From a3ce85b3b8e8fcc8778faf30924e73a8805ab5c6 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Thomas=20M=C3=BCller?= Date: Mon, 28 May 2018 20:50:13 +0200 Subject: [PATCH 01/15] Substitude space indentation with tabs --- Template/SSxx Klausurvorlage.tex | 39 ++++++++++++++++---------------- 1 file changed, 19 insertions(+), 20 deletions(-) diff --git a/Template/SSxx Klausurvorlage.tex b/Template/SSxx Klausurvorlage.tex index a489a2f..97824a9 100644 --- a/Template/SSxx Klausurvorlage.tex +++ b/Template/SSxx Klausurvorlage.tex @@ -1,26 +1,25 @@ \input{../settings/settings} \begin{document} + \klausur{Kürzel Modulname (z.B. KTR-Datkomm-B Datenkommunikation)} + {Professor (z.B. Prof. Dr. U. Krieger)} + {Semester (z.B. Wintersemester 13/14)} + {Bearbeitungs zeit (z.B. 90)} + {Zugelassene Hilfsmittel (z.B. Taschenrechner)} - \klausur{Kürzel Modulname (z.B. KTR-Datkomm-B Datenkommunikation)} - {Professor (z.B. Prof. Dr. U. Krieger)} - {Semester (z.B. Wintersemester 13/14)} - {Bearbeitungs zeit (z.B. 90)} - {Zugelassene Hilfsmittel (z.B. Taschenrechner)} + \begin{enumerate} + \item Aufgabe 1 - \begin{enumerate} - \item Aufgabe 1 + \item{ + \begin{enumerate} + \item Aufgabe 2 a) + \item Aufgabe 2 b) + \end{enumerate} + } - \item{ - \begin{enumerate} - \item Aufgabe 2 a) - \item Aufgabe 2 b) - \end{enumerate} - } - - \item{ - Hier könnte dein Bild stehen: - %\image{1}{Capture3.PNG}{DNS-Anfrage}{DNS-Anfrage} - } - \end{enumerate} -\end{document} \ No newline at end of file + \item{ + Hier könnte dein Bild stehen: + %\image{1}{Capture3.PNG}{DNS-Anfrage}{DNS-Anfrage} + } + \end{enumerate} +\end{document} From 954be68ac9a14edc16a2e1f06f1dae78fc1b40e2 Mon Sep 17 00:00:00 2001 From: Anna Date: Mon, 28 May 2018 21:18:50 +0200 Subject: [PATCH 02/15] SEDA-MobIS-B WS 16/17 --- SEDA-MobIS-B/Capture1.PNG | Bin 0 -> 269743 bytes SEDA-MobIS-B/Capture2.PNG | Bin 0 -> 166833 bytes SEDA-MobIS-B/SEDA-MobIS-B.tex | 64 ++++++++++++++++++++++++++++++++++ 3 files changed, 64 insertions(+) create mode 100644 SEDA-MobIS-B/Capture1.PNG create mode 100644 SEDA-MobIS-B/Capture2.PNG create mode 100644 SEDA-MobIS-B/SEDA-MobIS-B.tex diff --git a/SEDA-MobIS-B/Capture1.PNG b/SEDA-MobIS-B/Capture1.PNG new file mode 100644 index 0000000000000000000000000000000000000000..282e24501e91b3a6ae91c4437bbb3dae2748dc4b GIT binary patch literal 269743 zcmWh!1yob-8{TM8Kt+FaC`!Yml@^qc7?M&0MMPS9W26!yF_0D*ARrBbG$RxpF}fz5 zBSvoYsQ>==ob8^wvvck__q*@&JO6g70)@n0IAGs3B+3P^&X&s_>=xb4`saCFQks1H|S5B~z%K1q?@iK~6=GN6-Co`stH$+MLd_5o0 zOUoN*sB*fJguGl?{)W791?NT0Z<>-8@|ig!XdhHq3yFugN-xs#w%RW5%HHDL=tPJP zM(vshsk}RwgYS;j&%2%S1`%If>r8v!$BS8s`ki)OD;|=Z2p!3fuPDbqRXrednam&nSE~k*5h5&zFGdB)?7N%uqj(cv|xpS`t7Ek7*wJv zP6wPKi+@Ng;1(04F)x1Q0ewq54k9GJxrHn#Wt_{-q8bEh1 zdW|fU9j@4g|`Ke0HkCX6;&sO!?q7!0caYC%a$~A_g z@DCvSX9ksSn6hQN>*2et^22Z4uCg0!JroK+OB6}!G!amu|v%fuD< z-9pYiw{_j3vTCs-bvrGlw69tuSd^qso#*sUS(*#nq0+%iJ}!p>BM1GBks+~&+bRU> z(JN0kXKPMkl0kO0J}z;XC_qYY`#xa8?gTnCLi#{8MwG!;NnUT~IEgbJECfFfz2qiV zFGQc1cgQ$&=LL$l<@?Ux2}o;$3uLE`CudB$S#vP?!4A(RF^<2D-dd8NXV5FNYTDkr 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b/SEDA-MobIS-B/SEDA-MobIS-B.tex @@ -0,0 +1,64 @@ +\input{../settings/settings} + +\begin{document} + + \klausur{SEDA-MobIS-B Modellierung betrieblicher Informationssysteme)} + {Prof. Dr. H.-G. Fill} + {Wintersemester 16/17} + {nicht genannt} + {nicht genannt} + + \begin{enumerate} + \item Pflichtteil + \begin{enumerate} + \item Methodische Grundlagen der objektorientierten und der prozessorientierten Modellierung [45 Punkte] + \begin{enumerate} + \item a) Hinsichtlich der Modellaussage bzw. mit der mit einem Modell verfolgten Zielsetzung lassen sich \underline{vier} Formen von Modellen unterscheiden. Nennen und erläutern Sie diese Formen kurz. \textit{[8]} + \item b) Erläutern Sie die Begriffe \textit{Metapher}, \textit{Metamodell} und \textit{Modellsystem} sowie deren Beziehung zueinander. \textbf{[9]} + \item c) Erläutern Sie sowohl die \textit{Metapher} der \textit{objektorientierten} als auch der \textit{prozessorientierten} Modellierung und zeigen Sie die Unterschiede auf. \textbf{[10]} + \item d) Modellieren Sie sowohl die wesentlichen Teile des Metamodells des UML-Klassendiagramms als systemorientiertes Metamodell auf Basis des nachfolgend angegebenen Meta-Modells. \textbf{[8]} + \image{1}{Capture1.PNG}{Meta-Modell}{Meta-Modell} + \textit{Hinweis: Auf die Spezifikation der Metaobjekte Stereotyp, Merkmal und Bedingung kann verzichtet werden.} + \item e) Gegeben ist das nachfolgende Petri-Netz: + \image{2}{Capture2.PNG}{Petri-Netz}{Petri-Netz} + Erläutern Sie \underline{kurz} allgemein das Schaltverhalten eines Petri-Netzes. Finden Sie zudem eine Schaltreihenfolge für das angegebene Petri-Netz, mit dem Ziel, den Zustand P6 mit vier Marken zu belegen. Spezifizieren Sie diese Schaltreihenfolge schrittweise und geben Sie nach jedem Schaltvorgang die Markierung aller Zustände (P1 bis P6) sowie die zulässigen Übergänge (T1 bis T4) an. Orientieren Sie sich dazu an nachfolgendem Muster: \textbf{[10]} + \begin{table}[H] + \begin{tabular}{|c|c|c|c|c|c|c|c|} + \hline + \textit{geschachtelter Übergang} & \textit{P1} & \textit{P2} & \textit{P3} & \textit{P4} & \textit{P5} & \textit{P6} & \textit{zulässige Übergänge} \\ + \hline + & & & & & & & \\ + \hline + \end{tabular} + \end{table} + \end{enumerate} + \end{enumerate} + \item Wahlteil + + Bearbeiten Sie \textbf{entweder} Thema 1 \textbf{oder} Thema 2. + \begin{enumerate} + \item Thema 1: Datenmodellierung [45 Punkte] + + Gegeben ist nachfolgender Sachverhalt einer Kfz-Werkstatt: + + \textit{Mitarbeiter einer Kfz-Werkstatt sind entweder Mechaniker oder Servicekräfte. Servicekräfte übernehmen die Serviceannahme und fertigen hierzu Serviceaufträge. Ein Serviceauftrag verweist neben der verantwortlichen Servicekraft auch auf das Fahrzeug sowie auf den beauftragenden Kunden. Der Kunde kann die identische Person wie der Halter des Fahrzeugs sein, muss dies aber nicht. Jedes Fahrzeug hat genau einen Halter, wobei ein Halter auch mehrere Fahrzeuge besitzen kann. Zur Durchführung des Serviceauftrages wird jeder Arbeitsschritt mit dem durchführenden Mechaniker und einem Tätigkeitsschlüssel dokumentiert. Ein Tätigkeitsschlüssel verweist auf eine standardisierte Tätigkeitsbeschreibung mit geschätzter Dauer. Optional kann auch ein Ersatzteil in einem Arbeitsschritt verwendet werden und ist diesem zuzuordnen. Für die Dauer des Service kann der Kunde ein Leihfahrzeug aus dem Fuhrpark der Werkstatt buchen. Für diesen Fall wird dem Serviceauftrag ein Leihauftrag mit Verweis auf das Leihfahrzeug zugeordnet.} + \begin{enumerate} + \item a) Skizzieren und erläutern Sie anhand eines praktischen Beispiels die vier Formen der Generalisierung, die im \textit{Strukturierten Entity-Relationship-Modell} (SERM) unterschieden werden. Zur Erläuterung können Sie auch eine Mengendarstellung verwenden. \textbf{[10]} + \item b) Modellieren Sie den gegebenen Sachverhalt (Kfz-Werkstatt) unter Verwendung des \textit{Strukturierten Entity-Relationship-Modell} (SERM). \textbf{[15]} + \item c) Das \textit{Entity-Relationship-Modell} (ERM) weist eine Reihe von Schwächen auf, die sich sowohl auf das ERM als Darstellungsform wie auch auf das ERM als Analyseinstrument beziehen. Erläutern Sie Vorteile des SERM gegenüber dem ERM. Belegen Sie Ihre Aussagen jeweils mit einem aussagekräftigen Beispiel aus dem gegebenen Sachverhalt der Kfz-Werkstatt. \textbf{[12]} + \item d) Zentrale Anforderungen an ein Modellsystem stellen \textit{Konsistenz} und \textit{Vollständigkeit} sowie \textit{Struktur- und Verhaltenstreue} dar. Erläutern Sei diese Anforderungen allgemein und überprüfen Sie deren Einhaltung für das Modellsystem aus Teilaufgabe b) (SER-Schema zum Sachverhalt der Kfz-Werkstatt). \textbf{[8]} + \end{enumerate} + \item Thema 2: Objekt- und prozessorientierte Modellierung [45 Punkte] + Gegeben ist nachfolgender Sachverhalt des Beratungsunternehmens \textit{ProSOM}: + + \textit{Sie sind Beschäftigter des Beratungsunternehmens ProSOM, das sich auf die vollständige Spezifikation von Geschäftsprozessen spezialisiert hat. Auf Fachmessen stellen Sie potentielle Kunden Ihre Leistungen anhand einer Fallstudie vor. Nach Anfrage eines Kunden wird ein Angebot mit Stundensatz sowie zur Verfügung gestellten Ressourcen unterbreitet. Kommt es zum Vertragsabschluss, wird das Modellierer-Team mit der Prozessspezifikation beauftragt. Die Prozessspezifikation beinhaltet die Erfassung der ist-Prozesse, die Spezifikation der Soll-Prozesse sowie die abschließende Übergabe der Prozessdokumention. Nach erfolgter Übergabe wird dies dem Vertrieb zurückgemeldet und durch diesen die Finanzabteilung mit Rechnungsstellung und Überwachung des Zahlungseinganges beauftragt. Hierzu werden die erbrachten Stunden durch die Finanzabteilung beim zuständigen Modellierer-Team abgefragt.} + \begin{enumerate} + \item a) Definieren Sie mit dem \textit{Semantischen Objektmodell} (SOM) ein \textit{initiales Interaktionsschema} mti Diskurswelt- und Umweltobjekten sowie eine geeignete Durchführungstransaktion zur Leistungserbringung des Beratungsunternehmens \textit{ProSOM}. \textbf{[3]} + \item b) Verfeinern Sie das Interaktionsschema aus Teilaufgabe a) durch eine geeignete \underline{Transaktionszerlegung}. Erläutern Sie das der Zerlegung zugrundeliegende Koordinationsprinzip. \textbf{[10]} + \item c) Decken Sie die Lenkung im Inneren des Diskursweltobjektes aus Teilaufgabe a) durch seine Zerlegung auf und erstellen Sie das entsprechende Interaktionsschema. Erläutern Sie das von den resultierenden Objekten verwendete Koordinationsprinzip. \textbf{[10]} + \item d) Ein betriebliches Informationssystem kann durch \underline{vier} allgemeine Sichten vollständig beschrieben werden. Nennen und erläutern Sie diese Sichten kurz. Zeigen Sie, belegt anhand des Fallbeispiels \textit{ProSOM} sowie Ihrer Lösungen zu den Teilaufgaben a) bis c), welche Sichten durch die SOM-Methodik auf Aufgabenebene unterstützen werden. \textbf{[12]} + \item e) Bei der Bildung von Sichten kann grundsätzlich zwischen \textit{integraler Modellbildung} und \textit{getrennter Erfassung von Sichten} unterschieden werden. Stellen Sie die beiden Arten der Sichtenbildung kurz vor und erläutern Sie typische Vor- und Nachteile. Welche Form der Sichtenbildung wird im Rahmen der SOM-Methodik verwendet? Begründen Sie Ihre Antwort. \textbf{[10]} + \end{enumerate} + \end{enumerate} + \end{enumerate} +\end{document} \ No newline at end of file From 6ccf7ced2b3978c096da11ffc9dc73070dbe8757 Mon Sep 17 00:00:00 2001 From: jnoehl Date: Mon, 28 May 2018 21:34:14 +0200 Subject: [PATCH 03/15] Add ISDL-ITCHANGE-M-SS17 --- .../ISDL-ITCHANGE-M-SS17.tex | 96 +++++++++++++++++++ 1 file changed, 96 insertions(+) create mode 100644 ISDL-ITCHANGE-M - Management IT-bedingter Veränderungen/ISDL-ITCHANGE-M-SS17.tex diff --git a/ISDL-ITCHANGE-M - Management IT-bedingter Veränderungen/ISDL-ITCHANGE-M-SS17.tex b/ISDL-ITCHANGE-M - Management IT-bedingter Veränderungen/ISDL-ITCHANGE-M-SS17.tex new file mode 100644 index 0000000..9037238 --- /dev/null +++ b/ISDL-ITCHANGE-M - Management IT-bedingter Veränderungen/ISDL-ITCHANGE-M-SS17.tex @@ -0,0 +1,96 @@ +\input{../settings/settings} + +\begin{document} + +\klausur{ISDL-ITCHANGE-M: Management IT-bedingter Veränderungen} +{Prof. Dr. Tim Weitzel} +{Sommersemester 17} +{Bearbeitungszeit 90 Minuten} +{} + +\part{Pflichtteil} + +\section{Aufgabe 1 - IT-bedingte Veränderungen von Arbeitssystemen (45 Punkte)} + +Sie sind von der ISS Bank als Berater zur Einführung des neuen Kernbankensystems angestellt worden. Das System soll alle Prozesse des Unternehmens abbilden. Der Assistent der Geschäftsführung, Herr Pflichtbewusst, ist der Verantwortliche des Unternehmens und bittet Sie um einen Vorschlag für das weitere Vorgehen. Für das weitere Vorgehen der Bank ist es Ihre Aufgabe ein Change-Management-Konzept zu entwickeln. + +\begin{enumerate} +\item \textbf{Erklären Sie daher Herrn Pflichtbewusst drei Schritte der \glqq Work System Method\grqq und wie diese anzupassen sind, sodass ein Change-Management-Konzept entwickelt werden kann. (15 Punkte)} + + +Basierend auf Ihrem Vorschlag beschreibt Ihnen Herr Pflichtbewusst das zu ändernde Arbeitssystem der Wertpapierkundenberatung wie folgt: + +\textit { \glqq In einem Beratungstermin werden das Anlagerisiko und der finanzielle Hintergrund des Kunden erfasst. Hierzu füllt der Kundenberater einen papierbasierten Fragebogen aus. Basierend auf diesen Informationen wird dem Kunden eine Wertpapieranlagestrategie vorgeschlagen, welche in einem weiteren Termin mit dem Kunden diskutiert wird. Für die Anlagestrategie entwickelt der Kundenberater einen eigenen Vorschlag, dokumentiert diesen in einem Excel-Dokument und druckt die Ergebnisse für den Kunden aus. Zusätzlich werden dem Kunden unterschiedliche Informationsdokumente über Aktien und Wertpapierfonds ausgehändigt, die der Kundenberater in der Marketingabteilung abholen muss. Nimmt der Kunde den Vorschlag an, wird ein Beratungsprotokoll erstellt, welches der Kunde und der Kundenberater unterschreiben. Hierzu füllt der Kundenberater im Gespräch ein entsprechendes Formular händisch aus. Das Beratungsprotokoll wird im Anschluss archiviert. Die unterschriebene Wertpapieranlagestrategie wird zur Ausführung an einen Sachbearbeiter der Wertpapierabteilung weitergegeben. Der Kunde bekommt nach erfolgreicher Ausführung eine E-Mail, welche der Sachbearbeiter manuell über sein E-Mail-Postfach verschicken muss.