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\begin{document}

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

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\textbf{Blatt 10}\hfill % Nr des Blatts
\textbf{Abgabe 28.01.2016}\\

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\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}
