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KTR-MfI-2-B Mathematik für Informatiker 2/SS17 MfI2.tex
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KTR-MfI-2-B Mathematik für Informatiker 2/SS17 MfI2.tex
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\input{../settings/settings}
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% Mathematik
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\usepackage{amsmath}
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\usepackage{amssymb}
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\usepackage{amsthm}
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\usepackage{mathtools}
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% Referenzen
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\usepackage{hyperref}
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\begin{document}
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\klausur{KTR-MfI2-B Mathematik für Informatik 2)}
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{Prof. Dr. U. Krieger}
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{Sommersemester 17}
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{90}
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{Taschenrechner, zwei von Hand geschriebene DIN-A4-Seiten}
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% Aufgabe 1
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\section{Lineare Gleichungssysteme und Matlab/Octave (8 + 12 Punkte)}
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\begin{enumerate}
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% Aufgabe 1.1
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\item \begin{enumerate}
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\item Gegeben ist das folgende lineare Gleichungssystem:
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\begin{align}
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x_1 + x_3 = 1 \\
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2x_1 + x_2 = 0 \\
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4x_1 + 2x_2 + 4x_3 = 2
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\end{align}
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Geben Sie eine Darstellung in der Form eines Matrix-Vektor-Produkts $A \cdot x = b$ mithilfe einer geeigneten Koeffizientenmatrix $A$, des Vektors $x$ der Variablen sowie der rechten Seite $b$ des Gleichungssystems an.
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% Aufgabe 1.1a
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\item Berechnen Sie die Lösung $x = (\begin{array}{ccc}x_1 & x_2 & x_3\end{array}) \in \mathbb{R}^{3 \times 1}$. Wandeln Sie hierzu das System $(\begin{array}{cc}A & b\end{array})$ in Treppennormalform um und ermitteln Sie anschließend die Lösung mithilfe des Gaußschen Algorithmus.
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% Aufgabe 1.1b
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\item Ist das inhomogene lineare System $A \cdot x = b$ eindeutig lösbar? Begründen Sie.
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\end{enumerate}
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% Aufgabe 1.2
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\item Wir betrachten den gerichteten Graphen $G = (V, E)$ des Google-Rangbildungsverfahrens. Dabei sei $V = \{ 1,2,3,4 \}$ die Menge der Knoten (Webseiten) und $E \subset V \times V$ die Menge der Kanten (Webseitenverknüpfungen durch Hyperlinks). Betrachten Sie hierzu Abbildung~\ref{fig:googlePagerank}.
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$e^t = (1,1,1,1) \in \mathbb{R}^{1 \times 4}$ sei ein Zeilenvektor aus Einsen und $0_4 = (0,0,0,0)^t \in \mathbb{R}^{4 \times 1}$ sei ein Spaltenvektor aus Nullen.
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$x = (x_1, x_2, x_3, x_4)^t \in \mathbb{R}^{4 \times 1}$ ist der Spaltenvektor des Rangwertes $x_i \in [0,1]$ der Seiten $i \in V$.
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\begin{enumerate}
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% Aufgabe 1.2a
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\item Betrachten Sie den Graphen $G$ in Abbildung~\ref{fig:googlePagerank}. Geben Sie die stochastische Matrix $P \in \mathbb{R}^{4 \times 4}$ an, welche auf Basis der gewichteten Adjenzmatrix des Graphen $G$ bei der Berechnung des Google-Rangbildungsverfahrens verwendet wird.
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\begin{figure}[h]
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\centering
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\includegraphics[width=4cm]{./img/google-pagerank.pdf}
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\caption{Graph einer Google-Seitenrangbildung}
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\label{fig:googlePagerank}
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\end{figure}
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% Aufgabe 1.2b
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\item Geben Sie den Code eines Matlab/Octave-Programms an, mit dessen Hilfe diese soeben erzeugte Matrix $P$ dargestellt wird.