\grqq} + +\item \textbf{Erklären sie kurz die einzelnen Elemente eines Arbeitssystems und erstellen Sie für das im Zitat beschriebene Arbeitssystem einen Work-System-Snapshot. (20 Punkte)} +\\Hinweis: Planen Sie für den Snapshot ausreichend Platz ein, da dieser in Aufgabe 1c weiterentwickelt werden soll. + + +Herr Pflichtbewusst beschreibt Ihnen außerdem wie sich das Arbeitssystem durch das neue Kernbankensystem verändern wird. + +\glqq Mit dem neuen System sollen die Informationen des Kunden direkt im System erfasst werden, die Wertpapieranlagestrategie soll automatisiert durch das System vorgeschlagen werden und der Kundenberater soll nur noch kleine Änderungen manuell vornehmen können. Mit dem System kann eine Informationsmappe mit allen Informationen direkt ausgedruckt werden. Ebenso kann das Beratungsprotokoll automatisch erstellt werden, welches im Anschluss der Kundenberater scannt und im System archiviert. Nach Unterschrift durch den Kunden kann der Kundenberater den Auftrag direkt zur Ausarbeitung freigeben, welcher im Anschluss automatisiert ausgeführt wird. Der Kunde wird nach erfolgreicher Ausführung automatisch per E-Mail durch das System benachrichtigt.\grqq + +\item \textbf{Markieren und erläutern Sie in dem erstellten Snapshot die Veränderungen, welche sich durch das neue System in diesem Arbeitssystem ergeben. Diskutieren Sie, welche Veränderungen aus Change-Management-Sicht die wichtigsten sind und auf welche Veränderungen Sie den Fokus einer Change-Management-Strategie legen würden. (10 Punte)} +\end{enumerate} + +\part {Wahlteil - 3 aus 5 Aufgaben} + +\section{Aufgabe 2 - Change Management (15 Punkte)} + +Herr Pflichtbewusst ist besorgt darüber, wie die Einführung des Kernbankensystems gestaltet werden kann. Er bittet Sie um Hilfe, um ihm aufzuzeigen, welche unterschiedlichen Phasen im Rahmen einer Systemeinführung wichtig sind. + +\begin{enumerate} +\item \textbf{ Erläutern sie Herrn Pflichtbewusst die fünf Phasen einer IT-Implementierung nach Schwarz et al. (2014) und den acht Phasen nach Kotter (1996). Nennen Sie jeweils die Phase und erläutern Sie jeweils kurz die wesentlichen Eigenschaften der Phase. Diskutieren Sie Unterschiede und Gemeinsamkeiten der beiden Modelle. Welche unterschiedlichen Perspektiven beschreiben die beiden Modelle? (15 Punkte)} +\end{enumerate} + +\section{Aufgabe 3 - Top-Management-Support (15 Punkte)} + +Herr Pflichtbewusst schlägt vor, dass die Führung des Projektteams aus dem Leiter der IT-Abteilung und ihm selbst als Assistent der Geschäftsführung bestehen soll. Herr Pflichtbewusst gibt an, dass der Leiter der IT-Abteilung eine hohe IT-Expertise, aber geringe Prozesskenntnisse besitzt. Er selbst denkt, dass er ausreichende Prozesskenntnisse besitzt. Er ist sich jedoch nicht sicher, ob sowohl der IT-Leiter als auch er bei allen Mitarbeitern der Bank akzeptiert sind. + +\begin{enumerate} +\item \textbf{ Nennen Sie vier Kriterien die für ein gutes Change-Management-Team erfüllt sein müssen (nach Kotter 1996). Erfüllen Herr Pflichtbewusst und der IT-Leiter diese Kriterien? Welche Eigenschaften müssten weitere Mitglieder des Change-Management-Teams besitzen? Begründen Sie Ihre Antwort. (10 Punkte)} +\end{enumerate} +Der Vorstand der Bank möchte über den Projektverlauf informiert werden, jedoch selbst nicht aktiv eingebunden werden. + +\begin{enumerate} +\item \textbf{Erläutern Sie die grundlegende Bedeutung von Top-Management-Support für IT-bedingte Veränderungen im Unternehmen. Ist die Einführung des Kernbankensystems (=hohe Abhängigkeit zwischen einzelnen Prozessen und Abteilungen) die aktive Einbindung des Vorstands notwendig? Begründen Sie Ihre Antwort. (5 Punkte)} +\end{enumerate} + +\section{Aufgabe 4 - Akzeptanz von IT-bedingten Veränderungen (15 Punkte)} + +Um geeignete Change-Management-Maßnahmen vorschlagen zu können, führen Sie Gespräche mit den Mitarbeitern der ISS-Bank. Eine Sachbearbeiterin der Wertpapierabteilung (Alter 25 Jahre, wenig Erfahrung im Umgang mit IT) äußert sich dabei wie folgt: +\\\glqq Mein Chef sagt, das neue System wird unsere Arbeit gut unterstützen. Ich bin mir hier noch unsicher. In den Schulungen hatte ich das Gefühl, dass das neue System nicht einfach z nutzen sein wird und dass es nicht wirklich meine Arbeit als Sachbearbeiterin in der Wertpapierabteilung unterstützen wird. Wobei das Projektteam viel Hilfestellung anbietet. Das ist auch besonders wichtig, weil das neue System wirklich sehr kompliziert ist.\grqq + +\begin{enumerate} +\item \textbf{Erläutern Sie bezugnehmen auf das Zitat die Reaktion der Mitarbeiterin mit Hilfe der Unified Theory of Acceptance and Use of Technology (UTAUT). (5 Punkte)} + +\item \textbf {Erläutern Sie zudem, welche Faktoren zur Erläuterung der individuellen Akzeptanz in welchen Phasen einer IT-Implementierung relevant sind (beziehen Sie sich hierzu auf das Fünf-Phasen-Modell von Schwarz et al. (2014). (10 Punkte)} +\end{enumerate} + +\section{Aufgabe 5 - Widerstände gegen IT-bedingten Veränderungen (15 Punkte)} + +Herr Pflichtbewusst ist besorgt, dass die Mitarbeiter in der ISS-Bank Widerstände gegenüber dem neuen Kernbanksystem zeigen könnten. + +\begin{enumerate} +\item \textbf{Erklären Sie Herrn Pflichtbewusst vier unterschiedliche Arten von Widerstand und geben Sie je nach Kategorie ein Beispiel für eine Verhaltensweise die Mitarbeiter zeigen können. Welche Art von Widerständen sind eher positiv und welche eher negativ einzuschätzen? Begründen Sie Ihre Antwort. (7,5 Punkte)} + + +Herr Pflichtbewusst bittet Sie zudem zu erklären, wie Widerstände entstehen. + +\item \textbf{Erklären Sie Herrn Pflichtbewusst daher das \glqq Multilevel-Model of User Resistance\grqq (Lapointe \& Rivard 2005). Erläutern Sie das Model am Beispiel der ISS Bank und er geplanten Kernbankensystemeinführung wie oben beschrieben. Nehmen Sie Bezug auf das Beispiel und erläutern Sie das Modell beispielhaft am möglichen Widerstand einer der beteiligten Personen. (7,5 Punkte)} +\end{enumerate} + +\section{Aufgabe 6 - Training und Unterstützung während der Einführung (15 Punkte)} + +Herr Pflichtbewusst bittet Sie um einen Vorschlag, wie das Training und die Schulungen im Rahmen der Systemeinführung gestaltet werden sollten. + +\begin{enumerate} +\item \textbf{Erklären Sie Herrn Pflichtbewusst die Eigenschaft und den Unterschied zwischen \glqq Self-Learning\grqq , \glqq Collaborative Learning \grqq und \glqq Expert trainer instruction learning\grqq . Wann ist welche Lernart sinnvoll? Welche Lernart würden Sie für die Einführung des Kernbankensystems empfehlen (hohe technische Komplexität und Abhängigkeit zwischen Prozessen bzw. Abteilungen)? \textbf {Begründen Sie Ihre Antwort.} (7,5 Punkte)} + + +Als eine weitere Maßnahme schlagen Sie \glqq Peer Advice Networks\grqq vor. + +\item \textbf{Erklären Sie Herrn Pflichtbewusst diese Maßnahme und welche positiven Effekte dieser Maßnahme während der Einführung von Systemen haben kann. Welche Personen würden Sie zur Übernahme dieser Rollel auswählen? Begründen Sie Ihre Antwort mit Bezug auf die Rolle von sozialen Netzwerken während der Einführung von Systemen. (7,5 Punkte)} +\end{enumerate} +\end{document} + From 06fbbdb8db141ff08e1f06837a03b04eb436eee7 Mon Sep 17 00:00:00 2001 From: Anna Date: Mon, 28 May 2018 21:44:07 +0200 Subject: [PATCH 04/15] =?UTF-8?q?Bildgr=C3=B6=C3=9Fe=20ge=C3=A4ndert?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- SEDA-MobIS-B/SEDA-MobIS-B.tex | 49 +++++++++++++++++++---------------- 1 file changed, 26 insertions(+), 23 deletions(-) diff --git a/SEDA-MobIS-B/SEDA-MobIS-B.tex b/SEDA-MobIS-B/SEDA-MobIS-B.tex index fb00f54..40563c4 100644 --- a/SEDA-MobIS-B/SEDA-MobIS-B.tex +++ b/SEDA-MobIS-B/SEDA-MobIS-B.tex @@ -5,7 +5,7 @@ \klausur{SEDA-MobIS-B Modellierung betrieblicher Informationssysteme)} {Prof. Dr. H.-G. Fill} {Wintersemester 16/17} - {nicht genannt} + {unbekannt} {nicht genannt} \begin{enumerate} @@ -13,23 +13,26 @@ \begin{enumerate} \item Methodische Grundlagen der objektorientierten und der prozessorientierten Modellierung [45 Punkte] \begin{enumerate} - \item a) Hinsichtlich der Modellaussage bzw. mit der mit einem Modell verfolgten Zielsetzung lassen sich \underline{vier} Formen von Modellen unterscheiden. Nennen und erläutern Sie diese Formen kurz. \textit{[8]} - \item b) Erläutern Sie die Begriffe \textit{Metapher}, \textit{Metamodell} und \textit{Modellsystem} sowie deren Beziehung zueinander. \textbf{[9]} - \item c) Erläutern Sie sowohl die \textit{Metapher} der \textit{objektorientierten} als auch der \textit{prozessorientierten} Modellierung und zeigen Sie die Unterschiede auf. \textbf{[10]} - \item d) Modellieren Sie sowohl die wesentlichen Teile des Metamodells des UML-Klassendiagramms als systemorientiertes Metamodell auf Basis des nachfolgend angegebenen Meta-Modells. \textbf{[8]} + \item Hinsichtlich der Modellaussage bzw. mit der mit einem Modell verfolgten Zielsetzung lassen sich \underline{vier} Formen von Modellen unterscheiden. Nennen und erläutern Sie diese Formen kurz. \textbf{[8]} + \item Erläutern Sie die Begriffe \textit{Metapher}, \textit{Metamodell} und \textit{Modellsystem} sowie deren Beziehung zueinander. \textbf{[9]} + \item Erläutern Sie sowohl die \textit{Metapher} der \textit{objektorientierten} als auch der \textit{prozessorientierten} Modellierung und zeigen Sie die Unterschiede auf. \textbf{[10]} + \item Modellieren Sie sowohl die wesentlichen Teile des Metamodells des UML-Klassendiagramms als systemorientiertes Metamodell auf Basis des nachfolgend angegebenen Meta-Modells. \textbf{[8]} \image{1}{Capture1.PNG}{Meta-Modell}{Meta-Modell} + \textit{Hinweis: Auf die Spezifikation der Metaobjekte Stereotyp, Merkmal und Bedingung kann verzichtet werden.} - \item e) Gegeben ist das nachfolgende Petri-Netz: - \image{2}{Capture2.PNG}{Petri-Netz}{Petri-Netz} + \item Gegeben ist das nachfolgende Petri-Netz: + \image{1}{Capture2.PNG}{Petri-Netz}{Petri-Netz} Erläutern Sie \underline{kurz} allgemein das Schaltverhalten eines Petri-Netzes. Finden Sie zudem eine Schaltreihenfolge für das angegebene Petri-Netz, mit dem Ziel, den Zustand P6 mit vier Marken zu belegen. Spezifizieren Sie diese Schaltreihenfolge schrittweise und geben Sie nach jedem Schaltvorgang die Markierung aller Zustände (P1 bis P6) sowie die zulässigen Übergänge (T1 bis T4) an. Orientieren Sie sich dazu an nachfolgendem Muster: \textbf{[10]} \begin{table}[H] - \begin{tabular}{|c|c|c|c|c|c|c|c|} - \hline - \textit{geschachtelter Übergang} & \textit{P1} & \textit{P2} & \textit{P3} & \textit{P4} & \textit{P5} & \textit{P6} & \textit{zulässige Übergänge} \\ - \hline - & & & & & & & \\ - \hline - \end{tabular} + \begin{center} + \begin{tabular}{|c|c|c|c|c|c|c|c|} + \hline + \textbf{geschachtelter Übergang} & \textbf{P1} & \textbf{P2} & \textbf{P3} & \textbf{P4} & \textbf{P5} & \textbf{P6} & \textbf{zulässige Übergänge} \\ + \hline + & & & & & & & \\ + \hline + \end{tabular} + \end{center} \end{table} \end{enumerate} \end{enumerate} @@ -43,21 +46,21 @@ \textit{Mitarbeiter einer Kfz-Werkstatt sind entweder Mechaniker oder Servicekräfte. Servicekräfte übernehmen die Serviceannahme und fertigen hierzu Serviceaufträge. Ein Serviceauftrag verweist neben der verantwortlichen Servicekraft auch auf das Fahrzeug sowie auf den beauftragenden Kunden. Der Kunde kann die identische Person wie der Halter des Fahrzeugs sein, muss dies aber nicht. Jedes Fahrzeug hat genau einen Halter, wobei ein Halter auch mehrere Fahrzeuge besitzen kann. Zur Durchführung des Serviceauftrages wird jeder Arbeitsschritt mit dem durchführenden Mechaniker und einem Tätigkeitsschlüssel dokumentiert. Ein Tätigkeitsschlüssel verweist auf eine standardisierte Tätigkeitsbeschreibung mit geschätzter Dauer. Optional kann auch ein Ersatzteil in einem Arbeitsschritt verwendet werden und ist diesem zuzuordnen. Für die Dauer des Service kann der Kunde ein Leihfahrzeug aus dem Fuhrpark der Werkstatt buchen. Für diesen Fall wird dem Serviceauftrag ein Leihauftrag mit Verweis auf das Leihfahrzeug zugeordnet.} \begin{enumerate} - \item a) Skizzieren und erläutern Sie anhand eines praktischen Beispiels die vier Formen der Generalisierung, die im \textit{Strukturierten Entity-Relationship-Modell} (SERM) unterschieden werden. Zur Erläuterung können Sie auch eine Mengendarstellung verwenden. \textbf{[10]} - \item b) Modellieren Sie den gegebenen Sachverhalt (Kfz-Werkstatt) unter Verwendung des \textit{Strukturierten Entity-Relationship-Modell} (SERM). \textbf{[15]} - \item c) Das \textit{Entity-Relationship-Modell} (ERM) weist eine Reihe von Schwächen auf, die sich sowohl auf das ERM als Darstellungsform wie auch auf das ERM als Analyseinstrument beziehen. Erläutern Sie Vorteile des SERM gegenüber dem ERM. Belegen Sie Ihre Aussagen jeweils mit einem aussagekräftigen Beispiel aus dem gegebenen Sachverhalt der Kfz-Werkstatt. \textbf{[12]} - \item d) Zentrale Anforderungen an ein Modellsystem stellen \textit{Konsistenz} und \textit{Vollständigkeit} sowie \textit{Struktur- und Verhaltenstreue} dar. Erläutern Sei diese Anforderungen allgemein und überprüfen Sie deren Einhaltung für das Modellsystem aus Teilaufgabe b) (SER-Schema zum Sachverhalt der Kfz-Werkstatt). \textbf{[8]} + \item Skizzieren und erläutern Sie anhand eines praktischen Beispiels die vier Formen der Generalisierung, die im \textit{Strukturierten Entity-Relationship-Modell} (SERM) unterschieden werden. Zur Erläuterung können Sie auch eine Mengendarstellung verwenden. \textbf{[10]} + \item Modellieren Sie den gegebenen Sachverhalt (Kfz-Werkstatt) unter Verwendung des \textit{Strukturierten Entity-Relationship-Modell} (SERM). \textbf{[15]} + \item Das \textit{Entity-Relationship-Modell} (ERM) weist eine Reihe von Schwächen auf, die sich sowohl auf das ERM als Darstellungsform wie auch auf das ERM als Analyseinstrument beziehen. Erläutern Sie Vorteile des SERM gegenüber dem ERM. Belegen Sie Ihre Aussagen jeweils mit einem aussagekräftigen Beispiel aus dem gegebenen Sachverhalt der Kfz-Werkstatt. \textbf{[12]} + \item Zentrale Anforderungen an ein Modellsystem stellen \textit{Konsistenz} und \textit{Vollständigkeit} sowie \textit{Struktur- und Verhaltenstreue} dar. Erläutern Sei diese Anforderungen allgemein und überprüfen Sie deren Einhaltung für das Modellsystem aus Teilaufgabe b) (SER-Schema zum Sachverhalt der Kfz-Werkstatt). \textbf{[8]} \end{enumerate} \item Thema 2: Objekt- und prozessorientierte Modellierung [45 Punkte] Gegeben ist nachfolgender Sachverhalt des Beratungsunternehmens \textit{ProSOM}: \textit{Sie sind Beschäftigter des Beratungsunternehmens ProSOM, das sich auf die vollständige Spezifikation von Geschäftsprozessen spezialisiert hat. Auf Fachmessen stellen Sie potentielle Kunden Ihre Leistungen anhand einer Fallstudie vor. Nach Anfrage eines Kunden wird ein Angebot mit Stundensatz sowie zur Verfügung gestellten Ressourcen unterbreitet. Kommt es zum Vertragsabschluss, wird das Modellierer-Team mit der Prozessspezifikation beauftragt. Die Prozessspezifikation beinhaltet die Erfassung der ist-Prozesse, die Spezifikation der Soll-Prozesse sowie die abschließende Übergabe der Prozessdokumention. Nach erfolgter Übergabe wird dies dem Vertrieb zurückgemeldet und durch diesen die Finanzabteilung mit Rechnungsstellung und Überwachung des Zahlungseinganges beauftragt. Hierzu werden die erbrachten Stunden durch die Finanzabteilung beim zuständigen Modellierer-Team abgefragt.