvektoren von $f$. \end{theorem} \begin{proof} - Induktion über$n=\dim_K(V)$. \\ + Induktion über $n=\dim_K(V)$. \\ \emph{$n=0$}: klar \\ \emph{$n-1\to n$}: Da $K$ algebraisch abgeschlossen ist, hat $\chi_f$ eine Nullstelle $\lambda$, es gibt also einen Eigenvektor $x_1$ von $f$ zum Eigenwert $\lambda$. Ohne Einschränkung nehmen wir $\Vert x\Vert=1$ an. Sei $W=K\cdot x_1$. Nach \propref{6_4_11} ist dann $V=W\oplus W^\perp$. Für $v\in W^\perp, w\in W$ ist \begin{align} diff --git a/2. Semester/LAAG 1+2 Semester_Fehm/TeX_files/Orthogonalitaet.tex b/2. Semester/LAAG 1+2 Semester_Fehm/TeX_files/Orthogonalitaet.tex index c3c23f8..74212b9 100644 --- a/2. Semester/LAAG 1+2 Semester_Fehm/TeX_files/Orthogonalitaet.tex +++ b/2. Semester/LAAG 1+2 Semester_Fehm/TeX_files/Orthogonalitaet.tex @@ -82,11 +82,12 @@ Sei $V$ ein euklidischer bzw. unitärer Vektorraum. \proplbl{6_4_8} Sei $U\subseteq V$ ein Untervektorraum und $B=(x_1,...,x_k)$ eine Orthonormalbasis von $U$. Es gibt genau einen Epimorphismus $\pr_U:V\to U$ mit $\pr_U\vert_U=\id_U$ und $\Ker(\pr_U)\perp U$, insbesondere also $x-\pr_U\perp U$ für alle $x\in V$, genannt die \begriff{orthogonale Projektion} auf $U$, und dieser ist geben durch \begin{align} + \label{gl1} x\mapsto\sum_{i=1}^k \skalar{x}{x_i}x_i \end{align} \end{proposition} \begin{proof} - Sei zunächst $pr_U$ durch (1) gegeben. Die Linearität von $\pr_U$ folgt aus (S1) und (S3). Für $u=\sum_{i=1}^k \lambda_i x_i\in U$ ist $\skalar{u}{x_j}=\skalar{\sum_{i=1}^k \lambda_i x_i}{x_j}=\sum_{i=1}^k \lambda_i\skalar{x_i}{x_j}=\lambda_j$, woraus $\pr_U(u)=u$. Somit ist $\pr_U\vert_U=\id_U$, und insbesondere ist $pr_U$ surjektiv. Ist $\pr_U(x)=0$, so ist $\skalar{x}{x_i}=0$ für alle $i$., woraus mit (S2) und (S4) sofort $x\perp U$ folgt. Somit ist $\Ker(\pr_U)\perp U$. \\ + Sei zunächst $\pr_U$ durch \cref{gl1} gegeben. Die Linearität von $\pr_U$ folgt aus (S1) und (S3). Für $u=\sum_{i=1}^k \lambda_i x_i\in U$ ist $\skalar{u}{x_j}=\skalar{\sum_{i=1}^k \lambda_i x_i}{x_j}=\sum_{i=1}^k \lambda_i\skalar{x_i}{x_j}=\lambda_j$, woraus $\pr_U(u)=u$. Somit ist $\pr_U\vert_U=\id_U$, und insbesondere ist $\pr_U$ surjektiv. Ist $\pr_U(x)=0$, so ist $\skalar{x}{x_i}=0$ für alle $i$., woraus mit (S2) und (S4) sofort $x\perp U$ folgt. Somit ist $\Ker(\pr_U)\perp U$. \\ Für $x\in V$ ist $\pr_U(x-\pr_U(x))=\pr_U(x)-\pr_U(\pr_U(x))=\pr_U(x)-\pr_U(x)=0$, also $x-\pr_U(x)\in\Ker(\pr_U)\subseteq U^\perp$. \\ Ist $f:V\to U$ ein weiterer Epimorphismus mit $f\vert_U=\id_U$ und $\Ker(f)\perp U$, so ist \begin{align} diff --git a/2. Semester/LAAG 1+2 Semester_Fehm/TeX_files/Trigonalisierbarkeit.tex b/2. Semester/LAAG 1+2 Semester_Fehm/TeX_files/Trigonalisierbarkeit.tex index a7d9f7c..6b0dd8f 100644 --- a/2. Semester/LAAG 1+2 Semester_Fehm/TeX_files/Trigonalisierbarkeit.tex +++ b/2. Semester/LAAG 1+2 Semester_Fehm/TeX_files/Trigonalisierbarkeit.tex @@ -65,7 +65,6 @@ $(\Leftarrow)$: Induktion nach $n=\dim_K(V)$. \\ \emph{$n=1$}: trivial \\ \emph{$n-1\to n$}: Nach Annahme ist $\chi_f(t)=\prod_{i=1}^n (t-\lambda_i)$ mit $\lambda_1,...,\lambda_n\in K$. Sei $x_1$ ein EV zum EW $\lambda_1$. Dann ist $V_1=K\cdot x_1$ ein $f$-invarianter UVR. Ergänze $B_1=(x_1)$ zu einer Basis $B=(x_1,...,x_n)$ von $V$ und setze $B_2=(x_2,...,x_n)$, $V_2=\Span_K(B_2)$. - \emph{$n-1\to n$}: Nach Annahme ist $\chi_f(t)=\prod_{i=1}^n (t-\lambda_i)$ mit $\lambda_1,...,\lambda_n\in K$. Sei $x_1$ ein EV zum EW $\lambda_1$. Dann ist $V_1=K\cdot x_1$ ein $f$-invarianter UVR. Ergänze $B_1=(x_1)$ zu einer Basis $B=(x_1,...,x_n)$ von $V$ und setze $B_2=(x_2,...,x_n)$, $V_2=\Span_K(B_2)$. \begin{align} \Rightarrow M_B(f)&=\begin{pmatrix}\lambda_1&*\\0&A_2\end{pmatrix}\quad A_2\in\Mat_{n-1}(K)\notag\\ \chi_f(t)&=\chi_{\lambda_1\mathbbm{1}_1}\cdot \chi_{A_2}=(t-\lambda_1)\cdot\chi_{A_2}(t)\notag \\ diff --git a/2. Semester/LAAG 1+2 Semester_Fehm/Vorlesung LAAG.pdf b/2. 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