%\documentclass[12pt,a4paper, german]{article}
%\usepackage{a4kopka}
\documentclass[12pt,a4paper]{article}
%\usepackage{ngerman}
%\usepackage{fleqn}
\usepackage{array}
\usepackage[latin1]{inputenc}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
%\usepackage{mathbbol}
\usepackage{amstext}
%\usepackage{fancyhdr}
\usepackage{relsize}
%\usepackage{epic}
\usepackage{graphics}
\usepackage{graphicx}
\usepackage[T1]{fontenc}
\usepackage{a4wide}
%\usepackage{makeidx}\usepackage{german}
%\usepackage{textcomp}
\usepackage{ulem}


\newcommand{\D}{\displaystyle}
%%%%%%%%%%%%%%%%%%%%

\newcommand{\IR}{{\mathbb{R}}}
\newcommand{\IN}{{\mathbb{N}}}
\newcommand{\IZ}{{\mathbb{Z}}}
\newcommand{\IF}{{\mathbb{F}}}
\newcommand{\IK}{{\mathbb{K}}}
\newcommand{\IQ}{{\mathbb{Q}}}
\newcommand{\IC}{{\mathbb{C}}}
\newcommand{\IP}{{\mathbb{P}}}
\newcommand{\IE}{{\mathbb{E}}}

\newcommand{\ep}{{\varepsilon}}
\newcommand{\ph}{{\varphi}}
\newcommand{\thet}{{\vartheta}}
\newcommand{\rh}{{\varrho}}
\newcommand{\de}{{\delta}}
\newcommand{\la}{{(}}
\newcommand{\ra}{{)}}
\newcommand{\Om}{{\Omega}}
\newcommand{\al}{{\alpha}}
\newcommand{\be}{{\beta}}
\newcommand{\ga}{{\gamma}}
\newcommand{\gwh}{{\widehat{\gamma}}}
\newcommand{\om}{{\omega}}
\newcommand{\La}{{\Lambda}}
\newcommand{\Ga}{{\Gamma}}
\newcommand{\De}{{\Delta}}
\newcommand{\wh}{{\widehat}}
\newcommand{\Kern}{{\operatorname{Kern}}}
\newcommand{\Bild}{{\operatorname{Bild}}}
\newcommand{\foh}{{\mathfrak{h}}}

\newcommand{\nach}{{\rightarrow}}
\newcommand{\Nach}{{\,\rightarrow\,}}
\newcommand{\Fou}{{\mathcal{F}}}
\newcommand{\sk}{{\,|\,}}

\def\be{\begin{eqnarray*}}
\def\ee{\end{eqnarray*}}
\def\bel{\begin{eqnarray}}
\def\eel{\end{eqnarray}}

\def\ex{\exists}
\def\all{\forall}

\def\a{\alpha}
\def\eh{\frac{1}{2}}
\def\A{\mathcal{A}}
\def\R{\mathbb{R}}
\def\N{\mathbb{N}}\def\NN{\mathbb{N}}
\def\E{\mathbb{E}}
\def\P{\mathbb{P}}
\def\C{\mathbb{C}}
\def\B{\mathcal{B}}
\def\F{\mathcal{F}}
\def\S{\mathcal{S}}
\def\Z{\mathbb{Z}}
\def\O{\mathcal{O}}
\def\e{\epsilon}
\def\ep{\tilde{\delta}}
\def\d{\delta}
\def\lm{\lambda}
\def\id{\mathbb{I}}
\def\Sp{\textrm{Sp}}
\def\lili{\lim\limits}
\def\nn{\nonumber}
\def\fa{\bigwedge\limits}
\def\one{1}
\def\chiop{\chi_{\{\delta>L\}}}
\def\eqiv{\Longleftrightarrow}
\def\lra{\longrightarrow}
\newcommand{\ab}[1]{\left( #1\right)}