} \begin{enumerate} - \item a) Definieren Sie mit dem \textit{Semantischen Objektmodell} (SOM) ein \textit{initiales Interaktionsschema} mti Diskurswelt- und Umweltobjekten sowie eine geeignete Durchführungstransaktion zur Leistungserbringung des Beratungsunternehmens \textit{ProSOM}. \textbf{[3]} - \item b) Verfeinern Sie das Interaktionsschema aus Teilaufgabe a) durch eine geeignete \underline{Transaktionszerlegung}. Erläutern Sie das der Zerlegung zugrundeliegende Koordinationsprinzip. \textbf{[10]} - \item c) Decken Sie die Lenkung im Inneren des Diskursweltobjektes aus Teilaufgabe a) durch seine Zerlegung auf und erstellen Sie das entsprechende Interaktionsschema. Erläutern Sie das von den resultierenden Objekten verwendete Koordinationsprinzip. \textbf{[10]} - \item d) Ein betriebliches Informationssystem kann durch \underline{vier} allgemeine Sichten vollständig beschrieben werden. Nennen und erläutern Sie diese Sichten kurz. Zeigen Sie, belegt anhand des Fallbeispiels \textit{ProSOM} sowie Ihrer Lösungen zu den Teilaufgaben a) bis c), welche Sichten durch die SOM-Methodik auf Aufgabenebene unterstützen werden. \textbf{[12]} - \item e) Bei der Bildung von Sichten kann grundsätzlich zwischen \textit{integraler Modellbildung} und \textit{getrennter Erfassung von Sichten} unterschieden werden. Stellen Sie die beiden Arten der Sichtenbildung kurz vor und erläutern Sie typische Vor- und Nachteile. Welche Form der Sichtenbildung wird im Rahmen der SOM-Methodik verwendet? Begründen Sie Ihre Antwort. \textbf{[10]} + \item Definieren Sie mit dem \textit{Semantischen Objektmodell} (SOM) ein \textit{initiales Interaktionsschema} mit Diskurswelt- und Umweltobjekten sowie eine geeignete Durchführungstransaktion zur Leistungserbringung des Beratungsunternehmens \textit{ProSOM}. \textbf{[3]} + \item Verfeinern Sie das Interaktionsschema aus Teilaufgabe i. durch eine geeignete Transaktionszerlegung. Erläutern Sie das der Zerlegung zugrundeliegende Koordinationsprinzip. \textbf{[10]} + \item Decken Sie die Lenkung im Inneren des Diskursweltobjektes aus Teilaufgabe i. durch seine Zerlegung auf und erstellen Sie das entsprechende Interaktionsschema. Erläutern Sie das von den resultierenden Objekten verwendete Koordinationsprinzip. \textbf{[10]} + \item Ein betriebliches Informationssystem kann durch \underline{vier} allgemeine Sichten vollständig beschrieben werden. Nennen und erläutern Sie diese Sichten kurz. Zeigen Sie, belegt anhand des Fallbeispiels \textit{ProSOM} sowie Ihrer Lösungen zu den Teilaufgaben i. bis ii., welche Sichten durch die SOM-Methodik auf Aufgabenebene unterstützen werden. \textbf{[12]} + \item Bei der Bildung von Sichten kann grundsätzlich zwischen \textit{integraler Modellbildung} und \textit{getrennter Erfassung von Sichten} unterschieden werden. Stellen Sie die beiden Arten der Sichtenbildung kurz vor und erläutern Sie typische Vor- und Nachteile. Welche Form der Sichtenbildung wird im Rahmen der SOM-Methodik verwendet? Begründen Sie Ihre Antwort. \textbf{[10]} \end{enumerate} \end{enumerate} \end{enumerate} From e4cfef38c2bf19f6107db6941814c929952d90e3 Mon Sep 17 00:00:00 2001 From: clemens Date: Mon, 28 May 2018 22:20:01 +0200 Subject: [PATCH 05/15] add WATkom ws1617 --- .../WS1617 Datkomm.tex | 111 ++++++++++++++++++ .../WS1617/arp-text.png | Bin 0 -> 51692 bytes .../WS1617/arp.png | Bin 0 -> 40357 bytes .../WS1617/dns.png | Bin 0 -> 36852 bytes .../WS1617/http.png | Bin 0 -> 148739 bytes .../WS1617/ping.png | Bin 0 -> 113099 bytes 6 files changed, 111 insertions(+) create mode 100644 KTR-Datkomm-B Datenkommunikation/WS1617 Datkomm.tex create mode 100644 KTR-Datkomm-B Datenkommunikation/WS1617/arp-text.png create mode 100644 KTR-Datkomm-B Datenkommunikation/WS1617/arp.png create mode 100644 KTR-Datkomm-B Datenkommunikation/WS1617/dns.png create mode 100644 KTR-Datkomm-B Datenkommunikation/WS1617/http.png create mode 100644 KTR-Datkomm-B Datenkommunikation/WS1617/ping.png diff --git a/KTR-Datkomm-B Datenkommunikation/WS1617 Datkomm.tex b/KTR-Datkomm-B Datenkommunikation/WS1617 Datkomm.tex new file mode 100644 index 0000000..ed4e5a2 --- /dev/null +++ b/KTR-Datkomm-B Datenkommunikation/WS1617 Datkomm.tex @@ -0,0 +1,111 @@ +\input{../settings/settings} +\usepackage{amsmath} +\usepackage{amssymb} + +\begin{document} + + \klausur{KTR-Datkomm-B Datenkommunikation} + {Prof. Dr. U. Krieger} + {Wintersemester 16/17} + {90} + {Taschenrechner} + + \begin{enumerate} + \item Aufgabe ? + \begin{enumerate} + \item Bei der Datenübertragung werde die in der Vorlesung skizzierte Technik des Bitstopfens verwendet. Rekonstruieren sie für die empfangene Bitfolge $$y(x) = 111110011101001111110101$$, welche fehlerfrei übertragen wurde, die zugehörige gesendete Nutzdatenfolge $x$. + \item Betrachten sie den linearen Blockcode $$ C=\{0000, 1010, 1101, 0111\}$$. + \begin{enumerate} + \item Berechnen sie die Hamming-Distanz $d(c)$ des Codes. + \item Wie viele Übertragungsfehler $r$ können in $C$ erkannt werden? + \item Wie viele Übertragungsfehler $s$ können in $C$ korrigiert werden? + \item Geben sie ein binäres Wort $z \in \mathbb{B}^4$ an, bei dem man Übertragungsfehler zwar erkennen kann, diese Fehler aber nicht korrigieren kann. + \end{enumerate} + \end{enumerate} + \item Nehmen sie an, bei einer Datenübertragung erhält der Empfänger die binäre Zeichenfolge: + $$y=111100110010$$ + Führen sie eine Fehlererkennung nach dem Prinzip zyklischer Blockcodes (CRC) unter Verwendung des Generatorpolynoms $$G(X)=x^3+x^2+x$$ durch. Zu welchem Ergebnis kommt das Verfahren bei der Prüfung des Auftretens eines Übertragungsfehlers? + + \item Das Socket-Programmiermodell + \begin{enumerate} + \item Nennen sie die Kurzdefinition des Internet Sockets, die in der Domäne des Internets verwendet wird. + \item Welcher Socket lässt sich mit der Java-API (java.net) nicht programmieren? + \item Nennen sie den blockierenden Methodenaufruf innerhalb der run-Methode aus Codeausschnitt 1. + \item Mit welcher Javaklassen lassen sich IPAdressen einfach verwalten? + \item Nennen sie das Transportschichtprotokoll, dass bei der Programmierung eines MulticastSocket zum Einsatz kommt + \end{enumerate} + + \newpage + \item Aufgabe 5 (5+5+5+5 Punkte) + + Wir betrachten die Auswertung von Internetverkehrsmessungen mit Wireshark. + \begin{enumerate} + \item Betrachten sie die folgende Messung von Ethernet-Rahmen einer ping Anwendung in \autoref{img:ping} + \image{0.6}{WS1617/ping.png}{Ping}{img:ping} + Geben sie einen Display-Filter an, der in dem dargestellten IPNachrichtenaustausch diejenigen ICMPPakete aus den Ethernetrahmen herausfilter, welche die in \autoref{img:ping} angezeigte IPAdresse des Zielrechners als IPQuellen oder IPZieladresse eines Nachrichtenaustausches verwenden. + \item Betrachten sie die Messung zwischen HTTPServer und HTTPClient in \autoref{img:http}. + \image{0.6}{WS1617/http.png}{HTTP}{img:http} + \begin{enumerate} + \item Welche Funktion des HTTPProtokolls wird durch den Austausch der zweiten HTTPNachricht, die vom Rechner mit der IPAdresse 128.119.245.12 gesendet wird, realisiert. Wie lautet eine vollständige URL des im Server gesuchten Objekts 'lab2-4.html' der entsprechenden ersten GET-Anfrage? + \item Wie viele HTTP-Server werden in der in \autoref{img:http} gezeigten HTTP-Messung angefragt? Welche Bedeutung haben die Nummern 4307 bzw. 80 in der dargestellten 1. HTTP-Transakton mit der Methode GET? Begründen sie kurz ihre Antworten. + \end{enumerate} + \end{enumerate} + \item Aufgabe ? + + \autoref{img:dns} zeigt eine DNS-Anfrage zur Namensauflösung der Domäne tum.de auf dem LINUX-Rechner des Kommunikationslabors. + \image{0.6}{WS1617/dns.png}{DNS}{img:dns} + \begin{enumerate} + \item Wie lautet die verwendete Befehlsequenz, die dieses Ergebnis erzeugt hat? + \item Von welchem Server wurde die dargestellte Antwort erzeugt? Handelt es sich hierbei um einen autoritativen DNS Server? Begründen sie kurz ihre Antwort. + \item Welcher Unterschied besteht zwischen einem NS- und einem A-Datensatz in der ausgegeben Antwort der DNS-Anfrage? + \item Betrachten sie den in \autoref{img:arp} dargestellten EthernetRahmen einer ARP-Anfrage des Rechners mit der IPAdresse 192.168.1.105, der eine IPNachricht an den Rechner mit der IPadresse 192.168.1.1 senden will. + \image{0.6}{WS1617/arp-text.png}{ARP}{img:arp} + \begin{enumerate} + \item Geben sie die hexadezimale Darstellung der Zieladresse (Destination) A1 an, die im Paketkopf des Ethernetrahmens verwendet wird. + \item Geben sie die hexadezimale Darstellung der physikalische Zieladresse (Target MAC address) MAC1 an, die in der ARP Anfrage-Nachricht verwendet wird. + \item Geben sie die IPZieladresse (Target IP address) IP1 an, die in der ARP Anfragenachricht verwendet wird. + \end{enumerate} + \end{enumerate} + \newpage + \item Aufgabe 1 (4+4+6+6 Punkte) + \begin{enumerate} + \item Geben sie in punktierter Dezimalnotation der Form a.b.c.d die erste und letzte IP-Adresse innerhlab der Adressraums der IP-Subnetzes 124.55.64.9/29 an, die in diesem Subnetz einem Endsystem zugeweisen werden kann. + \item Wir betrachten die klassengebunden Adressierung eines IPv4-Netzes. Geben sie für die IPv4-Adressen 192.168.0.2 und 139.1.0.1 jeweils mit Begründung an, zu welcher Klassen die Adresse gehört, und ob es sich dabei um eine öffentliche oder private Ipadresse handelt. + \item Eine Firma besitzt die IPAdressblock 212.18.181.0/24 und hat 29 Subnetze mit jeweils gleicher Anzahl n von Adressen gebildet, beginnend bei Adresse 212.18.181.0 und ohne zwischen den Subnetzen Adressen ungenutzt zu lassen. Geben sie die Subnetzmaske in punktierter Dezimalnotation der Form a.b.c.d und die Anzahl n aller vorhandenen Adressen pro Subnetz an. + \item Wir betrachten jetzt zufällige Medienzugriffsverfahren der Datensicherungsschicht eines IP-Protokolstapels. Geben sie kurze Erklärungen als Antworten auf die folgenden Fragen: + \begin{enumerate} + \item Erläutern sie das prinzip des klassischen Aloha-Protokolls und seiner Kollisionsauflösung und verwenden sie dazu nach Möglichkeit eine geeignete Skizze + \item Welche wesentlichen Vorteil hat Slotted Aloha gegenüber reinem Aloha? + \item Wodurch wird dieser Vorteil erreicht? + + \end{enumerate} + \end{enumerate} + \item Aufgabe ? + + \begin{enumerate} + \item + wir betrachten nun die wichtigesten Protokolle des TCP/IP Protokollstapels. Ordnen sie die Eigenschaften aus der folgenden Liste den Protokollen TCP und UDP zu. Geben sie nur die Eigenschaften an, die auf das jeweilige Protokoll zutreffen. + + \begin{itemize} + \item Protokoll der Darstellungsschicht + \item Protokoll der Vermittlungsschicht + \item Protokoll der Sicherungsschicht + \item Bereitstellung eines flussgesteuerten Bytetransport + \item Bereitstellung eines verbindungslosen Dienstes + \item Bereitstellung eines verbindungsorientierten Dienstes + \item Bereitstellung eines zuverlässigen Dienstes + \item Bereitstellung eines unzuverlässigen Dienstes + \item Bereitstellung eines Dienstes mit Überlastungskontrolle + \item Standardlänge des Paketkopfs von 20 Byte + \item Standardlänge des Paketkopfs von 8 Byte + \item Standardlänge des Paketkopfs von 128 Byte + \end{itemize} + \item ??? + \item Internetnutzer verwenden meist symbolische Rechnernamen bei der Nutzung moderner Dienste. In diesem Zusammenhang stellen IPAdressen der an einem solchen Nachrichtenaustaschprogramm beteiligten Rechner logische Adressen der Vermittlungsschicht dar. Innerhalb eines LANs oder WANs werden dagegegen die Rechner mit ihren physikalischen Adressen bezeichnet. + \begin{enumerate} + \item Nennen sie die dienste bzw. Protokolle, die die schrittweise Übersetzung von symbolischen Namen in physikalische Adresse eines Rechner bewerkstelligen. + \item Geben sie dabei für jeden genannten Dienst bzw jedes genannte Protkoll an, welcher Teil der Übersetzung dadurch geregelt wird. + \end{enumerate} + \end{enumerate} + \end{enumerate} +\end{document} \ No newline at end of file diff --git a/KTR-Datkomm-B Datenkommunikation/WS1617/arp-text.png b/KTR-Datkomm-B Datenkommunikation/WS1617/arp-text.png new file mode 100644 index 0000000000000000000000000000000000000000..c08cf24890154e25eded20c4dae80d3f019af287 GIT binary patch literal 51692 zcmWh!1yCDp5Do6`PH=aZ7I!DX-QB&oySo%E?pEC0DNdodyA><#pMNsBx%ra0%kKB~ z?Rz_y7&R4HG-P6A004j{FDIn|06_hKoUbCnLyne_FzEmQ!>xo+@r1E}XI& zFwW3oZq6Ro+&=9L?