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% Aufgabe 1.2c
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\item Betrachten Sie das zum Rangbildungsverfahren gehörende lineare Gleichungssystem
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\begin{equation}
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A \cdot x = b
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\label{eq:lgs}
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\end{equation}
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und geben Sie ein Codesegment an, das sowohl die Systemmatrix $A$ mit der folgenden $2 \times 1$-Blockmatrixstruktur
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\begin{equation}
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A = \left( \begin{array}{c} A_{11} \\ A_{21} \end{array} \right)
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= \left( \begin{array}{c} I_4 - P^t \\ e^t \end{array} \right)
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\in \mathbb{R}^{5 \times 4},
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\end{equation}
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als auch die rechte Seite
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\begin{equation}
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b = \left( \begin{array}{c} 0_4 \\ 1 \end{array} \right)
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\in \mathbb{R}^{5 \times 1}
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\end{equation}
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von \eqref{eq:lgs} in Matlab/Octave-Syntax wiedergibt.
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% Aufgabe 1.2d
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\item Benennen Sie eine Matlab/Octave-Funktion, welche durch die Anwendung des Gaußalgorithmus die Treppennormalform der erweiterten Systemmatrix $(\begin{array}{cc}A & b\end{array})$ berechnet.
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% Aufgabe 1.2e
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\item Geben Sie einen Matlab/Octave-Funktionsanruf am, mit dem die Lösung $x \in \mathbb{R}^4$ des linearen Systems \eqref{eq:lgs} in Matlab/Octave berechnet werden kann.
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\end{enumerate}
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\end{enumerate}
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% Aufgabe 2
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\section{Vektorräume (5 + 7 + 8 Punkte)}
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\begin{enumerate}
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% Aufgabe 2.1
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\item Betrachten Sie die Vektoren
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\begin{equation}
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a = \left( \begin{array}{c} 1 \\ 1 \\ 0 \end{array} \right),
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b = \left( \begin{array}{c} 0 \\ 1 \\ 0 \end{array} \right),
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c = \left( \begin{array}{c} 1 \\ 1 \\ 1 \end{array} \right),
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\end{equation}
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des $\mathbb{F}_2$-Vektorraums $\mathbb{F}_2^3$ auf dem Körper der binären Elemente $\mathbb{F}_2 = \{0,1\}$. Dieser enthält die Operationen $\{\oplus, \odot\}$, definiert als
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$\forall x,y \in \mathbb{F}_2.
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x \oplus y = (x + y) ~\mathrm{mod}~2,
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x \odot y = (x \cdot y)~\mathrm{mod}~2$.
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Sind diese drei Vektoren $\{a, b, c\} \subset \mathbb{F}_2^3$ linear unabhängig oder linear abhängig? Begründen Sie Ihre Antwort durch geeignete algebraische Argumente.
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% Aufgabe 2.2
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\item Betrachten Sie die Monome $e_1(x) = 1$, $e_2(x) = x$ als Elemente des Vektorraums
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\begin{equation}
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V = \mathbb{Q}_2[x]
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= \{p:\mathbb{Q} \rightarrow \mathbb{Q},
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x \mapsto p(x) = b_0 + b_1x + b_2x^2~|~b_0, b_1, b_2 \in \mathbb{Q}\}
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\end{equation}
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der Polynome mit maximalem Grad 2 über dem Körper $\mathbb{K} = \mathbb{Q}$ der rationalen Zahlen. Als Grundoperation stehen in $V$ die punktweise Addition und die Skalarmultiplikation zur Verfügung:
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\begin{align}
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(p + q)(x) &= p(x) + q(x)
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~&\mathrm{mit}~x \in \mathbb{Q}, p(x), q(x) \in V \\
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(\lambda \cdot p)(x) &= (\lambda \cdot b_0) + (\lambda \cdot b_1)x + (\lambda \cdot b_2)x^2
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~&\mathrm{mit}~x, \lambda \in \mathbb{Q}, p(x) \in V
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\end{align}
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\begin{enumerate}
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% Aufgabe 2.2a
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\item Sind die Polynome $v_1 = e_1(x)$ und $v_2 = e_2(x)$ linear abhängige oder linear unabhängige Elemente $v_1, v_2$ des $\mathbb{Q}$-Vektorraums $V = \mathbb{Q}_2[x]$? Begründen Sie Ihre Antwort durch algebraische Argumente.