\renewcommand\Re{\operatorname{Re}}
\renewcommand\Im{\operatorname{Im}}

%\newcommand{\cosh}{\mbox{cosh}}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remarks}[theorem]{Remarks}
%%%%%%%%%%%%%%%%%%%%

%\addtolength{\voffset}{-15pt} \addtolength{\textheight}{-10pt}
%\addtolength{\textheight}{-11pt} \addtolength{\headsep}{6pt}

%\setlength{\parskip}{10pt plus 2pt minus 1pt}
\setlength{\parindent}{0pt}
\newcommand{\wk}{\mbox{$\,<$\hspace{-5pt}\footnotesize )$\,$}}
\thispagestyle{empty} %\addtolength{\voffset}{15pt}

%\setlength{\textheight}{10in}

\begin{document}

\rule{\textwidth}{0.3pt}\\

\begin{center}
\textbf{\large Mathematik für Wirtschaftsinformatik}\\
\end{center}
\textbf{Wintersemester 2015/2016\hfill Prof. Dr. M. Keller}\\

\rule{\textwidth}{0.3pt}\\

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Hier geht's los
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\textbf{Blatt 10}\hfill % Nr des Blatts
\textbf{Abgabe 28.01.2016}\\

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%








\begin{itemize}
\item[(1)] Seien $n,m\in \IN$ und $A$ eine Matrix mit $m$ Zeilen und $n$ Spalten. Zeigen Sie, dass \[\Kern(A):=\{x\in\IR^n:Ax=0\}\] ein Untervektorraum des $\IR^n$ und 
\[\Bild(A):=\{Ax: x\in\IR^n\}\] ein Untervektorraum des $\IR^m$ ist. Hinweis: Sie dürfen Verwenden, dass Matrixmultiplikation linear ist, d.h. für $\alpha,\beta\in\IR,u,v\in \IR^n$ gilt 
$A(\alpha u+\beta v)=\alpha Au+\beta Av$.

\item[(2)] Untersuchen Sie, ob die folgenden Mengen von Vektoren Basen des $\IR^3$ sind.
\begin{itemize}
\item[(a)]\[\begin{pmatrix}
1\\0\\ 1\\
\end{pmatrix},
\begin{pmatrix}
2\\1\\ 0\\
\end{pmatrix},
\begin{pmatrix}
3\\1\\ 1\\
\end{pmatrix}\]
\item[(b)]\[\begin{pmatrix}
1\\0\\ 0\\
\end{pmatrix},
\begin{pmatrix}
1\\1\\ 1\\
\end{pmatrix},
\begin{pmatrix}
0\\1\\ 1\\
\end{pmatrix},
\begin{pmatrix}
0\\0\\ 1\\
\end{pmatrix}\]
\item[(c)]
\[\begin{pmatrix}
1\\0\\ 1\\
\end{pmatrix},
\begin{pmatrix}
1\\1\\ 0\\
\end{pmatrix},
\begin{pmatrix}
3\\1\\ 0\\
\end{pmatrix}\]
\end{itemize}

\item[(3)] Bringen Sie das folgende lineare Gleichungssystem auf Stufenform und bestimmen Sie anschließend alle Lösungen. 
 \begin{align*}
  \begin{matrix}
     x_1&+x_2 &+2x_3 &+x_4&= & 0 & \\
    x_1 &+2x_2 & & &= & 0 &\\
3x_1 &+2x_2&+x_3 &+4x_4&= & 0 & \\
 5x_1 & +5x_2 &+3x_3 &+5x_4&= & 0 & \\  
  \end{matrix}
\end{align*} 


\item[(4)]
Gegeben Sei die folgende Matrix: \begin{align*}
A=\begin{pmatrix}
1& 1& 1& 1\\
1& 0& 1& 0\\
0& 1& 0& 1\\
\end{pmatrix}.
\end{align*}
Geben Sie eine Basis von $\Kern(A)$ an.  
\end{itemize}
\end{document}