|Q!9h2-`d6@Ko&K7Z>{dGAtt3EX>TKAU@cSbN`Gi_rMfL*DoI z`fcq*=I`)<{|Cj|^Z1LwdshF+l!e2AA7ht~hTPYk9YbttIaahjlK%1KN4h@vhNx$S zKObA2H$i_a@2fuoum8OG$-enA<^SW`ILZ=y^AlZXZwTrxvN}zoM&9{L81#1#>F%fO 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zSc{a{2S{Xt&a|~9&*)a|6=L0BE)%^!01_a=scl@{c>Zce(*^{5s#L<0;70ETHvh0< zPrSYmI(%XFjz!0lR4Q&{+}OfG`2R!$C>g)|lSC|Bt>q0~Ia01r*+J3qONH4#z3X7T zDOz8iG96nqso$k7ERN?jk3d?o`_TdpbF=b&@~hYgNzYr_>bH^7QA2bKIw~%uW$y~C zDbo<07Lq52$ON!qdI4Y@UX0v@kdjaXgPrg`ucj$-hFqvFep(^n1mcc@IxHpKblXNB zg(8&#fbXr>8wd~uv1DXchYC*d$uZRHyMFAw>pMlpJ`rGW%7QJya`{E-9pEnIAyu-` zTW_5zDFL*wftXsn|BA3S%5Q=<1VN0PTs2|QCMIAA4Ojdoo9s7Ure0L}A!nNytgDL;}tuf2e^^ao@&G z5_IaNOHceV;j05{b;~-*7{q_8R*q zjyx3iTzlX{U4!!jh7uoYgYrYxFE~ Date: Mon, 28 May 2018 22:20:21 +0200 Subject: [PATCH 06/15] =?UTF-8?q?=09gel=C3=B6scht:=20=20=20=20=20=20=20../?= =?UTF-8?q?SNA-ASN-M=20Analyse=20sozialer=20Netzwerke/graphs-2015/.graph6.?= =?UTF-8?q?dot.swp?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../graphs-2015/.graph6.dot.swp | Bin 1024 -> 0 bytes 1 file changed, 0 insertions(+), 0 deletions(-) delete mode 100644 SNA-ASN-M Analyse sozialer Netzwerke/graphs-2015/.graph6.dot.swp diff --git a/SNA-ASN-M Analyse sozialer Netzwerke/graphs-2015/.graph6.dot.swp b/SNA-ASN-M Analyse sozialer Netzwerke/graphs-2015/.graph6.dot.swp deleted file mode 100644 index 1c7498cfd6ceaa5ede7fdc3f1a3c9640178d6ada..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 1024 zcmeH@K@P$o5Jg#f3@!juN!7UQ0c6v7fYDAUDFp-6#-3iIE?D#koz=_#$^7KqDvv&J zB~P+u#qG|F-5rhOQ<1J?u2t1N*Sj)>K8ytauV8G3*`kHP;gqIGUZ8X4Gig1vx)DvQ w#f3K>-767KVX^2aa!q-eDGmfC;-gPORr|w_kFpE7T- Date: Mon, 28 May 2018 22:23:11 +0200 Subject: [PATCH 07/15] Add GEI exam --- EESYS-GEI-B/WS1718 EESYS-GEI-B.tex | 106 +++++++++++++++++++++++++++++ EESYS-GEI-B/schaltbild.png | Bin 0 -> 25377 bytes 2 files changed, 106 insertions(+) create mode 100644 EESYS-GEI-B/WS1718 EESYS-GEI-B.tex create mode 100644 EESYS-GEI-B/schaltbild.png diff --git a/EESYS-GEI-B/WS1718 EESYS-GEI-B.tex b/EESYS-GEI-B/WS1718 EESYS-GEI-B.tex new file mode 100644 index 0000000..208f566 --- /dev/null +++ b/EESYS-GEI-B/WS1718 EESYS-GEI-B.tex @@ -0,0 +1,106 @@ +\input{../settings/settings} + +\begin{document} + \klausur{EESYS-GEI-B Grundlagen der Energieinformatik} + {Prof. Dr. Thorsten Staake} + {Wintersemester 17/18} + {90} + {Taschenrechner} + + \begin{enumerate} + \item \textbf{Grundlagen der Energietechnik (total = 10)} + + \begin{enumerate} + \item Wie viel Energie ist in folgenden System enthalten? (6)? + \begin{enumerate} + \item Elektrofahrrad mit Fahrerin, $m=105 kg$, $v=25 km/h$ (kinetische Energie; einfacher Fall, also nur Translation ohne Berücksichtigung der Drehung der Räder) + \item Vollständig geladene Smartphone-Batterie mit einer Spannung von $3,8 V$ und einer Batteriekapazität von $2900 mAh$ (elektrische Energie) + \item Einer Tasse Tee Inhalt Wasser mit $c=4182 J/(kg \cdot K)$, $Volumen=0,4 L$, Temperatur $95°C$, Raumtemperatur $22°C$ (Wärmeenergie gegenüber Raumtemeperatur) + \end{enumerate} + \item Wie viel Energie (Heizwert oder Brennwert in kWh) ist in einem Liter Heizöl enthalten? Wie viel $CO_{2}$ entsteht bei dessen vollständiger Verbrennung? (2) + \item Wie hoch ist der Gesamtwirkungsgrad des folgenden Systems: Der Wirkungsgrad eines Elektromotors eines Elektroautos beträgt 95\%, und der Wirkungsgrad des Kohlekraftwerks, das den Strom erzeugt, beträgt 40\%. (2) + \end{enumerate} + + \item \textbf{Elektrizitätsversorgung (total = 14)} + \begin{enumerate} + \item Ungefähr 18\% des Energieverbrauchs („final energy consumption“) in OECD-Ländern wird mit Strom gedeckt. Zugleich verursacht der Stromverbrauch rund 30\% des Primärenergieverbrauchs aus. Wodurch erklärt sich der große Unterschied? (2) + \item Insbesondere bei der Kohleverstromung spricht man häufig von „hohen nicht-internalisierten Kosten“. Nennen Sie drei wesentliche, nicht-internalisierte Kosten bei der Verstromung von Kohle und schlagen Sie konkrete Maßnahmen vor, um diese Kosten zu internalisieren. (3) + \item Nach welchem technischen Prinzip „erkennen“ Kraftwerke in einem konventionellen Wechselstromnetz, dass zur Aufrechterhaltung der Netzstabilität geringfügig mehr oder weniger Strom eingespeist werden sollte? Benennen und erklären Sie die Prinzip kurz. (3) + \item Elektrische Energie lässt sich in großem Maßstab nur schwer unmittelbar speichern. Dennoch sind Energiespeicher für die Stromversorgung wichtig. Welche technischen Ansätze für den Aufbau entsprechender Speicher gibt es, und in welcher Form liegt die gespeicherte Energie jeweils vor? Nennen Sie vier gängige Ansätze. (4) + \item Warum sind mit der Vorhaltung von Regelenergie sehr hohe Kosten verbunden? (2) + \end{enumerate} + + \newpage + \item \textbf{Stromnetzen (total = 14)}\\ + Gegeben sei das Ersatzschaltbild einer Taschenlampe (Abbildung 1) mit folgenden Parametern: + \begin{itemize}[noitemsep] + \item Spannung der Batteriezelle $U_{B} = 3.6V$ + \item Innenwiderstand der Batterie $R_{i} = 2 \Omega$ + \item Widerstand zur Modellierung der Selbstentladung $R_{e} = 1000 \Omega$ + \item Widerstand der Glühbirne $R_{g} = 24 \Omega$ + \end{itemize} + \begin{enumerate} + \item Bestimmen Sie dir Ströme $I_{B}$,$I_{e}$ und $I_{g}$. (6) + \item Bestimmen Sie die Verluste in der Batterie im Betrieb. (3) + \item Bestimmen Sie den Gesamtwirkungsgrad des Systems im Betrieb. (3) + \item Ist ein hoher oder ein niedriger Widerstand für $R_{i}$ wünschenswert? Was gilt für $R_{e}$?\\ + Begründen Sie Ihre Antwort kurz qualitativ. (2) + \end{enumerate} + + \image{0.7}{schaltbild.png}{Ersatzschaltbild einer Taschenlampe}{Ersatzschaltbild einer Taschenlampe} + + \item \textbf{IT (total = 12)} + \begin{enumerate} + \item Zeichen Sie ein Schaltbild eines CMOS-Inverters auf Transistoren-Ebene. (2) + \item Zeichnen Sie ein Schaltbild eines CMOS-NOR-Gatters auf Transistoren-Ebene. (2) + \item Der Energieverbrauch eines CMOS-Inverters setzt sich – vereinfacht betrachtet – zusammen aus den Großen $P_{Switching}$, $P_{Short_Circuit}$ und $P_{Leakage}$. Erklären Sie, wie die Verbräuche entstehen. (3) + \item Welchen Einfluss hat die Strukturgröße von integrierten Schaltungen auf den Energieverbrauch je Transistor? Was ist der Mechanismus dahinter? (2) + \item Moderne Prozessoren passen ihre Versorgungsspannung dynamisch an die erforderliche Rechenleistung an, um Energie zu sparen. Geben Sie eine Formel an, die den Zusammenhang zwischen Versorgungsspannung und Energieverbrauch von CMOS-Logik beschreibt. Warum wählt man nicht gleich das niedrigste Spannungsniveau und spart sich die dynamische Anpassung im Betrieb? (3) + \end{enumerate} + + \newpage + \item \textbf{Demand Side Management (total = 10)} + \begin{enumerate} + \item Warum ist Demand Side Management sinnvoll... + \begin{enumerate} + \item ...aus der Sicht der Erzeugerseite? Nennen Sie zwei Gründe. (2) + \item ...aus Sicht der Netzbetreiber? Nennen Sie zwei Gründe? (2) + \end{enumerate} + \item In wie weit können Smart Meter helfen, Lastverschiebungsprogramme in Haushalten durchzusetzen, auch ohne dass der Energieversorger direkt auf einzelne elektrische Verbraucher zugreift? (3) + \item An den Strombörsen treten mitunter negative Strompreise auf. Was sind die Voraussetzungen für dieses Phänomen? Bei negativen Strompreisen werden Marktakteure für den Energieverbrauch „belohnt“; warum ist das nicht notwendigerweise ein Fehler des Systems? (3) + \end{enumerate} + + \item \textbf{Energiemärkte (total = 22)} + \begin{enumerate} + \item Was versteht man bei Strommärkten unter dem Begriff „Unbundling“? (2) + \item Welche Aspekte eines „Unbundling“ könnten zu niedrigeren, welche zu höheren Endverbraucherpreisen führen? Geben Sie jeweils zwei Aspekte an. (4) + \item Ein wettbewerblich organisierter Strommarkt, in dem Kunden den Anbieter frei wählen können, erfordert häufig Finanzprodukte zum Handel von Energie/Strom. Weshalb? (3) + \item Elektrizität wird in Deutschland zum Teil außerbörslich gehandelt (auch Freiverkehrs- oder OTC-Handel genannt) und zum Teil an der Börse gehandelt. Nennen Sie je einen Vor- und einen Nachteil des börslichen Handels. (2) + \item Was versteht man im Bereich der Stromerzeugung unter dem Merit-Order-Prinzip? Erklären Sie das Prinzip kurz. (2) + \item Zeichnen Sie einen Merit-Order-Graphen für die folgenden Kraftwerke mit den gegebenen Grenzkosten und Kapazitäten: (6)\\ + \begin{table}[h] + \centering + \begin{tabular}{clccc} + i. & Kernkraft & 5 EUR/MWh; & 15 GW \\ + ii. & Braunkohle & 40 EUR/MWh; & 17 GW \\ + iii. & Steinkohle & 60 EUR/MWh; & 20 GW \\ + iv. & Gas „Combined Cycle“ Typ 1 & 50 EUR/MWh; & 5 GW \\ + v. & Gas „Combined Cycle“ Typ 2 & 65 EUR/MWh; & 10 GW \\ + vi. & Gas konventionell & 70 EUR/MWh; & 5 GW \\ + vii. & Öl & 100 EUR/MWh; & 2 GW \\ + \end{tabular} + \end{table} + \end{enumerate} + + \item \textbf{Rebound-Effekte (total = 8)} + \begin{enumerate} + \item Erklären Sie, was in Bezug auf Technologien zur Steigerung der Energieeffizienz unter den folgenden Begriffen zu verstehen ist und nennen Sie je ein Beispiel. (6) + \begin{enumerate} + \item Direkte Rebound-Effekte + \item Indirekte Rebound-Effekte + \item Makroökonomische Rebound-Effekte + \end{enumerate} + \item Stellen Sie sich vor, Sie arbeiten bei einer Lobby-Organisation für Windkraftanlagen. Haben Sie Interesse daran, den Rebound-Effekt möglichst kleinzureden oder dessen Bedeutung möglichst stark zu betonen? Begründen Sie Ihre Antwort. (2) + \end{enumerate} + \end{enumerate} +\end{document} \ No newline at end of file diff --git a/EESYS-GEI-B/schaltbild.png b/EESYS-GEI-B/schaltbild.png new file mode 100644 index 0000000000000000000000000000000000000000..2ca87c438f0797435a2a78d125545c32328e54d0 GIT binary patch literal 25377 zcmZ^KWmFtovMo*rcXzko?ry;e?(XhE8+V7`?(PJF1$Tl3cXxMvop0vOopslG{9tt- zj!RYTy^9DX1t~;0JUB2gFhm(?aTPEy@H)`vNdOG!w*jgK6Brndn3b5Al8l%biITH} zxs|OM7?@^AiWanH^4LPIM^cDh`_G?zML&!1foA2!#S-xtlr@I=5Ru8@$%X;p0O@ct zX#fBwFOP%{CQ@YszU<-s;=`-!;>kSMYhxpa$7O$Go(;HZ;ns5tL{^q;w?0LUT(CnB&vi`_#1^!hXnD_*_fXhy8~Ee0H1z#V4>~ zyuo$Bgd!9i16Z(PIk>oR2(V75S!mfrlNO*ee+3*#t4B5xgE z9B*)8AT=8KD|p`ed=yvcF7jCZByiNa4Z5c715)A#?A*L>96VTA(Q}mL6Y8nckjs+cX&9q zjlsI}-)~A6o$AabVOKOu7GeJQQ?TNh+^^dPloH2%oRH&SB$tBXkS;hby=g-1p0xu^ z?Wx2|!x)2gXQ=nPG^#IaU4Yzh<`si$RHv?lQ1@;Y7GsMlyPT>+64*5ZRwk&&jQ7+f zG-AVxrb_B6R7VK0zsnZUFg;9#N5@^zMZh z=^x195^sN_s4_6`q_r+-WHcu!E_EE(XGvb2*ZJ9@g&s!_{o$`0N?``ity9vgc!p7m#w*wDR&Rc9Vl8R&UQB-urBttix)t!{t< 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Schmid, Prof. Dr. C. Schlieder, Prof. Dr. T. Gross, Prof. Dr. A. Henrich} + {Wintersemester 17/18} + {90} + {-} + \section{Aufgabenteil 1 - Prof. Schlieder} + \begin{enumerate} + \item Aufgabe 1 (KInf: Geoinformationsverarbeitung; 10P) + \begin{enumerate} + \item Schreibe die Definition von Georeferenzierung auf. Erkläre den Unterschied zwischen direkter und indirekter Georeferenzierung (2P) + \item Bestimme die kürzeste Route von Aurach (A) nach Gerolzhofen (G) durch Anwendung des Djikstra-Algorithmus und gebe die Länge der Strecke sowie alle Zwischenschritte an (8P) + \image{0.5}{A1.png}{Wandertour}{Wandertour} + + \end{enumerate} + \item Aufgabe 2 (KInf: Digitale Bibliotheken; 10P) + \begin{enumerate} + \item Das GroupLens-Empfehlungsverfahren prädiziert die Bewertung r(p,i), die eine Person p für ein Item i abgeben wird. Dabei bezeichnet $\bar{r}(p)$ die über alle Items bestimmte durchschnittliche von p abgegebene Bewertung und sim(p,q) das Ähnlichkeitsmaß, welches angibt, wie ähnlich zwei Personen p und q bewerten:\\ $r(p,i)=\bar{r}(p)+c(p,i)$\\\\ $c(p,i)=\frac{\sum_{q \in P tex\ (p)}(r(q,i)-\bar{r}(q))*sim(p,q)}{\sum_{q \in P tex\ (p))}|sim(p,q)|} $\\ + Erkläre, warum die Bestimmung von r(p,i) auf $\bar{r}$(p,i) zurückgreift. In welchem Wertebereich liegen die Werte für sim(p,q)? Welche Bedeutung haben die beiden Extremwerte? (3P) + \item Vier Websites sind wie abgebildet durch Hyperlinks verbundem (siehe Graph). Stelle die linearen Gleichungen auf, die den PageRank von A, B, C und D durch Lösen der Gleichungen. Verwende einen Dämpfungsfaktor von d=0.5 (7P) + \item Schreibe die Definition von Georeferenzierung auf. Erkläre den Unterschied zwischen direkter und indirekter Georeferenzierung (2P) + \image{0.5}{A2.png}{Hyperlinks}{Hyperlinks} + \end{enumerate} + \newpage + \section{Aufgabenteil 2 - Prof. Henrich} + +\item Aufgabe 3 (Suchen, Suchmaschinen; 10P) + \begin{enumerate} + \item Schreibe die Definition von Suchmaschinenoptimierung. In Suchmaschinenoptimierung sollte man drei Tipps (Aufgabenbereiche) adressieren, welche sind dies? (5P) + \item (Auf Basis d. Abbildung:) Wie kann man das rote Fragezeichen in einer Enterprise Search Lösung ersetzen. \\ + Anders ausgedrückt: Welche grundlegenden Ansätze gibt es um eine unternehmensweite Suchlösung zu realisieren. (5P) + \image{0.5}{A3.png}{Abbildung}{Abbildung} + \end{enumerate} + \item Aufgabe 4 (Museum Digital \& Visualisierung; 10P) + \begin{enumerate} + \item In einer Ausstellung soll ein wertvolles historisches Buch als zentrales Objekt ausgestellt werden. Was sind die Probleme? Welche digitale Unterstützung ist möglich? Welche Funktionen sollten angeboten werden? Falls nötig treffe sinnvolle Annahmen und erläutere diese (5P) + \item Die Datei test1.gnu besteht aus drei Zeilen mit folgenden Gnuplot-Anweisungen:\\ + set terminal png size 1600,1200\\ + set output 'test1.png'\\ + plot 'silver.dat' using 1:2 with lines\\ + Beschreibe möglichst genau was die Anweisungen bewirken. Treffe falls nötig sinnvolle Annahmen. (5P) + \end{enumerate} + \newpage + \section{Aufgabenteil 3 - Prof. Schmid} + \item Aufgabe 5 (Problemlösen; 10P) + \begin{enumerate} + \item Problemraum (7P)\\ + Robi ist im Arbeitszimmer und dort geht die Glühlampe in der Deckenleuchte kaputt. Robi muss erstmal genau schauen, um welchen Typ Glühlampe es sich handelt, bevor er eine neue besorgt. Da es sich um eine hohe Altbauwohnung handelt, muss Robi eine Leiter holen. Diese steht in der Küche. Zur Vereinfachung nehmen wir an, dass es nur die räumliche Position Küche und Arbeitszimmer gibt. Das heißt, Robi steht immer schon unter der Lampe, wenn er im Arbeitszimmer ist und direkt an der Leiter, wenn er in der Küche ist. Robi hat folgende Aktionen zur Verfügung. Er kann von einem Raum in den anderen gehen (go(from, to)). Er kann die Leiter von einem Raum in den anderen transportieren (move(from, to)) und er kann auf die Leiter steigen (climb). Zum Transportieren der Leiter von einem Raum zum anderen muss er im gleichen Raum sein, wie die Leiter. Zum Transportieren der Leiter und zum Gehen von einem in den anderen muss Robi sich auf dem Boden (und nicht auf der Leiter) befinden. Außerdem kann er, falls er im Arbeitszimmer ist und auf der Leiter steht, die Glühbirne nehmen (takeBulb).\\\\ + Zeichne den Problemraum-Graph für das oben beschriebene Problem (alle zulässigen Zustände als Knoten, alle Zustandsübergänge als gerichtete und beschriftete Kanten). + \item Konzepte und Algorithmen (3P)\\ + \begin{itemize} + \item Greeno (1978) unterscheidet Transformationsprobleme, Neuordnungsprobleme und Induktionsprobleme. Um welchen Problemtyp handelt es sich bei einem Anagramm (einem Wort, das aus einem anderen Wort allein durch eine Umstellung der Buchstaben gebildet werden kann)? Begründe deine Auswahl kurz. (1P) + \item Was versteht man unter Problemlösen unter einer Heuristik? (1P) + \item Nenne die allgemeine Form zur Berechnung der minimalen Zahl von Aktionen zur Lösung eines Turm von Hanoi Problems mit n Scheiben und 3 Stiften. (1P) + \end{itemize} + \end{enumerate} + \item Aufgabe 6 (Lernen; 10P) + \begin{enumerate} + \item Lernalgorithmen (2P)\\ + Um zu erkennen, ob ein Pilz giftig ist oder nicht hast du die folgenden Attribute: Farbe des Hutes mit den Ausprägungen \{rot, gelb, braun\}, die durchschnittliche Größe \{2cm, 5cm, 8cm\} und die Form des Hutes \{kopfig, schwammähnlich\}.\\ + In der Veranstaltung wurden Perceptron, Cal2 und ID3 behandelt. + Welches der Lernverfahren ist für diese Daten (ohne weitere Aufbereitung der Daten) nicht geeignet. Begründe die Antwort. + \item Perzeptron (2P)\\ + Gegeben sind folgende Daten:\\ + \begin{tabular}{ccc} + X & Y & Klasse \\ + 0 & 0 & + \\ + -1 & 0 & + \\ + 0 & 1 & + \\ + 2 & 0 & - \\ + 1 & -1 & - \\ + 0 & -2 & -\\ + \end{tabular} + + Kann ein Perzeptron trainiert werden, das alle Beispiele dieses Datensatzes korrekt klassifiziert? Begründe die Antwort mit Hilfe einer Skizze. + + \item CAL2 (6P)\\ + Führe eine Handsimulation des Entscheidungsbaum-Algorithmus CAL2 für die in der Tabelle angegebene Trainingsmenge durch. Führe die Merkmale in der vorgegebenen Reihenfolge (beginnend mit x1) in den Baum ein. Zeichne den Baum in jeder Entwicklungsstufe.\\ + + \begin{tabular}{ccccc} + Bsp & x1 & x2 & x3 & Klasse \\ + 1 & 0 & 0 & 0 & B \\ + 2 & 1 & 0 & 1 & B \\ + 3 & 0 & 1 & 1 & A \\ + 4 & 0 & 0 & 1 & B \\ + 5 & 1 & 0 & 0 & B \\ + 6 & 0 & 1 & 0 & A \\ + 7 & 1 & 1 & 0 & A \\ + 8 & 1 & 1 & 1 & B \\ + \end{tabular} + + + \end{enumerate} + \newpage + \section{Aufgabenteil 4 - Prof. Gross} + \item Aufgabe 7 (Mensch-Computer-Interaktion; 10P) + \begin{enumerate} + \item Human-Computer Interaction and Usability: bitte definiere die Begriffe Mensch-Computer-Interaktion und Gebrauchstauglichkeit (4P). + \item Consistency Maintenance of Replicated Documents: Werden zeitgleich Änderungen an einem replizierten Dokument in einem verteilten Editor vorgenommen, können verschiedene Arten von Konsistenzproblemen auftreten, die behandelt werden müssen.\\ + Gegeben ist die Ausgangssequenz "klm". NutzerInnen an zwei verteilten Orten (Sites $S_{1}$ und $S_{2}$) führen in etwa zeitgleich darauf jeweils eine Operation ($o_{1}$ und $o_{1}$) lokal aus. Danach wird die jeweils entfernt ausgeführte Operationen lokal ausgeführt. Bitte benenne jeweils welches Konsistenzproblem durch das Ausführen der Operationen I), II) und III) auftritt und erläutere es am jeweiligen Beispiel (6P).\\\\ + I) $o_{1}$ =insert['x',1]\\ + $o_{2}$ =insert['y',2]\\\\ + II) $o_{1}$ =insert['y',4]\\ + $o_{2}$ =delete[1]\\\\ + III) $o_{1}$ =insert['y',2]\\ + $o_{2}$ =insert['y',2]\\ + \end{enumerate} + \item Aufgabe 8 (Mensch-Computer-Interaktion; 10P) + \begin{enumerate} + \item Interaction: Bitte beschreibe den Verlauf der Interaktion zwischen Benutzerinnen und Benutzern einerseits und dem System andererseits (6P). + \item Dimensions: Bitte beschrifte den Sensor und Aktuator im folgenden Diagramm (2P). + \image{0.7}{A8.png}{Sensor und Aktuator}{Sensor und Aktuator} + \item Building Blocks: Bitte nenne die Funktion von Sensor und Aktuator in einer sensorbasierten Infrastruktur (2P). + \end{enumerate} + \end{enumerate} +\end{document} \ No newline at end of file From b7b9029f89106f55ed8fd541f0eef470c8b00451 Mon Sep 17 00:00:00 2001 From: Fabian Lamprecht Date: Mon, 28 May 2018 23:15:09 +0200 Subject: [PATCH 10/15] MfI 2 WS 17/18 --- .../WS17 MfI2.tex | 191 ++++++++++++++++++ 1 file changed, 191 insertions(+) create mode 100644 KTR-MfI-2-B Mathematik für Informatiker 2/WS17 MfI2.tex diff --git a/KTR-MfI-2-B Mathematik für Informatiker 2/WS17 MfI2.tex b/KTR-MfI-2-B Mathematik für Informatiker 2/WS17 MfI2.tex new file mode 100644 index 0000000..238b93c --- /dev/null +++ b/KTR-MfI-2-B Mathematik für Informatiker 2/WS17 MfI2.tex @@ -0,0 +1,191 @@ +\input{../settings/settings} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{multirow} + +\begin{document} + +\klausur{KTR-MfI 2} +{Prof. Dr. U. Krieger)} +{Wintersemester 17/18} +{90} +{Taschenrechner, 1 Din A4 Seite doppelseitig handbeschrieben} + +\begin{enumerate} + \item Aufgabe 1 (8+12 Punkte) Lineare Gleichungssysteme und Matlab/Octave + + Geben Sie die Ergebnisse folgender Aufgaben zu linearen Gleichungssystemen in den daf\"ur vorgegebenen Feldern an. + \begin{enumerate} + \item [a)] Betrachten Sie das folgende lineare Gleichungssystem: + \[\begin{array}{lllllll} + 2x_1 & + & 4x_2 & & & = 0 \\ + & & 2x_2 & + & 2x_3 & = 2 \\ + 2x_1 & + & 4x_2 & + & 4x_3 & = 1 + \end{array}\] + \begin{enumerate} + \item [1)] Geben Sie eine Darstellung dieses linearen Gleicungssystems in der Form eines Matrix-Vektor-Produkts $$A\cdot x = b$$ mit Hilfe einer geeigneten Koeffizientenmatrix $A$ und dem Vektor $x$ der Variablen sowie der rechten Seite $b$ des Gleicungssystems an. + \item[2)] Berechnen Sie die L\"osung $x = \left(\begin{array}{c}x_1\\ x_2\\ x_3\end{array}\right) \in \mathbb{R}^3$ durch die \"Uberf\"uhrung des Systems (A~~~b) in Treppennormalform und anschlie{\ss}ende L\"osung unter Verwendung des Gau{\ss}schen Algorithmus. + + + \end{enumerate} + + \item [b)] Berechnen Sie da lineare Gleichungssystem + $$L\cdot x = \left( \begin{array}{cccc} + L_{11} & 0 & \dots & 0 \\ + L_{21} & L_{22} & \cdots & \vdots \\ + \vdots & \ddots & \ddots & 0 \\ + L_{n1} & \cdots & \cdots & L_{nn} + \end{array}\right)\cdot\left(\begin{array}{c} + x_1 \\x_2\\\vdots\\x_n + \end{array}\right)=\left(\begin{array}{c} + b_1 \\b_2\\\vdots\\b_n + \end{array}\right)=b$$ + mit der unteren Dreiecksmatrix + $$L = \left( \begin{array}{cccc} + L_{11} & 0 & \dots & 0 \\ + L_{21} & L_{22} & \cdots & \vdots \\ + \vdots & \ddots & \ddots & 0 \\ + L_{n1} & \cdots & \cdots & L_{nn} + \end{array}\right)\in \mathbb{R}^{n\times n}, n\in\mathbb{N}, n\geq 2$$ + und nehmen Sie an, dass alle Elemente $L_{ii}\neq 0,i=1,\dots,n$, sind. + + Der Vektor $x=\left(\begin{array}{c} + x_1 \\x_2\\\vdots\\x_n + \end{array}\right)$ kann mit Hilfe des folgenden \textit{Forward-Substitution} Algorithmus + \begin{equation} + x_i=\left(b_i-\sum^{i-1}_{j=1}~~L_{ij}\cdot x_j\right)/ L_{ii}, i=1,\dots,n + \end{equation} + bestimmt werden, der sukzessive die Elemente $x_i,i=1,\dots,n,$ berechnet. + \newpage + \begin{enumerate} + \item[1)] Implementieren Sie den \textit{Forward-Substitution} Algorithmus als MATLAB oder Octave Funktion $ForSub(b, L, y)$, die f\"ur eine gegebene rechte Seite $b\in\mathbb{R}^n$ und eine invertierbare untere Dreiecksmatrix $L\in\mathbb{R}^{n\times n}$ der Dimension $ n \in \mathbb{N}, n\geq 2$, mit Diagonalelementen $L_{ii}\neq 0, i=1,\dots,n$, als Eingabeparametern des L\"osungsvektor $x\in\mathbb{R}^n$ in (1) berechnet und diesen w\"ahrend des L\"osungsverfahrens sukzessive in den Speicherplatz des Ausgabevektors $y\in\mathbb{R}^n$ schreibt, nachdem $y$ zu Beginn des Algorithmus mit dem Nullvektor initialisieert wurde. + \item[2)] Schreiben Sie ein MATLAB oder Octave Programm, das diese erstellte MATLAB oder Octave Funktion $ForSub$ zur L\"osung des Gleichungssystems $L \cdot x = b$ mit folgenden Eingangsdaten + $$L=\left(\begin{array}{ccc} + 2 & 0 & 0 \\ + 1 & 5 & 0 \\ + 7 & 9 & 8 + \end{array}\right), b = \left(\begin{array}{c} + 6 \\2\\5 + \end{array}\right)$$ + anwendet und dann den L\"osungsvektor $x$ komponentenweise ausgibt. + \end{enumerate} + \end{enumerate} + \item Aufgabe 2 (6+6+8 Punkte) Vektorr\"aume + Geben Sie die Ergebnisse folgender Aufgaben zur Theorie der Vektorr\"aume in den daf\"ur vorgesehenen Feldern an. + \begin{enumerate} + \item [a)] Betrachten Sie die Vektoren + $$a=\left(\begin{array}{c} + 0 \\1\\1 + \end{array}\right),b=\left(\begin{array}{c} + 1 \\1\\1 + \end{array}\right),c=\left(\begin{array}{c} + 1 \\0\\1 + \end{array}\right)$$ + des $\mathbb{F}_2$-Vektorraums $\mathbb{F}^3_2$ auf dem K\"orper deer bin\"aren Elemente $\mathbb{F}_2={0,1}$ mit den Operationen $\{\oplus, \odot\}$ der Addition und Multiplikation modulo 2, d. h. f\"ur $x, y\in\mathbb{F}_2,x\oplus y = (x+y)~mod~2,\\ x \odot y = (x\cdot y)~mod~2$.\\ + Sind diese drei Vektoren $\{a,b,c\}\subset\mathbb{F}^3_2$ linear unabh\"angig oder linear abh\"angig? Begr\"unden Sie Ihre Antwort durch geeignete algebraische Argumente. + \item [b)] Betrachten Sie die Matrizen + $$A=\left(\begin{array}{cc} + 0 & 1 \\ + 0 & 1 \\ + \end{array}\right), B= \left(\begin{array}{cc} + 0 & 0 \\ + 1 & 0 + \end{array}\right), C = \left(\begin{array}{cc} + 0 & 1 \\ + 1 & 0 + \end{array}\right)$$ + des Vektorraums $\mathbb{R}^{2\times 2}$.\\ + Sind diese drei Matrizen $\{A,B,C\}\subset\mathbb{R}^{2\times 2}$ linear unabh\"angig oder linear abh\"angig? Begr\"unden Sie Ihre Antwort mit Hilfe einer geeigneten algebraischen Argumentation. + \item[c)] Weisen Sie durch eine geeignete algebraische Argumentation nach, dass die Menge + $$U=\{v=\left(\begin{array}{c} + x_1 \\x_2\\x_3 + \end{array}\right)\in \mathbb{R}^{3} | 4x_1+3x_2=0\} \subset \mathbb{R}^3$$ + ein Untervektorraum des reellen Vektorraums $V = \mathbb{R}^3$ ist. + \end{enumerate}\newpage + \item Aufgabe 3 (6+6+4+4 Punkte) Lineare Abbildungen + + Betrachten Sie die lineare Abbildung + $$ f: V \longrightarrow W $$ + $$ v=\left(\begin{array}{c}x \\ y \\ z \end{array}\right)\rightarrow f(v)=\left(\begin{array}{c} y\oplus z \\ x\oplus z \\ x\oplus y \end{array}\right) $$ + zwischen den $\mathbb{F}_2$-Vektorr\"aumen $V=\mathbb{F}^3_2$ und $W=\mathbb{F}^3_2$ \"uber dem K\"orper der bin\"aren Elemente $\mathbb{F}_2 = {0,1}$ mit den Operationen $\{\oplus,\odot\}$ der Addition und Multiplikation modulo 2, d.h. $x\oplus y = (x+y) mod 2, x\odot y = (x\cdot y) mod 2$ f\"ur $x,y\in\mathbb{F}_2$. + \begin{itemize} + \item [a)] W\"ahlen Sie jeweils die kanonische Basis $\mathcal{B} = \{e_1,e_2,e_3\}$ im $\mathbb{F}_2$-Vektorraum $V = \mathbb{F}^3_2$ bzw. $W=\mathbb{F}^3_2$ aus und geben Sie die der Abbildung $f$ zugeordnete Matrix $A_f$ bzgl. dieser beiden Basen an. + \item [b)] Bestimmen Sie mit Hilfe einer geeigneten algebraischen Argumentation oder eines geeigneten Berechnungsverfahrens den Rang $rg(f)$ der Abbildung $f$. + \item [c)] Bestimmen Sie mit Hilfe einer geeigneten Formel, die Sie explizit angeben sollen, die Dimension $k=dim(Ker(f))$ des Kerns $Ker(f)$ der Abbildung $f$. + \item [d)] Geben Sie eine Basis $\mathcal{C} = \{v_1,\dots,v_k\}$ des Kerns $Ker(f)$ der Abbildung $f$ an. + \end{itemize} + + \item Aufgabe 4 (10+4+6 Punkte) Gruppen- und Matrizenalgebra + \begin{enumerate} + \item [a)] Beantworten Sie folgende Fragen zur Algebra einfacher Gruppen $(G,\oplus,e)$ + \begin{enumerate} + \item [1)] Betrachten Sie die Gruppe $(G,\oplus,e)=(\mathbb{F}^3_2,\oplus,e)$ aller bin\"arer Tupel + $$(a,b)\in\mathbb{F}^3_2 = \{(0,0),(1,0),(0,1),(1,1)\}$$ + mit den Komponenten $a,b\in\mathbb{F}_2= \{ 0,1\}$ und der komponentenweise definierten Addition modulo 2 als Gruppenverkn\"upfung $\oplus$. + + Geben Sie das neutrale Element $e\in G$ an und berechnen Sie f\"ur $f=(1,0), g=(0,1),h=(1,1)$ die folgenden Gruppenelemente $k,l\in G$ + + $$k = g \oplus f = $$ + $$l = g \oplus h = $$ + $$e = $$ + + \item[2)] Definiert die Teilmenge + $$U=\{(0,0),(1,0),(0,1)\}\subset \mathbb{F}^2_2=G$$ + der Untergruppe von $(\mathbb{F}^2_2,\oplus,e)$? Geben Sie eine algebraische Begr\"undung an. + \item[3)] Die aus drei verschiedenen Elementen besthende Menge $G={e,a,h}$ mit der additiven Verkn\"upfung $z=x\oplus y \in G $ auf zwei Gruppenelementen $x,y \in G$ kann durch eine entsprechende Verkn\"upfungstabelle T beschrieben werden (siehe unten), z.B. ergibt sich aus der Additon der Elemente $x = b$ in der 5. Zeile und $y=a$ in der 4. Spalte das Element + $$z=x\oplus y = b \oplus a = e$$ + Erg\"anzen Sie die fehlenden Elemente der 5. Zeile hinter $x=b$ und der 5. Spalte unterhalb von $y=b$ in der Verkn\"upfungstabelle $T$ sowie das fehlende Element der 5. Zeile und 3. Spalte f\"ur $x=b$ und $y=e$ derart, dass $T$ eine kommutative Gruppe $(G,\oplus,e)$ definiert. + + \begin{tabular}{|c|c||c|c|c|}\hline + $\oplus$ & \multicolumn{4}{|c|}{}\\\hline + &y= & e & a & b \\\hline\hline + \multirow{3}{*}{b=} & e & e & a & \\\hline + & a & a & b & e \\\hline + & b & & e & \\\hline + \end{tabular} + \end{enumerate} + \item[b)] Es sei $A=\left(\begin{array}{ccc} + 2 & 1 & 4\\ + 1 & -1 & -5 + \end{array}\right), B = \left(\begin{array}{ccc} + 4 & 1 & 0\\ + 5 & 2 & 1\\ + 10& 7 & 6 + \end{array}\right)$ + \begin{enumerate} + \item[1)] Berechnen Sie das Element $C_{23}$ der Produktmatrix $C=A\cdot B$ + \item[2)] Wie lautet die transponierte Matrix $A^t$ zu $A$? + \item[3)] Es sei + $$A=\left(\begin{array}{ccc} + 1 & 0 & 0\\ + 0 & 1 & -1\\ + 2 & -4 & 3 + \end{array}\right)$$ + Berechnen Sie die inverse Matrix $C=A^{-1}$ zu $A$ + \end{enumerate} + \item Aufgabe 5(4+6+10 Punkte) Lineare Abbildungs- und Eigenwerttheorie + + Geben Sie die Ergebnisser folgender Aufgaben in den daf\"ur vorgesehenen Feldern an. + \begin{enumerate} + \item [a)] Berechnen Sie die Determinante der folgenden Matrix + $$A=\left(\begin{array}{ccc} + 0.3 & 0 & 0\\ + 0.7 & 0 & 1\\ + 0 & 2 & 0 + \end{array}\right)$$ + $det(A)=$ + \item[b)] Berechnen Sie einen Eigenvektor $v\in\mathbb{R}^2$ zum Eigenwert $\lambda = 2$ der folgenden Matrix $B$: + $$B=\left(\begin{array}{cc} + 1 & -1\\ + -1 & 1 + \end{array}\right)\in\mathbb{R}^{2\times 2}$$ + \item[c)] Bestimmen Sie das charakteristische Polinom $p(\lambda) = \lambda^3+p_2\lambda^2+p_1\lambda+p_0$ der folgenden Matrix $A$ und berechnen Sie mit dessen Hilfe alle Eigenwerte $\lambda_i\in\mathbb{R}, i\in\{1,2,3\}$, der Matrix $A$: + $$A=\left(\begin{array}{ccc} + 0.9 & 0 & 0\\ + 0 & -1 & 1\\ + 0.1 & -2 & 2 + \end{array}\right)$$ + \end{enumerate} + \end{enumerate} +\end{enumerate} +\end{document} \ No newline at end of file From ca9a90c919dea0a7676f286adcf2871fa6985a48 Mon Sep 17 00:00:00 2001 From: Anika Date: Mon, 28 May 2018 23:50:32 +0200 Subject: [PATCH 11/15] add WiMa-B-01a exam (task 12 missing?) --- WiMa-B-01a/WS17 WiMa-B-01a.tex | 154 +++++++++++++++++++++++++++++++++ 1 file changed, 154 insertions(+) create mode 100644 WiMa-B-01a/WS17 WiMa-B-01a.tex diff --git a/WiMa-B-01a/WS17 WiMa-B-01a.tex b/WiMa-B-01a/WS17 WiMa-B-01a.tex new file mode 100644 index 0000000..a8f304f --- /dev/null +++ b/WiMa-B-01a/WS17 WiMa-B-01a.tex @@ -0,0 +1,154 @@ +\input{../settings/settings} +\usepackage{amsmath, amssymb} + +\begin{document} + \klausur{WiMa-B-01a Wirtschaftsmathematik 1} + {Prof. Dr. Christian Aßmann} + {Wintersemester 17/18} + {60} + {Taschenrechner, Formelsammlung, 1 handbeschriebenes DIN-A-4-Blatt} + + \begin{enumerate} + \item Aufgabe 1 (5 Punkte)\\ + Geben sie die Lösungsmenge der Gleichung + \begin{equation*} + | x- 3 | = 2 + \end{equation*} + an. + \begin{enumerate} + \item $L={x|x<3} $ + \item $L={} $ + \item $L={1;5} $ + \item $L={x|x>3} $ + \item $L={x|x<1} $ + \end{enumerate} + + \item Aufgabe 2 (6 Punkte)\\ + Welche der folgenden Aussagen ist falsch? + \begin{enumerate} + \item Sei $f(x)=log_a(x)$. Dann gilt $f'(x)=\frac{1}{x\ln a}$ mit $a>0$ und $a\neq1$. + \item Sei $h(x)=exp\{g(x)\}$. Dann gilt $h'(x)=g'(x)exp\{g(x)\}$. + \item $\lim_{h \rightarrow 0}\frac{f(x+h-f(x)}{h} = f'(x)$, sofern der Grenzwert existiert. + \item Sei $ g(x) = f(x) + c $ für $c \in \mathbb{R}$. Dann gilt $g'(x) = f'(x)+c$. + \item Sei $ f(x)=\ln x $. Dann gilt $f'(x)=\frac{1}{x}$. + \end{enumerate} + + \item Aufgabe 3 (6 Punkte)\\ + Geben sie den natürlichen Definitionsbereich $D_f$ der Funktion + \begin{equation*} + f(x) = \ln(a-x) + \end{equation*} + für $a > 0$ an. + \begin{enumerate} + \item $D_f=\{\}$. + \item $D_f=\{x|x<0\}$. + \item $D_f=\{x|xa\}$. + \end{enumerate} + \item Aufgabe 4 (5 Punkte)\\ + Seien $x_1,x_2,\dots, x_n \in \mathbb{R}$ fest gegeben. Wie lautet die Minimalstelle $y_{min}$ der Funktion + \begin{equation*} + f(y)=\sum_{i=1}^{n}(x_1-y)^2\mathrm{?} + \end{equation*} + \begin{enumerate} + \item $y_{min}=0$. + \item $y_{min}=x_n-x_1$. + \item $y_{min}=x_1$. + \item $y_{min}=\frac{1}{n}\sum_{i=1}^{n}x_i$. + \item $y_{min}=\sum_{i=1}^{n}x_i$. + \end{enumerate} + + \item Aufgabe 5 (5 Punkte)\\ + Bilden Sie die erste Ableitung der Funktion + \begin{equation*} + f(x)=(1+e^{-2x})^{-3} + \end{equation*} + und berechnen Sie die Ableitung an der Stelle $x_0=1$. + \begin{enumerate} + \item $f'(x_0)=-0,3887$. + \item $f'(x_0)=0,3887$. + \item $f'(x_0)=0,4887$. + \item $f'(x_0)=0,5887$. + \item $f'(x_0)=-0,4887$. + \end{enumerate} + + \item Aufgabe 6 (5 Punkte)\\ + Wie lautet die Maximalstelle $x_{max}$ der Funktion + \begin{equation*} + f(x)=exp\{-\frac{1}{2}(x-\mu)^2\} \mathrm{?} + \end{equation*} + Welchen Wert nimmt die Funktion an der Maximalstelle an? + \begin{enumerate} + \item $x_{max}=\inf$, $f(x_{max}=0)$.\item $x_{max}=\mu$, $f(x_{max}=0)$. + \item $x_{max}=0$, $f(x_{max}=1)$. + \item $x_{max}=\mu$, $f(x_{max}=1)$. + \item $x_{max}=1$, $f(x_{max}=\mu)$. + \end{enumerate} + + \item Aufgabe 7 (5 Punkte)\\ + Sei $D_f\subset \mathbb{R}$ und $f : D_f \rightarrow \mathbb{R}$ zweimal differenzierbar in einer offenen Umgebung von $x_0 \in D_f$. Ferner sei $f(x_0)=0$. Welche der folgenden Aussagen ist falsch? + \begin{enumerate} + \item $f''(x)<0 \Rightarrow f$ hat ein striktes lokales Maximum in $x_0$. + \item $f''(x)>0 \Rightarrow f$ hat ein striktes lokales Minimum in $x_0$. + \item $f''(x)<0$ für alle $x \in D_f \Rightarrow f$ hat ein striktes globales Maximum in $x_0$. + \item $f''(x)>0$ für alle $x \in D_f \Rightarrow f$ hat ein striktes globales Minimum in $x_0$. + \item $x_0$ ist ein striktes lokales Minimum $\Rightarrow x_0$ ist auch ein striktes globales Minimum. + \end{enumerate} + \item Aufgabe 8 (5 Punkte)\\ + Bestimmen sie den Grenzwert + \begin{equation*} + \lim_{n \rightarrow \inf}1-\left[1-\frac{y}{c}\right]^n + \end{equation*} + für $00$. + \end{enumerate} + + \item Aufgabe 11 (6 Punkte)\\ + Bestimmen Sie mit der Lagrangemethode die kritische Stelle der Kostenfunktion + \begin{equation*} + K : \mathbb{R}^2_+ \rightarrow \mathbb{R}, \quad K(x,y)=x^2-y^2 + \end{equation*} + unter der Nebenbedingung + \begin{equation*} + g: \mathbb{R}^2_+ \rightarrow \mathbb{R}, \quad g(x,y)=x^3+y^3-16=0. + \end{equation*} + Die hinreichenden Bedingungen müssen nicht geprüft werden. + \begin{enumerate} + \item Kritische Stelle $ (x,y)=(2,2)$. + \item Kritische Stelle $ (x,y)=(0,0)$. + \item Kritische Stelle $ (x,y)=(12,4)$. + \item Kritische Stelle $ (x,y)=(0,4)$. + \item Kritische Stelle $ (x,y)=(4,0)$. + \end{enumerate} + \end{enumerate} +\end{document} From e577707b0f8a2d84c89e6067a79ccadefedaf257 Mon Sep 17 00:00:00 2001 From: Anika Date: Tue, 29 May 2018 00:30:24 +0200 Subject: [PATCH 12/15] correct some mistakes --- WiMa-B-01a/WS17 WiMa-B-01a.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/WiMa-B-01a/WS17 WiMa-B-01a.tex b/WiMa-B-01a/WS17 WiMa-B-01a.tex index a8f304f..88c99ec 100644 --- a/WiMa-B-01a/WS17 WiMa-B-01a.tex +++ b/WiMa-B-01a/WS17 WiMa-B-01a.tex @@ -17,7 +17,7 @@ an. \begin{enumerate} \item $L={x|x<3} $ - \item $L={} $ + \item $L=\{\} $ \item $L={1;5} $ \item $L={x|x>3} $ \item $L={x|x<1} $ From 5aed26bad26398b48d7212d1b85ebffd596835ab Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Thomas=20M=C3=BCller?= Date: Tue, 29 May 2018 00:45:13 +0200 Subject: [PATCH 13/15] Add ss17 wima exam --- .../SS17 WiMa-B-02a.tex | 335 ++++++++++++++++++ 1 file changed, 335 insertions(+) create mode 100644 WiMa-B-02a Wirtschaftsmathematik 2/SS17 WiMa-B-02a.tex diff --git a/WiMa-B-02a Wirtschaftsmathematik 2/SS17 WiMa-B-02a.tex b/WiMa-B-02a Wirtschaftsmathematik 2/SS17 WiMa-B-02a.tex new file mode 100644 index 0000000..a87bd1e --- /dev/null +++ b/WiMa-B-02a Wirtschaftsmathematik 2/SS17 WiMa-B-02a.tex @@ -0,0 +1,335 @@ +\input{../settings/settings} + +\begin{document} + \klausur{WiMa-B-02a Wirtschatfsmathematik II} + {Prof. Dr. Christian Aßmann} + {Sommersemester 17} + {90} + {Taschenrechner, Formelsammlung, Handbeschriebenes DIN A4 Blatt (beidseitig)} + + \begin{enumerate} + \item \textbf{Aufgabe 1 (5 Punkte)}\\ + Berechnung Sie das Skalarprodukt $x' \cdot y$ für die Vektoren\\ + + % pmatrix requires amsmath package + $ x = + \begin{pmatrix} + 1 \\ + 2 \\ + 3 + \end{pmatrix} + $ + und + $ y = + \begin{pmatrix} + 1 \\ + 2 \\ + 4 + \end{pmatrix} + $\\ + + \begin{enumerate} + \item $x' \cdot y = 13$. + \item $x' \cdot y = 14$. + \item $x' \cdot y = 15$. + \item $x' \cdot y = 16$. + \item $x' \cdot y = 17$. + \end{enumerate} + + \item \textbf{Aufgabe 2 (5 Punkte)}\\ + Berechnen Sie $x \cdot y'$ für Vektoren\\ + + $ x = + \begin{pmatrix} + 1 \\ + 2 \\ + 3 + \end{pmatrix} + $ + und + $ y = + \begin{pmatrix} + 1 \\ + 2 \\ + 4 + \end{pmatrix} + $\\ + + \begin{enumerate} + \item + $ x \cdot y' = + \begin{pmatrix} + 1&2 \\ + 2&4 + \end{pmatrix} + $ + \item + $ x \cdot y' = + \begin{pmatrix} + 1&2&4 \\ + 2&4&8 \\ + 3&6&12 + \end{pmatrix} + $ + \item + $ x \cdot y' = + \begin{pmatrix} + 1&2&3 \\ + 2&4&6 \\ + 4&6&9 + \end{pmatrix} + $ + \item + $ x \cdot y' = + \begin{pmatrix} + 1&3&4 \\ + 3&2&6 \\ + 4&6&16 + \end{pmatrix} + $ + \item + $ x \cdot y' = + \begin{pmatrix} + 1&4&4 \\ + 2&4&4 \\ + 6&4&16 + \end{pmatrix} + $ + \end{enumerate} + + \newpage + \item \textbf{Aufgabe 3 (6 Punkte)}\\ + Berechnen Sie das Integral + + $$ A = \int_{\frac{1}{2}}^{1} \ln(4x^2)dx. $$ + + \begin{enumerate} + \item $A = 0,2863$. + \item $A = 0,4201$. + \item $A = 0,3004$. + \item $A = 0,3863$. + \item $A = 0,2003$. + \end{enumerate} + + \item \textbf{Aufgabe 4 (5 Punkte)}\\ + Welche der folgenden Aussagen stimmt nicht? + + \begin{enumerate} + \item Das Skalarprodukt ist ein Spezialfall des Matrizenprodukts. + \item Beim Matrizenprodukt $x \cdot y$ muss die Spaltenzahl von $x$ mit der Zeilenanzahl von $y$ übereinstimmen. + \item Das Matrizenprodukt ist nur für quadratische Matrizen definiert. + \item Wenn das Ergebnis eines Matrizenprodukts existiert, kann es immer in Form einer Matrix dargestellt werden. + \item Beim Matrizenprodukt $x \cdot y$ muss die Zeilenanzahl von $x$ nicht mit der Spaltenanzahl von $y$ übereinstimmen. + \end{enumerate} + + \item \textbf{Aufgabe 5 (5 Punkte)}\\ + Berechnen Sie die Inverse $X^-1$ der Matrix + $$ X = + \begin{pmatrix} + 1&4 \\ + 2&5 + \end{pmatrix}, + $$ + sofern diese existiert und berechnen Sie die Summe der beiden Hauptdiagonalelemente der inversen Matrix, d.h. die Spur der inversen Matrix $tr(X^-1)$. + + \begin{enumerate} + \item $X^-1$ existiert nicht. + \item $tr(X^-1) = -2$. + \item $tr(X^-1) = 2$. + \item $tr(X^-1) = 0$. + \item $tr(X^-1) = 1$. + \end{enumerate} + + \newpage + \item \textbf{Aufgabe 6 (Punkte)}\\ + Die drei Vektoren\\ + + $ x = + \begin{pmatrix} + 1 \\ + 2 \\ + 3 + \end{pmatrix} + $ + , + $ y = + \begin{pmatrix} + -1 \\ + -3 \\ + -2 + \end{pmatrix} + $ + und + $ z = + \begin{pmatrix} + 1 \\ + -2 \\ + 7 + \end{pmatrix} + $\\ + + sind linear abhängig. Wie lautet demnach eine Linearkombination für die gilt + + $$ ax + by + cz = 0 ?$$ + + \begin{enumerate} + \item $a = 5, b = 4, c = -1$. + \item $a = 5, b = -4, c = -1$. + \item $a = 4, b = 5, c = 2$. + \item $a = -4, b = -5, c = 2$. + \item $a = -3, b = 4, c = -2$. + \end{enumerate} + + \item \textbf{Aufgabe 7 (5 Punkte)}\\ + Berechnen Sie $det(A)$ für + $$ A = + \begin{pmatrix} + 2&0&1&4 \\ + 3&0&-4&-2 \\ + 2&0&-1&0 \\ + 11&8&-4&6 \\ + \end{pmatrix} + $$ + + \begin{enumerate} + \item $det(A) = 94$. + \item $det(A) = 104$. + \item $det(A) = 96$. + \item $det(A) = 92$. + \item $det(A) = 88$. + \end{enumerate} + + \newpage + \item \textbf{Aufgabe 8 (5 Punkte)}\\ + Welche der folgenden Aussagen ist falsch? + + \begin{enumerate} + \item Der Nullvektor ist immer linear abhängig. + \item Ein Vektor $x \in R^n$ ist genau dann linear unabhängig, wenn $x \neq 0$. + \item Zwei Vektoren im $R^n$ sind genau dann linear unabhängig, wenn sie nicht auf einer Geraden liegen. + \item Drei Vektoren im $R^n$ sind genau dann linear unabhängig, wenn sie nicht auf einer Ebene liegen. + \item Die Einheitsvektoren $e_1,...,e_n \in R^n$ sind linear abhängig. + \end{enumerate} + + \item \textbf{Aufgabe 9 (6 Punkte)}\\ + Welche der folgenden Operationen ist für\\ + + $ A = + \begin{pmatrix} + 1&2&3 \\ + 6&5&4 \\ + 4&3&7 + \end{pmatrix} + $ + , + $ B = + \begin{pmatrix} + 5&-2&3 \\ + -1&2&9 \\ + 1&-2&-9 + \end{pmatrix} + $ + und + $ C = + \begin{pmatrix} + 4&5 \\ + 1&3 \\ + 2&7 + \end{pmatrix} + $\\ + + definiert? + + \begin{enumerate} + \item $(A' + B)\,C$. + \item $(A' + B)\,C'$. + \item $(A + B)\,C'$. + \item $BC'$. + \item $B^{-1} C$. + \end{enumerate} + + \item \textbf{Aufgabe 10 (? Punkte)}\\ + Wie lauten die Eigenwerte der Matrix\\ + + $ A = + \begin{pmatrix} + 2&-2&3 \\ + 0&3&-2 \\ + 0&-1&2 + \end{pmatrix}? + $\\ + + \begin{enumerate} + \item $\lambda_{1} = 1, \lambda_{2} = 4, \lambda_{3} = 7$. + \item $\lambda_{1} = 3, \lambda_{2} = 5, \lambda_{3} = 1$. + \item $\lambda_{1} = 2, \lambda_{2} = 4, \lambda_{3} = 6$. + \item $\lambda_{1} = 2, \lambda_{2} = 3, \lambda_{3} = 7$. + \item $\lambda_{1} = 2, \lambda_{2} = 4, \lambda_{3} = 1$. + \end{enumerate} + + \newpage + \item \textbf{Aufgabe 11 (6 Punkte)}\\ + Ein Verleger produziert ein neues Buch in drei Varianten, als Taschenbuch ($T$), als Deluxe-Ausgabe ($D$) und als Buchclub-Ausgabe ($B$). Die Produktion eines Taschenbuchs erfordert 1 Minute Heften ($H$) und 2 Minuten Leimen ($L$), eine Deluxe-Ausgabe 3 Minuten Heften und 5 Minuten Leimen und eine Buchclub-Ausgabe 2 Minuten Heften und 4 Minuten Leimen. Die Heftmaschine ist täglich 360 Minuten frei, die Leimmaschine täglich 660 Minuten. Das Produktionsproblem des Verlegers kann damit als inhomogenes lineares Gleichungssystem + \begin{equation*} + \begin{pmatrix} + 1 & 3 & 2 \\ + 2 & 5 & 4 + \end{pmatrix} + \begin{pmatrix} + T \\ + D \\ + B \\ + \end{pmatrix} + = + \begin{pmatrix} + 360 \\ + 660 \\ + \end{pmatrix} + \end{equation*} + dargestellt werden. Wie lautet die allgemeine Lösungsmenge $L$? Beachten Sie, dass nur positive Anzahlen produzierter Bücher ökonomisch sinnvoll sind. + \begin{enumerate} + \item $L=\left\{\begin{pmatrix} + T \\ D \\ B \\ + \end{pmatrix} | \begin{pmatrix} + T \\ D \\ B \\ + \end{pmatrix} + = + \begin{pmatrix} + 180 \\ 60 \\ 0 + \end{pmatrix} + +r \begin{pmatrix} + -2 \\ 0 \\ 1 \\ + \end{pmatrix},\ r \in [0,90]\right\}$ + \item $L=\{\} $ + \item $L=\left\{\begin{pmatrix} + T \\ D \\ B \\ + \end{pmatrix} | \begin{pmatrix} + T \\ D \\ B \\ + \end{pmatrix} + = + \begin{pmatrix} + 180 \\ 60 \\ 120 + \end{pmatrix} + +r \begin{pmatrix} + -2 \\ -1 \\ 1 \\ + \end{pmatrix},\ r \in [0,90]\right\}$ + \item $L=\begin{pmatrix} + T \\ D \\ B \\ + \end{pmatrix} = \begin{pmatrix} + 120 \\ 180 \\ 60 \\ + \end{pmatrix}$ + \item $L=\left\{\begin{pmatrix} + T \\ D \\ B \\ + \end{pmatrix} | \begin{pmatrix} + T \\ D \\ B \\ + \end{pmatrix} + = + \begin{pmatrix} + 180 \\ 60 \\ 0 + \end{pmatrix} + +r \begin{pmatrix} + -2 \\ 0 \\ 1 \\ + \end{pmatrix},\ r \in [-10,90]\right\}$ + \end{enumerate} + \end{enumerate} +\end{document} From 0d66f0ab4f296779f0f3cbc6e80f5affbf61271b Mon Sep 17 00:00:00 2001 From: Marius Hofmann Date: Tue, 29 May 2018 00:56:14 +0200 Subject: [PATCH 14/15] Add WiMa 2 WS1718 --- .../WS1718 WiMa-B-02a.tex | 429 ++++++++++++++++++ 1 file changed, 429 insertions(+) create mode 100644 WiMa-B-02a Wirtschaftsmathematik 2/WS1718 WiMa-B-02a.tex diff --git a/WiMa-B-02a Wirtschaftsmathematik 2/WS1718 WiMa-B-02a.tex b/WiMa-B-02a Wirtschaftsmathematik 2/WS1718 WiMa-B-02a.tex new file mode 100644 index 0000000..e4422ec --- /dev/null +++ b/WiMa-B-02a Wirtschaftsmathematik 2/WS1718 WiMa-B-02a.tex @@ -0,0 +1,429 @@ +\input{../settings/settings} +\usepackage{amsmath} +\usepackage{amssymb} +\begin{document} + \klausur{WiMa-B-02a Wirtschaftsmathematik 2} + {Prof. Dr. Christian Aßmann} + {Wintersemester 17/18} + {60} + {Taschenrechner, Formelsammlung, ein handbeschriebenes DIN A4 Blatt (beidseitig)} + + \section*{Aufgabe 1 (4 Punkte)} + Bestimmen Sie, falls möglich, den Homogenitätsgrad der Funktion + \begin{equation*} + f: \mathbb{R}^{+}\times \mathbb{R}^{+} \to \mathbb{R}, \qquad f(x,y) = y \ln(2^{x}) + \end{equation*} + + \begin{enumerate}[label=\Alph*)] + \item 0 + \item 1 + \item 2 + \item $\ln(2)$ + \item $f(x,y)$ ist nicht homogen + \end{enumerate} + + \section*{Aufgabe 2 (4 Punkte)} + Die Funktion + + \begin{equation*} + f(x) = \frac{x+2}{x-1}, + \end{equation*} + + \noindent + mit $x\neq 1$, besitzt an der Stelle $x_{0}=2$ die Tangente + + \begin{equation*} + g(x) = -3x+10, + \end{equation*} + + \noindent + was sie nicht zu prüfen brauchen. + Approximieren Sie den Funktionswert von $f(x)$ an der Stelle $x_{1}=3$ durch die Tangende + und geben Sie den absoluten Approximationsfehler $\delta f$ an. + + \begin{enumerate}[label=\Alph*)] + \item $\delta f = 3$ + \item $\delta f = 0$ + \item $\delta f = 1,5$ + \item $\delta f = 3,5$ + \item $\delta f= -1,5$ + \end{enumerate} + + \newpage + \section*{Aufgabe 3 (6 Punkte)} + Bestimmen Sie für die Funktion + + \begin{equation*} + f: = \mathbb{R} \to \mathbb{R}, \qquad f(x)=\sqrt{x^{2}+1} + \end{equation*} + + \noindent + das Taylorpolynom 2. Grades $T^{2}_{f}(x)$ mit der Entwicklungsstelle $x_{0}=0$. + + \begin{enumerate}[label=\Alph*)] + \item $1+\frac{1}{2}x^{2}$ + \item $1+\frac{1}{2}x-\frac{1}{4}x^{2}$ + \item $x+\frac{1}{2}x^{2}$ + \item $1-\frac{1}{4}x^{2}$ + \item $\frac{1}{2}x^{2}$ + \end{enumerate} + + \section*{Aufgabe 4 (5 Punkte)} + Welche der folgenden Aussagen ist falsch? + \begin{enumerate}[label=\Alph*)] + \item Das Matrizenprodukt $A \cdot B$ ist definiert für zwei beliebige Matrizen + $A \in \mathcal{M}_{m,n}$ und $B \in \mathcal{M}_{r,s}$ mit $n=r$. + \item Der Nullvektor ist immer linear unabhängig. + \item Ist eine Matrix $\tilde{A}$ durch endlich viele elementare Zeilenumformungen aus + der Matrix $A$ entstanden, so besitzen $A$ und $\tilde{A}$ den gleichen Rang. + \item Eine Matrix $A \in \mathcal{M}_{n}$ ist invertierbar, wenn $rg(A)=n$ gilt. + \item Die Lösungsmenge eines homogenen linearen Gleichungssystems enthält immer den Nullpunkt. + \end{enumerate} + + \section*{Aufgabe 5 (6 Punkte)} + Lösen Sie das bestimmte Integral + + \begin{equation*} + \int^{b}_{1} x\ln(x)dx, + \end{equation*} + + \noindent + mit $b>1$, mittels partieller Integration. + + \begin{enumerate}[label=\Alph*)] + \item $b\ln(b)-b+1$ + \item $\frac{1}{2}b^{2}\ln(b)-\frac{1}{4}b^{2}+\frac{1}{4}$ + \item $b-1$ + \item $b\ln(b)$ + \item $\frac{b^{2}-1}{2}\ln(b)$ + \end{enumerate} + + \section*{Aufgabe 6 (4 Punkte)} + Berechnen Sie + \begin{equation*} + (a+b)^{T}\cdot c + \end{equation*} + + \noindent + für die Vektoren + + \begin{equation*} + a= + \begin{pmatrix} + 1\\ + 4\\ + -2\\ + \end{pmatrix} + , \qquad + b= + \begin{pmatrix} + 0\\ + -3\\ + 3\\ + \end{pmatrix} + , \qquad + c= + \begin{pmatrix} + 8\\ + 5\\ + 2\\ + \end{pmatrix} + . + \end{equation*} + + \begin{enumerate}[label=\Alph*)] + \item 13. + \item 14. + \item 15. + \item 16. + \item 17. + \end{enumerate} + + \section*{Aufgabe 7 (5 Punkte)} + Seien die Matrizen + + \begin{equation*} + A \in \mathcal{M}_{m,n}, B \in \mathcal{M}_{p,q} und C \in \mathcal{M}_{r,s} + \end{equation*} + + \noindent + Welche Bedingungen müssen $m,n,p,q,r,s \in \mathbb{N}$ erfüllen, damit + + \begin{equation*} + A \cdot (B+C^{T}) + \end{equation*} + + \noindent + definiert ist? + + \begin{enumerate}[label=\Alph*)] + \item $n=p=r$ + \item $m=r$ und $p=q=s$ + \item $n=p=s$ und $q=r$ + \item $m=s$ und $p=q=r$ + \item $m=p=s$ und $n=q=r$ + \end{enumerate} + + \newpage + \section*{Aufgabe 8 (5 Punkte)} + Berechnen Sie die fehlenden Werte $\theta _{1}$ und $\theta _{2}$ in der Matrizenmultiplikation + + \begin{equation*} + \begin{pmatrix} + \theta _{1} & 3 & 5\\ + 2 & 4 & 6\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + 1 & 4\\ + 2 & 5\\ + 3 & 6\\ + \end{pmatrix} + = + \begin{pmatrix} + 20 & 41\\ + 28 & \theta _{2}\\ + \end{pmatrix} + \end{equation*} + + \begin{enumerate}[label=\Alph*)] + \item $\theta _{1}=20, \theta _{2}=36$ + \item $\theta _{1}=1, \theta _{2}=60$ + \item $\theta _{1}=12, \theta _{2}=51$ + \item $\theta _{1}=1, \theta _{2}=36$ + \item $\theta _{1}=-1, \theta _{2}=64$ + \end{enumerate} + + \section*{Aufgabe 9 (5 Punkte)} + Bestimmen Sie die Inverse $X^{-1}$ der Matrix + + \begin{equation*} + X = + \begin{pmatrix} + 2 & 4\\ + -5 & \theta\\ + \end{pmatrix} + \end{equation*} + + \noindent + in Abhängigkeit von $\theta$. Für welche Werte von $\theta$ existiert die Inverse der Matrix $X$? + Berechnen Sie auch die Summe der beiden Hauptdiagonalelemente der inversen Matrix, d.h. die Spur der inversen Matrix $tr(X^{-1})$. + + \begin{enumerate}[label=\Alph*)] + \item $\theta \neq 10, tr(X^{-1}) = \frac{2+\theta}{2\theta-20}$ + \item $\theta \neq -10, tr(X^{-1}) = 2 + \theta$ + \item $\theta \neq 5, tr(X^{-1}) = \frac{\theta}{\theta - 5}$ + \item $\theta \neq -10, tr(X^{-1}) = \frac{2 + \theta}{2\theta + 20}$ + \item $\theta \neq -5, tr(X^{-1}) = \frac{\theta - 3}{2\theta + 10}$ + \end{enumerate} + + \newpage + \section*{Aufgabe 10 (6 Punkte)} + Wie lautet die allgemeine Lösungsmenge des inhomogenen linearen Gleichungssystems? + + \begin{equation*} + \begin{pmatrix} + 1 & 3 & 2 & 0\\ + 0 & 3 & -3 & 3\\ + 1 & 2 & 3 & -1\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + x_{1}\\ + x_{2}\\ + x_{3}\\ + x_{4}\\ + \end{pmatrix} + = + \begin{pmatrix} + 0\\ + 6\\ + -2\\ + \end{pmatrix} + \end{equation*} + + \begin{enumerate}[label=\Alph*)] + \item $\mathbb{L} = \theta$ + \item $\mathbb{L} = + \left\{ + \begin{pmatrix} + x_{1}\\ + x_{2}\\ + x_{3}\\ + x_{4}\\ + \end{pmatrix} + \vert + \begin{pmatrix} + x_{1}\\ + x_{2}\\ + x_{3}\\ + x_{4}\\ + \end{pmatrix} + = + \begin{pmatrix} + -6\\ + 2\\ + 0\\ + 0\\ + \end{pmatrix} + +r_{1}\cdot + \begin{pmatrix} + -5\\ + 1\\ + 1\\ + 0\\ + \end{pmatrix} + +r_{2}\cdot + \begin{pmatrix} + 3\\ + -1\\ + 0\\ + 1\\ + \end{pmatrix} + ; \quad r_{1},2_{2} \in \mathbb{R} + \right\} + $ + + \item $\mathbb{L} = + \left\{ + \begin{pmatrix} + x_{1}\\ + x_{2}\\ + x_{3}\\ + x_{4}\\ + \end{pmatrix} + = + \begin{pmatrix} + -6\\ + 2\\ + 4\\ + -2\\ + \end{pmatrix} + \right\} + $ + \item $\mathbb{L} = + \left\{ + \begin{pmatrix} + x_{1}\\ + x_{2}\\ + x_{3}\\ + x_{4}\\ + \end{pmatrix} + \vert + \begin{pmatrix} + x_{1}\\ + x_{2}\\ + x_{3}\\ + x_{4}\\ + \end{pmatrix} + = + \begin{pmatrix} + 6\\ + -2\\ + 0\\ + 0\\ + \end{pmatrix} + +r_{1}\cdot + \begin{pmatrix} + 5\\ + -1\\ + 1\\ + 0\\ + \end{pmatrix} + +r_{2}\cdot + \begin{pmatrix} + -3\\ + 1\\ + 0\\ + 1\\ + \end{pmatrix} + ; \quad r_{1},2_{2} \in \mathbb{R} + \right\} + $ + \item $\mathbb{L} = + \left\{ + \begin{pmatrix} + x_{1}\\ + x_{2}\\ + x_{3}\\ + x_{4}\\ + \end{pmatrix} + \vert + \begin{pmatrix} + x_{1}\\ + x_{2}\\ + x_{3}\\ + x_{4}\\ + \end{pmatrix} + = + \begin{pmatrix} + -6\\ + 2\\ + -2\\ + 0\\ + \end{pmatrix} + +r \cdot + \begin{pmatrix} + -5\\ + 3\\ + 1\\ + 1\\ + \end{pmatrix} + ; \quad r_{1},2_{2} \in \mathbb{R} + \right\} + $ + \end{enumerate} + + \newpage + \section*{Aufgabe 11 (5 Punkte)} + Die Matrix + \begin{equation*} + A = + \begin{pmatrix} + a_{11} & a_{12} & a_{13}\\ + a_{21} & a_{22} & a_{23}\\ + a_{31} & a_{32} & a_{33}\\ + \end{pmatrix} + \end{equation*} + besitzt die Determinante $det(A)=10$. Betrachten Sie die Matrix + \begin{equation*} + A^{*} = + \begin{pmatrix} + a_{21} & a_{22} & a_{23}\\ + a_{11} & a_{12} & a_{13}\\ + 2a_{31} & 2a_{32} & 2a_{33}\\ + \end{pmatrix} + , + \end{equation*} + die durch elementare Zeilen- und Spaltenumformungen aus $A$ und + Transponieren entstanden ist. Wie lautet die Determinante $det(A^{*})$. + + \begin{enumerate}[label=\Alph*)] + \item $det(A^{*})=-5$ + \item $det(A^{*})=-20$ + \item $det(A^{*})=10$ + \item $det(A^{*})=20$ + \item $det(A^{*})=\frac{1}{20}$ + \end{enumerate} + + \section*{Aufgabe 12 (5 Punkte)} + Wie lauten die Eigenwerte der Matrix + \begin{equation*} + A = + \begin{pmatrix} + 7 & 0 & 0\\ + -1 & -3 & 0\\ + 2 & -4 & 5\\ + \end{pmatrix} + ? + \end{equation*} + + \begin{enumerate}[label=\Alph*)] + \item $\lambda_{1}=-4,\lambda_{2}=4,\lambda_{3}=6$ + \item $\lambda_{1}=3,\lambda_{2}=-5,\lambda_{3}=-7$ + \item $\lambda_{1}=1,\lambda_{2}=4,\lambda_{3}=2$ + \item $\lambda_{1}=-3,\lambda_{2}=5,\lambda_{3}=7$ + \item $\lambda_{1}=2,\lambda_{2}=3,\lambda_{3}=0$ + \end{enumerate} +\end{document} +% \image{1}{Capture3.PNG}{DNS-Anfrage}{DNS-Anfrage} \ No newline at end of file From 477d142491213a6418b7dacdb0ffc8d490738a9a Mon Sep 17 00:00:00 2001 From: Fabian Lamprecht Date: Tue, 29 May 2018 08:24:11 +0200 Subject: [PATCH 15/15] fixed indendtation --- .../WS17 MfI2.tex | 320 +++++++++--------- 1 file changed, 159 insertions(+), 161 deletions(-) diff --git a/KTR-MfI-2-B Mathematik für Informatiker 2/WS17 MfI2.tex b/KTR-MfI-2-B Mathematik für Informatiker 2/WS17 MfI2.tex index 238b93c..b8de04d 100644 --- a/KTR-MfI-2-B Mathematik für Informatiker 2/WS17 MfI2.tex +++ b/KTR-MfI-2-B Mathematik für Informatiker 2/WS17 MfI2.tex @@ -14,178 +14,176 @@ \begin{enumerate} \item Aufgabe 1 (8+12 Punkte) Lineare Gleichungssysteme und Matlab/Octave - Geben Sie die Ergebnisse folgender Aufgaben zu linearen Gleichungssystemen in den daf\"ur vorgegebenen Feldern an. - \begin{enumerate} - \item [a)] Betrachten Sie das folgende lineare Gleichungssystem: - \[\begin{array}{lllllll} - 2x_1 & + & 4x_2 & & & = 0 \\ - & & 2x_2 & + & 2x_3 & = 2 \\ - 2x_1 & + & 4x_2 & + & 4x_3 & = 1 - \end{array}\] - \begin{enumerate} - \item [1)] Geben Sie eine Darstellung dieses linearen Gleicungssystems in der Form eines Matrix-Vektor-Produkts $$A\cdot x = b$$ mit Hilfe einer geeigneten Koeffizientenmatrix $A$ und dem Vektor $x$ der Variablen sowie der rechten Seite $b$ des Gleicungssystems an. - \item[2)] Berechnen Sie die L\"osung $x = \left(\begin{array}{c}x_1\\ x_2\\ x_3\end{array}\right) \in \mathbb{R}^3$ durch die \"Uberf\"uhrung des Systems (A~~~b) in Treppennormalform und anschlie{\ss}ende L\"osung unter Verwendung des Gau{\ss}schen Algorithmus. + Geben Sie die Ergebnisse folgender Aufgaben zu linearen Gleichungssystemen in den daf\"ur vorgegebenen Feldern an. + \begin{enumerate} + \item [a)] Betrachten Sie das folgende lineare Gleichungssystem: + $$\begin{array}{lllllll} + 2x_1 & + & 4x_2 & & & = 0 \\ + & & 2x_2 & + & 2x_3 & = 2 \\ + 2x_1 & + & 4x_2 & + & 4x_3 & = 1 + \end{array}$$ + \begin{enumerate} + \item[1)] Geben Sie eine Darstellung dieses linearen Gleicungssystems in der Form eines Matrix-Vektor-Produkts $$A\cdot x = b$$ mit Hilfe einer geeigneten Koeffizientenmatrix $A$ und dem Vektor $x$ der Variablen sowie der rechten Seite $b$ des Gleicungssystems an. + \item[2)] Berechnen Sie die L\"osung $x = \left(\begin{array}{c}x_1\\ x_2\\ x_3\end{array}\right) \in \mathbb{R}^3$ durch die \"Uberf\"uhrung des Systems (A~~~b) in Treppennormalform und anschlie{\ss}ende L\"osung unter Verwendung des Gau{\ss}schen Algorithmus. + \end{enumerate} + \item [b)] Berechnen Sie da lineare Gleichungssystem + $$L\cdot x = \left( \begin{array}{cccc} + L_{11} & 0 & \dots & 0 \\ + L_{21} & L_{22} & \cdots & \vdots \\ + \vdots & \ddots & \ddots & 0 \\ + L_{n1} & \cdots & \cdots & L_{nn} + \end{array}\right)\cdot\left(\begin{array}{c} + x_1 \\x_2\\\vdots\\x_n + \end{array}\right)=\left(\begin{array}{c} + b_1 \\b_2\\\vdots\\b_n + \end{array}\right)=b$$ + mit der unteren Dreiecksmatrix + $$L = \left( \begin{array}{cccc} + L_{11} & 0 & \dots & 0 \\ + L_{21} & L_{22} & \cdots & \vdots \\ + \vdots & \ddots & \ddots & 0 \\ + L_{n1} & \cdots & \cdots & L_{nn} + \end{array}\right)\in \mathbb{R}^{n\times n}, n\in\mathbb{N}, n\geq 2$$ + und nehmen Sie an, dass alle Elemente $L_{ii}\neq 0,i=1,\dots,n$, sind. - \end{enumerate} - - \item [b)] Berechnen Sie da lineare Gleichungssystem - $$L\cdot x = \left( \begin{array}{cccc} - L_{11} & 0 & \dots & 0 \\ - L_{21} & L_{22} & \cdots & \vdots \\ - \vdots & \ddots & \ddots & 0 \\ - L_{n1} & \cdots & \cdots & L_{nn} - \end{array}\right)\cdot\left(\begin{array}{c} - x_1 \\x_2\\\vdots\\x_n - \end{array}\right)=\left(\begin{array}{c} - b_1 \\b_2\\\vdots\\b_n - \end{array}\right)=b$$ - mit der unteren Dreiecksmatrix - $$L = \left( \begin{array}{cccc} - L_{11} & 