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Henry Haustein 2018-08-07 19:34:46 +02:00
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2. Semester/Summary LAAG/Ubungsaufgaben LAAG.pdf Normal file
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2. Semester/Summary LAAG/Ubungsaufgaben LAAG.tex Normal file
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\documentclass[ngerman,a4paper]{article}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{enumitem}
\usepackage[left=2.1cm,right=3.1cm,bottom=3cm,top=1cm]{geometry}
\usepackage[ngerman]{babel}
\title{\textbf{\"Ubungsaufgaben f\"ur Lineare Algebra}}
\author{}
\date{}
\begin{document}
\maketitle
\renewcommand{\arraystretch}{1.5}
\section{Eigenwerte und Eigenvektoren}
\begin{align}
\begin{pmatrix}
3 & -1 & 0 \\ 2 & 0 & 0 \\ -2 & 2 & -1
\end{pmatrix},
\begin{pmatrix}
1 & 2 & 3 \\ 4 & 3 & -2 \\ 0 & 0 & 5
\end{pmatrix},
\begin{pmatrix}
8 & 12 & -4 \\ -40 & -60 & 20 \\ -100 & -150 & 50
\end{pmatrix}\notag
\end{align}
\section{Diagonalisierung}
\begin{align}
\begin{pmatrix}
3 & 4 & -3 \\ 2 & 7 & -4 \\ 3 & 9 & -5
\end{pmatrix},
\begin{pmatrix}
3 & 0 & 0 \\ 1 & 2 & 2 \\ 1 & 0 & 4
\end{pmatrix},
\begin{pmatrix}
1 & 0 & 1 \\ 0 & 2 & 0 \\ 1 & 0 & 1
\end{pmatrix}\notag
\end{align}
\section{Trigonalisierung}
\begin{align}
\begin{pmatrix}
3 & 4 & 3 \\ -1 & 0 & -1 \\ 1 & 2 & 3
\end{pmatrix},
\begin{pmatrix}
3 & 0 & -2 \\ -2 & 0 & 1 \\ 2 & 1 & 0
\end{pmatrix},
\begin{pmatrix}
2 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 1 & 2
\end{pmatrix}\notag
\end{align}
\section{Jordan-Normalform}
\begin{align}
\begin{pmatrix}
1 & 1 & 6 & -2 \\ 0 & 1 & -3 & 2 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & -2 & 2
\end{pmatrix},
\begin{pmatrix}
1 & 1 & 0 & 0 \\ -1 & 3 & 0 & 0 \\ -1 & 1 & 1 & 1 \\ -1 & 1 & -1 & 3
\end{pmatrix},
\begin{pmatrix}
1 & 1 & -3 \\ 0 & 1 & 1 \\ 1 & 0 & 4
\end{pmatrix}\notag
\end{align}
\section{Gram-Schmidt-Verfahren}
\begin{align}
B = \left(\begin{pmatrix}3\\1\end{pmatrix}, \begin{pmatrix}2\\2\end{pmatrix}\right),
C = \left(\begin{pmatrix}1\\1\\1\end{pmatrix}, \begin{pmatrix}1\\1\\0\end{pmatrix}, \begin{pmatrix}1\\0\\0\end{pmatrix}\right),
D = \left(\begin{pmatrix}1\\1\\0\\0\end{pmatrix}, \begin{pmatrix}1\\0\\1\\0\end{pmatrix}, \begin{pmatrix}0\\1\\0\\1\end{pmatrix}, \begin{pmatrix}0\\0\\1\\-1\end{pmatrix}\right)\notag
\end{align}
\end{document}