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%Aufgabe 2.2b
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\item Geben Sie einen Untervektorraum $U_1 \subset V$ von $V$ der Dimension $1$ an.
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\end{enumerate}
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% Aufgabe 2.3
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\item Weisen Sie durch eine geeignete algebraische Argumentation nach, dass die Menge
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\begin{equation}
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U = \{ v = \left( \begin{array}{c}
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x \\
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y \\
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z
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\end{array} \right) \in \mathbb{R}^3 ~|~ y - z = 0 \} \subset \mathbb{R}^3
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\end{equation}
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ein Untervektorraum des reellen Vektorraums $V = \mathbb{R}^3$ ist.
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\end{enumerate}
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% Aufgabe 3
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\section{Lineare Abbildungen (6 + 6 + 4 + 4 Punkte)}
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Betrachten Sie die lineare Abbildung
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\begin{align}
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f : V &\rightarrow W \\
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v = \left( \begin{array}{c} x \\ y \\ z \end{array} \right) &\mapsto
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f(v) = \left( \begin{array}{c} x \\ 2 \cdot (x + z) \\ 2 \cdot x + z \end{array} \right)
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\end{align}
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zwischen den $\mathbb{R}$-Vektorräumen $V = \mathbb{R}^3$ und $W = \mathbb{R}^3$.
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\begin{enumerate}
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% Aufgabe 3.1
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\item Wählen Sie die kanonische Basis $\mathcal{B} = \{ e_1, e_2, e_3 \}$ im $\mathbb{R}$-Vektorraum $V = \mathbb{R}^3$ bzw. $W = \mathbb{R}^3$ aus und geben Sie die der Abbildung $f$ zugeordnete Matrix $A_f$ bzgl. dieser beiden Basen an.
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% Aufgabe 3.2
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\item Bestimmen Sie mithilfe einer geeigneten algebraischen Argumentation oder eines geeigneten Berechnungsverfahrens den Rang $rg(f)$ der Abbildung $f$.
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% Aufgabe 3.3
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\item Bestimmen Sie mithilfe einer geeigneten Formel, die Sie explizit angeben sollen, die Dimension $k = \mathrm{dim}(\mathrm{Ker}(f))$ des Kerns $\mathrm{Ker}(f)$ der Abbildung $f$.
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% Aufgabe 3.4
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\item Geben Sie eine Basis $\mathcal{C} = \{v_1,...,v_k\}$ des Kerns $\mathrm{Ker}(f)$ der Abbildung $f$ an.
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\end{enumerate}
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% Aufgabe 4
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\section{Gruppentheorie und Matrizenalgebra (8 + 6 + 6 Punkte)}
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\begin{enumerate}
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% Aufgabe 4.1
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\item Wir betrachten die Einheitengruppe $E(\mathbb{Z}) = \{1, -1\}$, die aus dem Monoid $(\mathbb{Z}, \cdot, 1)$ der ganzen Zahlen $\mathbb{Z}$ bei Verwendung der Multiplikation $\cdot$ als Grundoperation hervorgeht. Dann definiert die komponentenweise Multiplikation auf $(\mathbb{Z}, \cdot)$ in der Form
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\begin{equation}
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x \otimes y = (a, b) \otimes (c, d)
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= (a \cdot c, b \cdot d)
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~\mathrm{mit}~
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x = (a, b),
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y = (c, d) \in \mathbb{Z} \times \mathbb{Z}
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\end{equation}
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eine neue Multiplikationsoperation $\otimes$ auf dem Produktmonoid $(\mathbb{Z} \times \mathbb{Z}, \otimes, e)$ und
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\begin{equation}
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G = E(\mathbb{Z} \times \mathbb{Z})
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= \{(1, 1), (1, -1), (-1, 1), (-1, -1)\}
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\end{equation}
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wird zur Einheitengruppe.
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\begin{enumerate}
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% Aufgabe 4.1.1
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\item Ergänzen Sie die folgende Verknüpfungstabelle der Teilmenge $U = \{g_1, g_2\} = \{ (1, 1), (1, -1) \} \subset G$, die sich bei der Multiplikation $z_{i,j} = g_i \otimes g_j \in G = E(\mathbb{Z} \times \mathbb{Z})$ der Elemente $g_i, g_j, i, j \in \{1,2\}$ ergeben:
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\begin{center}
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\begin{tabular}{|c|c|c|} \hline
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$\otimes$ & $g_1 = (1, 1)$ & $g_2 = (1, -1)$ \\ \hline
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$g_1 = (1, 1)$ & & \\ \hline
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$g_2 = (1, -1)$ & & \\ \hline
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\end{tabular}
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\end{center}
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% Aufgabe 4.1.2
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\item Wie lautet das inverse Element $g^{-1}_2$ zu $g_2 = (1, -1)$?
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% Aufgabe 4.1.3
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\item Um welche algebraische Teilstruktur der Gruppe $(G, \otimes)$ handelt es sich bei $(U, \otimes)$?
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\end{enumerate}
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% Aufgabe 4.2
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\item Es sei $A = \left( \begin{array}{cc}
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2 & 1 \\
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3 & -10
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\end{array} \right), B = \left( \begin{array}{ccc}
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4 & 1 & 0 \\
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5 & -2 & 1
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\end{array} \right)$.
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\begin{enumerate}
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% Aufgabe 4.2.1
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\item Berechnen Sie das Element $C_{12}$ der Produktmatrix $C = A \cdot B$.
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% Aufgabe 4.2.2
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\item Wie lautet die transponierte Matrix $B^t$ zu $B$?
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\end{enumerate}
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% Aufgabe 4.3
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\item Es sei $A = \left( \begin{array}{ccc}
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1 & 0 & 1 \\
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0 & 1 & -2 \\
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0 & 0 & 1
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\end{array} \right)$.
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Berechnen Sie die inverse Matrix $C = A^{-1}$ zu $A$.
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\end{enumerate}
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% Aufgabe 5
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\section{Lineare Abbildungs- und Eigenwerttheorie (2 + 8 + 10 Punkte)}
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\begin{enumerate}
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% Aufgabe 5.1
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\item Es seien feste Basen in den reellwertigen Vektorräumen $V = \mathbb{R}^k$ und $W = \mathbb{R}^m$ gewählt und
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\begin{equation}
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A_f = \left( \begin{array}{ccc}
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2 & 4 & - 4 \\
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-2 & -5 & 8 \\
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3 & 6 & -18 \\
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0 & 0 & 1
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\end{array} \right)
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\end{equation}
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sei die der linearen Abbildung $f : V \rightarrow W$ zugeordnete Matrix. Geben Sie die Dimensionen $k, m \in \mathbb{N}_0$ der zugrunde liegenden Vektorräume $V$ und $W$ an.
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% Aufgabe 5.2
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\item Berechnen Sie die Determinanten der folgenden Matrizen:
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\begin{equation}
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A = \left( \begin{array}{ccc}
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1 & 0 & 2 \\
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0 & 6 & 0 \\
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2 & 0 & 2
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\end{array} \right),
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B = \left( \begin{array}{ccc}
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6 & 0 & 0 \\
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5 & 4 & 0 \\
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3 & 2 & 1
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\end{array} \right)
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\end{equation}
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% Aufgabe 5.3
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\item Bestimmen Sie das charakteristische Polynom
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$p(\lambda) = \lambda^3 + p_2\lambda^2 + p_1\lambda + p_0$
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der folgenden Matrix $A$ und berechnen Sie mit dessen Hilfe alle Eigenwerte
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$\lambda_i, i \in \{1, 2, 3\}$ der Matrix.
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\begin{equation}
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A = \left( \begin{array}{ccc}
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0 & 3 & 0 \\
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4 & -1 & 0 \\
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0 & 0 & 1
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\end{array} \right)
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\end{equation}
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\end{enumerate}
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\end{document}
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After Width: | Height: | Size: 16 KiB |
Loading…
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Reference in New Issue
Block a user