0 & \dots & 0 \\ - L_{21} & L_{22} & \cdots & \vdots \\ - \vdots & \ddots & \ddots & 0 \\ - L_{n1} & \cdots & \cdots & L_{nn} - \end{array}\right)\in \mathbb{R}^{n\times n}, n\in\mathbb{N}, n\geq 2$$ - und nehmen Sie an, dass alle Elemente $L_{ii}\neq 0,i=1,\dots,n$, sind. - - Der Vektor $x=\left(\begin{array}{c} - x_1 \\x_2\\\vdots\\x_n + Der Vektor $x=\left(\begin{array}{c} + x_1 \\x_2\\\vdots\\x_n \end{array}\right)$ kann mit Hilfe des folgenden \textit{Forward-Substitution} Algorithmus - \begin{equation} - x_i=\left(b_i-\sum^{i-1}_{j=1}~~L_{ij}\cdot x_j\right)/ L_{ii}, i=1,\dots,n - \end{equation} - bestimmt werden, der sukzessive die Elemente $x_i,i=1,\dots,n,$ berechnet. - \newpage - \begin{enumerate} - \item[1)] Implementieren Sie den \textit{Forward-Substitution} Algorithmus als MATLAB oder Octave Funktion $ForSub(b, L, y)$, die f\"ur eine gegebene rechte Seite $b\in\mathbb{R}^n$ und eine invertierbare untere Dreiecksmatrix $L\in\mathbb{R}^{n\times n}$ der Dimension $ n \in \mathbb{N}, n\geq 2$, mit Diagonalelementen $L_{ii}\neq 0, i=1,\dots,n$, als Eingabeparametern des L\"osungsvektor $x\in\mathbb{R}^n$ in (1) berechnet und diesen w\"ahrend des L\"osungsverfahrens sukzessive in den Speicherplatz des Ausgabevektors $y\in\mathbb{R}^n$ schreibt, nachdem $y$ zu Beginn des Algorithmus mit dem Nullvektor initialisieert wurde. - \item[2)] Schreiben Sie ein MATLAB oder Octave Programm, das diese erstellte MATLAB oder Octave Funktion $ForSub$ zur L\"osung des Gleichungssystems $L \cdot x = b$ mit folgenden Eingangsdaten - $$L=\left(\begin{array}{ccc} - 2 & 0 & 0 \\ - 1 & 5 & 0 \\ - 7 & 9 & 8 - \end{array}\right), b = \left(\begin{array}{c} - 6 \\2\\5 - \end{array}\right)$$ - anwendet und dann den L\"osungsvektor $x$ komponentenweise ausgibt. - \end{enumerate} - \end{enumerate} + \begin{equation} + x_i=\left(b_i-\sum^{i-1}_{j=1}~~L_{ij}\cdot x_j\right)/ L_{ii}, i=1,\dots,n + \end{equation} + bestimmt werden, der sukzessive die Elemente $x_i,i=1,\dots,n,$ berechnet. + \newpage + \begin{enumerate} + \item[1)] Implementieren Sie den \textit{Forward-Substitution} Algorithmus als MATLAB oder Octave Funktion $ForSub(b, L, y)$, die f\"ur eine gegebene rechte Seite $b\in\mathbb{R}^n$ und eine invertierbare untere Dreiecksmatrix $L\in\mathbb{R}^{n\times n}$ der Dimension $ n \in \mathbb{N}, n\geq 2$, mit Diagonalelementen $L_{ii}\neq 0, i=1,\dots,n$, als Eingabeparametern des L\"osungsvektor $x\in\mathbb{R}^n$ in (1) berechnet und diesen w\"ahrend des L\"osungsverfahrens sukzessive in den Speicherplatz des Ausgabevektors $y\in\mathbb{R}^n$ schreibt, nachdem $y$ zu Beginn des Algorithmus mit dem Nullvektor initialisieert wurde. + \item[2)] Schreiben Sie ein MATLAB oder Octave Programm, das diese erstellte MATLAB oder Octave Funktion $ForSub$ zur L\"osung des Gleichungssystems $L \cdot x = b$ mit folgenden Eingangsdaten + $$L=\left(\begin{array}{ccc} + 2 & 0 & 0 \\ + 1 & 5 & 0 \\ + 7 & 9 & 8 + \end{array}\right), b = \left(\begin{array}{c} + 6 \\2\\5 + \end{array}\right)$$ + anwendet und dann den L\"osungsvektor $x$ komponentenweise ausgibt. + \end{enumerate} + \end{enumerate} \item Aufgabe 2 (6+6+8 Punkte) Vektorr\"aume - Geben Sie die Ergebnisse folgender Aufgaben zur Theorie der Vektorr\"aume in den daf\"ur vorgesehenen Feldern an. - \begin{enumerate} - \item [a)] Betrachten Sie die Vektoren - $$a=\left(\begin{array}{c} - 0 \\1\\1 - \end{array}\right),b=\left(\begin{array}{c} - 1 \\1\\1 - \end{array}\right),c=\left(\begin{array}{c} - 1 \\0\\1 - \end{array}\right)$$ - des $\mathbb{F}_2$-Vektorraums $\mathbb{F}^3_2$ auf dem K\"orper deer bin\"aren Elemente $\mathbb{F}_2={0,1}$ mit den Operationen $\{\oplus, \odot\}$ der Addition und Multiplikation modulo 2, d. h. f\"ur $x, y\in\mathbb{F}_2,x\oplus y = (x+y)~mod~2,\\ x \odot y = (x\cdot y)~mod~2$.\\ - Sind diese drei Vektoren $\{a,b,c\}\subset\mathbb{F}^3_2$ linear unabh\"angig oder linear abh\"angig? Begr\"unden Sie Ihre Antwort durch geeignete algebraische Argumente. - \item [b)] Betrachten Sie die Matrizen - $$A=\left(\begin{array}{cc} - 0 & 1 \\ - 0 & 1 \\ - \end{array}\right), B= \left(\begin{array}{cc} - 0 & 0 \\ - 1 & 0 - \end{array}\right), C = \left(\begin{array}{cc} - 0 & 1 \\ - 1 & 0 - \end{array}\right)$$ - des Vektorraums $\mathbb{R}^{2\times 2}$.\\ - Sind diese drei Matrizen $\{A,B,C\}\subset\mathbb{R}^{2\times 2}$ linear unabh\"angig oder linear abh\"angig? Begr\"unden Sie Ihre Antwort mit Hilfe einer geeigneten algebraischen Argumentation. - \item[c)] Weisen Sie durch eine geeignete algebraische Argumentation nach, dass die Menge - $$U=\{v=\left(\begin{array}{c} - x_1 \\x_2\\x_3 - \end{array}\right)\in \mathbb{R}^{3} | 4x_1+3x_2=0\} \subset \mathbb{R}^3$$ - ein Untervektorraum des reellen Vektorraums $V = \mathbb{R}^3$ ist. - \end{enumerate}\newpage + Geben Sie die Ergebnisse folgender Aufgaben zur Theorie der Vektorr\"aume in den daf\"ur vorgesehenen Feldern an. + \begin{enumerate} + \item [a)] Betrachten Sie die Vektoren + $$a=\left(\begin{array}{c} + 0 \\1\\1 + \end{array}\right),b=\left(\begin{array}{c} + 1 \\1\\1 + \end{array}\right),c=\left(\begin{array}{c} + 1 \\0\\1 + \end{array}\right)$$ + des $\mathbb{F}_2$-Vektorraums $\mathbb{F}^3_2$ auf dem K\"orper deer bin\"aren Elemente $\mathbb{F}_2={0,1}$ mit den Operationen $\{\oplus, \odot\}$ der Addition und Multiplikation modulo 2, d. h. f\"ur $x, y\in\mathbb{F}_2,x\oplus y = (x+y)~mod~2,\\ x \odot y = (x\cdot y)~mod~2$.\\ + Sind diese drei Vektoren $\{a,b,c\}\subset\mathbb{F}^3_2$ linear unabh\"angig oder linear abh\"angig? Begr\"unden Sie Ihre Antwort durch geeignete algebraische Argumente. + \item [b)] Betrachten Sie die Matrizen + $$A=\left(\begin{array}{cc} + 0 & 1 \\ + 0 & 1 \\ + \end{array}\right), B= \left(\begin{array}{cc} + 0 & 0 \\ + 1 & 0 + \end{array}\right), C = \left(\begin{array}{cc} + 0 & 1 \\ + 1 & 0 + \end{array}\right)$$ + des Vektorraums $\mathbb{R}^{2\times 2}$.\\ + Sind diese drei Matrizen $\{A,B,C\}\subset\mathbb{R}^{2\times 2}$ linear unabh\"angig oder linear abh\"angig? Begr\"unden Sie Ihre Antwort mit Hilfe einer geeigneten algebraischen Argumentation. + \item[c)] Weisen Sie durch eine geeignete algebraische Argumentation nach, dass die Menge + $$U=\{v=\left(\begin{array}{c} + x_1 \\x_2\\x_3 + \end{array}\right)\in \mathbb{R}^{3} | 4x_1+3x_2=0\} \subset \mathbb{R}^3$$ + ein Untervektorraum des reellen Vektorraums $V = \mathbb{R}^3$ ist. + \end{enumerate}\newpage \item Aufgabe 3 (6+6+4+4 Punkte) Lineare Abbildungen - Betrachten Sie die lineare Abbildung - $$ f: V \longrightarrow W $$ - $$ v=\left(\begin{array}{c}x \\ y \\ z \end{array}\right)\rightarrow f(v)=\left(\begin{array}{c} y\oplus z \\ x\oplus z \\ x\oplus y \end{array}\right) $$ - zwischen den $\mathbb{F}_2$-Vektorr\"aumen $V=\mathbb{F}^3_2$ und $W=\mathbb{F}^3_2$ \"uber dem K\"orper der bin\"aren Elemente $\mathbb{F}_2 = {0,1}$ mit den Operationen $\{\oplus,\odot\}$ der Addition und Multiplikation modulo 2, d.h. $x\oplus y = (x+y) mod 2, x\odot y = (x\cdot y) mod 2$ f\"ur $x,y\in\mathbb{F}_2$. - \begin{itemize} - \item [a)] W\"ahlen Sie jeweils die kanonische Basis $\mathcal{B} = \{e_1,e_2,e_3\}$ im $\mathbb{F}_2$-Vektorraum $V = \mathbb{F}^3_2$ bzw. $W=\mathbb{F}^3_2$ aus und geben Sie die der Abbildung $f$ zugeordnete Matrix $A_f$ bzgl. dieser beiden Basen an. - \item [b)] Bestimmen Sie mit Hilfe einer geeigneten algebraischen Argumentation oder eines geeigneten Berechnungsverfahrens den Rang $rg(f)$ der Abbildung $f$. - \item [c)] Bestimmen Sie mit Hilfe einer geeigneten Formel, die Sie explizit angeben sollen, die Dimension $k=dim(Ker(f))$ des Kerns $Ker(f)$ der Abbildung $f$. - \item [d)] Geben Sie eine Basis $\mathcal{C} = \{v_1,\dots,v_k\}$ des Kerns $Ker(f)$ der Abbildung $f$ an. - \end{itemize} + Betrachten Sie die lineare Abbildung + $$ f: V \longrightarrow W $$ + $$ v=\left(\begin{array}{c}x \\ y \\ z \end{array}\right)\rightarrow f(v)=\left(\begin{array}{c} y\oplus z \\ x\oplus z \\ x\oplus y \end{array}\right) $$ + zwischen den $\mathbb{F}_2$-Vektorr\"aumen $V=\mathbb{F}^3_2$ und $W=\mathbb{F}^3_2$ \"uber dem K\"orper der bin\"aren Elemente $\mathbb{F}_2 = {0,1}$ mit den Operationen $\{\oplus,\odot\}$ der Addition und Multiplikation modulo 2, d.h. $x\oplus y = (x+y) mod 2, x\odot y = (x\cdot y) mod 2$ f\"ur $x,y\in\mathbb{F}_2$. + \begin{itemize} + \item [a)] W\"ahlen Sie jeweils die kanonische Basis $\mathcal{B} = \{e_1,e_2,e_3\}$ im $\mathbb{F}_2$-Vektorraum $V = \mathbb{F}^3_2$ bzw. $W=\mathbb{F}^3_2$ aus und geben Sie die der Abbildung $f$ zugeordnete Matrix $A_f$ bzgl. dieser beiden Basen an. + \item [b)] Bestimmen Sie mit Hilfe einer geeigneten algebraischen Argumentation oder eines geeigneten Berechnungsverfahrens den Rang $rg(f)$ der Abbildung $f$. + \item [c)] Bestimmen Sie mit Hilfe einer geeigneten Formel, die Sie explizit angeben sollen, die Dimension $k=dim(Ker(f))$ des Kerns $Ker(f)$ der Abbildung $f$. + \item [d)] Geben Sie eine Basis $\mathcal{C} = \{v_1,\dots,v_k\}$ des Kerns $Ker(f)$ der Abbildung $f$ an. + \end{itemize} \item Aufgabe 4 (10+4+6 Punkte) Gruppen- und Matrizenalgebra - \begin{enumerate} - \item [a)] Beantworten Sie folgende Fragen zur Algebra einfacher Gruppen $(G,\oplus,e)$ - \begin{enumerate} - \item [1)] Betrachten Sie die Gruppe $(G,\oplus,e)=(\mathbb{F}^3_2,\oplus,e)$ aller bin\"arer Tupel - $$(a,b)\in\mathbb{F}^3_2 = \{(0,0),(1,0),(0,1),(1,1)\}$$ - mit den Komponenten $a,b\in\mathbb{F}_2= \{ 0,1\}$ und der komponentenweise definierten Addition modulo 2 als Gruppenverkn\"upfung $\oplus$. + \begin{enumerate} + \item [a)] Beantworten Sie folgende Fragen zur Algebra einfacher Gruppen $(G,\oplus,e)$ + \begin{enumerate} + \item [1)] Betrachten Sie die Gruppe $(G,\oplus,e)=(\mathbb{F}^3_2,\oplus,e)$ aller bin\"arer Tupel + $$(a,b)\in\mathbb{F}^3_2 = \{(0,0),(1,0),(0,1),(1,1)\}$$ + mit den Komponenten $a,b\in\mathbb{F}_2= \{ 0,1\}$ und der komponentenweise definierten Addition modulo 2 als Gruppenverkn\"upfung $\oplus$. - Geben Sie das neutrale Element $e\in G$ an und berechnen Sie f\"ur $f=(1,0), g=(0,1),h=(1,1)$ die folgenden Gruppenelemente $k,l\in G$ + Geben Sie das neutrale Element $e\in G$ an und berechnen Sie f\"ur $f=(1,0), g=(0,1),h=(1,1)$ die folgenden Gruppenelemente $k,l\in G$ - $$k = g \oplus f = $$ - $$l = g \oplus h = $$ - $$e = $$ + $$k = g \oplus f = $$ + $$l = g \oplus h = $$ + $$e = $$ - \item[2)] Definiert die Teilmenge - $$U=\{(0,0),(1,0),(0,1)\}\subset \mathbb{F}^2_2=G$$ - der Untergruppe von $(\mathbb{F}^2_2,\oplus,e)$? Geben Sie eine algebraische Begr\"undung an. - \item[3)] Die aus drei verschiedenen Elementen besthende Menge $G={e,a,h}$ mit der additiven Verkn\"upfung $z=x\oplus y \in G $ auf zwei Gruppenelementen $x,y \in G$ kann durch eine entsprechende Verkn\"upfungstabelle T beschrieben werden (siehe unten), z.B. ergibt sich aus der Additon der Elemente $x = b$ in der 5. Zeile und $y=a$ in der 4. Spalte das Element - $$z=x\oplus y = b \oplus a = e$$ - Erg\"anzen Sie die fehlenden Elemente der 5. Zeile hinter $x=b$ und der 5. Spalte unterhalb von $y=b$ in der Verkn\"upfungstabelle $T$ sowie das fehlende Element der 5. Zeile und 3. Spalte f\"ur $x=b$ und $y=e$ derart, dass $T$ eine kommutative Gruppe $(G,\oplus,e)$ definiert. + \item[2)] Definiert die Teilmenge + $$U=\{(0,0),(1,0),(0,1)\}\subset \mathbb{F}^2_2=G$$ + der Untergruppe von $(\mathbb{F}^2_2,\oplus,e)$? Geben Sie eine algebraische Begr\"undung an. + \item[3)] Die aus drei verschiedenen Elementen besthende Menge $G={e,a,h}$ mit der additiven Verkn\"upfung $z=x\oplus y \in G $ auf zwei Gruppenelementen $x,y \in G$ kann durch eine entsprechende Verkn\"upfungstabelle T beschrieben werden (siehe unten), z.B. ergibt sich aus der Additon der Elemente $x = b$ in der 5. Zeile und $y=a$ in der 4. Spalte das Element + $$z=x\oplus y = b \oplus a = e$$ + Erg\"anzen Sie die fehlenden Elemente der 5. Zeile hinter $x=b$ und der 5. Spalte unterhalb von $y=b$ in der Verkn\"upfungstabelle $T$ sowie das fehlende Element der 5. Zeile und 3. Spalte f\"ur $x=b$ und $y=e$ derart, dass $T$ eine kommutative Gruppe $(G,\oplus,e)$ definiert. - \begin{tabular}{|c|c||c|c|c|}\hline - $\oplus$ & \multicolumn{4}{|c|}{}\\\hline - &y= & e & a & b \\\hline\hline - \multirow{3}{*}{b=} & e & e & a & \\\hline - & a & a & b & e \\\hline - & b & & e & \\\hline - \end{tabular} - \end{enumerate} - \item[b)] Es sei $A=\left(\begin{array}{ccc} - 2 & 1 & 4\\ - 1 & -1 & -5 - \end{array}\right), B = \left(\begin{array}{ccc} - 4 & 1 & 0\\ - 5 & 2 & 1\\ - 10& 7 & 6 - \end{array}\right)$ - \begin{enumerate} - \item[1)] Berechnen Sie das Element $C_{23}$ der Produktmatrix $C=A\cdot B$ - \item[2)] Wie lautet die transponierte Matrix $A^t$ zu $A$? - \item[3)] Es sei - $$A=\left(\begin{array}{ccc} - 1 & 0 & 0\\ - 0 & 1 & -1\\ - 2 & -4 & 3 - \end{array}\right)$$ - Berechnen Sie die inverse Matrix $C=A^{-1}$ zu $A$ - \end{enumerate} - \item Aufgabe 5(4+6+10 Punkte) Lineare Abbildungs- und Eigenwerttheorie + \begin{tabular}{|c|c||c|c|c|}\hline + $\oplus$ & \multicolumn{4}{|c|}{}\\\hline + &y= & e & a & b \\\hline\hline + \multirow{3}{*}{b=} & e & e & a & \\\hline + & a & a & b & e \\\hline + & b & & e & \\\hline + \end{tabular} + \end{enumerate} + \item[b)] Es sei $A=\left(\begin{array}{ccc} + 2 & 1 & 4\\ + 1 & -1 & -5 + \end{array}\right), B = \left(\begin{array}{ccc} + 4 & 1 & 0\\ + 5 & 2 & 1\\ + 10& 7 & 6 + \end{array}\right)$ + \begin{enumerate} + \item[1)] Berechnen Sie das Element $C_{23}$ der Produktmatrix $C=A\cdot B$ + \item[2)] Wie lautet die transponierte Matrix $A^t$ zu $A$? + \item[3)] Es sei + $$A=\left(\begin{array}{ccc} + 1 & 0 & 0\\ + 0 & 1 & -1\\ + 2 & -4 & 3 + \end{array}\right)$$ + Berechnen Sie die inverse Matrix $C=A^{-1}$ zu $A$ + \end{enumerate} + \end{enumerate} + \item Aufgabe 5(4+6+10 Punkte) Lineare Abbildungs- und Eigenwerttheorie - