From d0d5af9d27ea3d682a7b66969059dc441a71443e Mon Sep 17 00:00:00 2001 From: henrydatei Date: Fri, 27 Jul 2018 18:11:56 +0200 Subject: [PATCH] 500 Commits! :smile: + added @epsilonbelowzero Zeug --- ...inatendarstellung_linearer_Abbildungen.tex | 2 +- .../TeX_files/Lineare_Gleichungssysteme.tex | 10 +++++----- .../LAAG 1+2 Semester/TeX_files/Matrizen.tex | 6 +++--- .../LAAG 1+2 Semester/TeX_files/Rang.tex | 6 +++--- .../LAAG 1+2 Semester/Vorlesung LAAG.pdf | Bin 1186433 -> 1186487 bytes .../LAAG 1+2 Semester/Vorlesung LAAG.tex | 4 ++++ 2. Semester/LAAG/Vorlesung LAAG.pdf | Bin 817021 -> 817019 bytes 2. Semester/LAAG/Vorlesung LAAG.tex | 4 ++++ .../Matrix_mit_Klammern_unten_und_links.pdf | Bin 58711 -> 65836 bytes .../Matrix_mit_Klammern_unten_und_links.tex | 4 ++-- 10 files changed, 22 insertions(+), 14 deletions(-) diff --git a/2. Semester/LAAG 1+2 Semester/TeX_files/Koordinatendarstellung_linearer_Abbildungen.tex b/2. Semester/LAAG 1+2 Semester/TeX_files/Koordinatendarstellung_linearer_Abbildungen.tex index 9264c4f..06ba105 100644 --- a/2. Semester/LAAG 1+2 Semester/TeX_files/Koordinatendarstellung_linearer_Abbildungen.tex +++ b/2. Semester/LAAG 1+2 Semester/TeX_files/Koordinatendarstellung_linearer_Abbildungen.tex @@ -84,7 +84,7 @@ Seien $V,W$ endlichdimensionale $K$-Vektorräume mit den Basen $B=(x_1,...,x_n)$ \begin{proof} Sei $A=M_C^B(f)$. Nach \propref{3_6_2} ist $f$ genau dann ein Isomorphismus, wenn $f_A$ einer ist, und in diesem Fall ist $m=n$. Zudem ist $f_A$ genau dann ein Isomorphismus, wenn $A\in \GL_(K)$ (\propref{3_5_12}). Ist $f$ ein Isomorphismus, so ist $M_B^C(f^{-1})\cdot - M_C^B(f)=M_C^C(f^{-1}\circ f)=1_n$, also $M_B^C(f^{-1})=M_C^B(f)^{-1}$ (\propref{3_6_4}). + M_C^B(f)=M_C^C(f^{-1}\circ f)=\mathbbm{1}_n$, also $M_B^C(f^{-1})=M_C^B(f)^{-1}$ (\propref{3_6_4}). \end{proof} \begin{conclusion} diff --git a/2. Semester/LAAG 1+2 Semester/TeX_files/Lineare_Gleichungssysteme.tex b/2. Semester/LAAG 1+2 Semester/TeX_files/Lineare_Gleichungssysteme.tex index cbc6773..cd3caa2 100644 --- a/2. Semester/LAAG 1+2 Semester/TeX_files/Lineare_Gleichungssysteme.tex +++ b/2. Semester/LAAG 1+2 Semester/TeX_files/Lineare_Gleichungssysteme.tex @@ -91,9 +91,9 @@ Sei $A\in \Mat_{m\times n}(K)$ und $b\in K^m$. Für $i,j\in \{1,...,m\}$, $\lambda \in K^{\times}$ und $\mu\in K$ definieren wir $m\times m$-Matrizen, die sogenannten \begriff{Elementarmatrizen}: \begin{itemize} - \item $S_i(\lambda):=1_m + (\lambda-1)E_{ii}$ - \item $Q_{ij}(\mu):= 1_m + \mu E_{ij}$ - \item $P_{ij}:= 1_m + E_{ij} + E_{ji} - E_{ii} - E_{jj}$ + \item $S_i(\lambda):=\mathbbm{1}_m + (\lambda-1)E_{ii}$ + \item $Q_{ij}(\mu):= \mathbbm{1}_m + \mu E_{ij}$ + \item $P_{ij}:= \mathbbm{1}_m + E_{ij} + E_{ji} - E_{ii} - E_{jj}$ \end{itemize} \end{definition} @@ -171,9 +171,9 @@ Sei $A\in \Mat_{m\times n}(K)$ und $b\in K^m$. Es gibt von diesem Verfahren verschiedene Varianten und weitere Anwendungen: So kann man z.B. die Invertierbarkeit einer Matrix $A\in \Mat_n(K)$ prüfen und ggf. das Inverse bestimmen: Ist $E_l\cdot ... \cdot E_1\cdot A$ in Zeilenstufenform, so ist $A$ genau dann invertierbar, wenn alle Zeilen von Null verschieden sind. Ist dies der Fall, so ist $r=n$ und $k_i=i$ für alle $i$, - und man findet weitere Elementarmatrizen $E_{l+1},...,E_s$ vom Typ I und II, für die $E_s\cdot ... \cdot E_1\cdot A=1_n$. Dann ist + und man findet weitere Elementarmatrizen $E_{l+1},...,E_s$ vom Typ I und II, für die $E_s\cdot ... \cdot E_1\cdot A=\mathbbm{1}_n$. Dann ist $S'=E_s\cdot ... \cdot E_1\cdot A=A^{-1}$ (vgl. \propref{3_8_11}). Praktisch erhält man $A^{-1}$, indem man die Zeilenumformungen an $A$ parallel dazu - auch an $1_n$ ausführt. + auch an $\mathbbm{1}_n$ ausführt. \end{remark} \begin{conclusion} diff --git a/2. Semester/LAAG 1+2 Semester/TeX_files/Matrizen.tex b/2. Semester/LAAG 1+2 Semester/TeX_files/Matrizen.tex index 7451193..717cf11 100644 --- a/2. Semester/LAAG 1+2 Semester/TeX_files/Matrizen.tex +++ b/2. Semester/LAAG 1+2 Semester/TeX_files/Matrizen.tex @@ -38,7 +38,7 @@ Sei $K$ ein Körper. \item Die \begriff{Nullmatrix} ist $0=(0)_{i,j} \in \Mat_{m\times n}(K)$. \item Für $k,l \in \{1,...,n\}$ ist die $(k,l)$-\begriff{Basismatrix} gegeben durch $E_{kl}=(\delta_{jk}\delta_{jl})\in \Mat_{m\times n}(K)$. - \item Die \begriff{Einheitsmatrix} ist $1_n=(\delta_{ii})\in \Mat_n(K)$. + \item Die \begriff{Einheitsmatrix} ist $\mathbbm{1}_n=(\delta_{ii})\in \Mat_n(K)$. \item Für $a_i,...,a_n \in K$ definiert man eine \begriff{Diagonalmatrix} $\diag(a_1,...,a_n)=(\delta_{ij}\cdot a_i)$. \item Für eine Permutation $\sigma\in S_n$ definiert man die \begriff{Permutationsmatrix} $P_\sigma := (\delta_{\sigma (i),j})$. @@ -175,7 +175,7 @@ Sei $K$ ein Körper. \begin{pmatrix}d & -b\\-c & a\\\end{pmatrix}\in \Mat_2(K)\notag \end{align} Man prüft nach, dass $A\cdot \tilde A=\tilde A\cdot A= - (ad-bc)\cdot 1_2$. Definiert man nun $\det(A)=ad-bc$ so sieht man: Ist $\det(A)\neq 0$, so ist $A$ invertierbar mit + (ad-bc)\cdot \mathbbm{1}_2$. Definiert man nun $\det(A)=ad-bc$ so sieht man: Ist $\det(A)\neq 0$, so ist $A$ invertierbar mit $A^{-1}=\det(A)^{-1}\cdot \tilde A$. Ist $\det(A)=0$ so $A$ ist Nullteiler und somit nicht invertierbar (\propref{1_4_13}). Mehr dazu in Kapitel IV. \end{example} @@ -197,5 +197,5 @@ Sei $K$ ein Körper. Für $A\in \GL_n(K)$ ist $A^t\in \GL_n(K)$ und $(A^{-1})^t = (A^t)^{-1}$ \end{proposition} \begin{proof} - Aus $AA^{-1}=1$ folgt nach \propref{3_1_14}, dass $(A^{-1})^tA^t=1_n^t=1_n$. Somit ist $(A^{-1})^t$ das Inverse zu $A^t$. + Aus $AA^{-1}=1$ folgt nach \propref{3_1_14}, dass $(A^{-1})^tA^t=\mathbbm{1}_n^t=\mathbbm{1}_n$. Somit ist $(A^{-1})^t$ das Inverse zu $A^t$. \end{proof} \ No newline at end of file diff --git a/2. Semester/LAAG 1+2 Semester/TeX_files/Rang.tex b/2. Semester/LAAG 1+2 Semester/TeX_files/Rang.tex index 99395a4..cc3c2ab 100644 --- a/2. Semester/LAAG 1+2 Semester/TeX_files/Rang.tex +++ b/2. Semester/LAAG 1+2 Semester/TeX_files/Rang.tex @@ -111,12 +111,12 @@ Seien $V,W$ zwei endlichdimensionale $K$-Vektorräume und $f\in \Hom_K(V,W)$. \proplbl{3_8_11} Für $A\in \Mat_n(K)$ sind äquivalent: \begin{itemize} - \item $A\in \GL_n(K)$, d.h. es gibt $S\in \GL_n(K)$ mit $SA=AS=1_n$ + \item $A\in \GL_n(K)$, d.h. es gibt $S\in \GL_n(K)$ mit $SA=AS=\mathbbm{1}_n$ \item $\rk(A)=n$ \item Die Spalten von $A$ sind linear unabhängig. \item Die Zeilen von $A$ sind linear unabhängig. - \item Es gibt $S\in \GL_n(K)$ mit $SA=1_n$. - \item Es gibt $T\in \GL_n(K)$ mit $AT=1_n$. + \item Es gibt $S\in \GL_n(K)$ mit $SA=\mathbbm{1}_n$. + \item Es gibt $T\in \GL_n(K)$ mit $AT=\mathbbm{1}_n$. \end{itemize} \end{conclusion} \begin{proof} diff --git a/2. Semester/LAAG 1+2 Semester/Vorlesung LAAG.pdf b/2. 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Semester/LAAG/Vorlesung LAAG.tex b/2. Semester/LAAG/Vorlesung LAAG.tex index c255bf3..ba4532b 100644 --- a/2. Semester/LAAG/Vorlesung LAAG.tex +++ b/2. Semester/LAAG/Vorlesung LAAG.tex @@ -15,6 +15,10 @@ \usepackage{calc} \usepackage{xstring} \usepackage[bb=boondox]{mathalfa} %spezielle Null mit \mathbb{0}, sollte auch mit 1 klappen, aber dafür haben wir \mathbbm{1} +\usepackage{bigdelim} +\newcommand\undermat[2]{% + \makebox[0pt][l]{$\smash{\underbrace{\phantom{% + \begin{matrix}#2\end{matrix}}}_{\text{$#1$}}}$}#2} \usepackage{scalerel,stackengine} \usepackage{tocloft} diff --git a/Material/Matrix_mit_Klammern_unten_und_links.pdf b/Material/Matrix_mit_Klammern_unten_und_links.pdf index 2cf0a5dffee6950b2e4033f3c6ccdfa7ceba35ef..7e747245d02dec934e5252e6224e85fb315c57d6 100644 GIT binary patch delta 27162 zcmV(&K;ggF$^)!|1dt>GGa!?2B`ANbSGjH!K@`mQD>}qN^XPZ|zR(k<_Ql&HInu zUEm{}ke2Lzj?l{rH&~4Bn|rr;yEz|eYdoq~N)fbWPpq~`0?4>T4w@#gkZV0x%uJErPl3e9hBfOco5;*Am^L zIKUS^wK%3EIlx#%TA?YE#t&(X80FUBu0Fdjb8T8T62UbBpKp&|e*fyY!I*L?$GRy3 z?qlobZ)dL!iv1U5ES5=XnM?~H{HFgrb{SNF(J)Ix9r_eEtc1P1loT1=l$VNu{F~W> 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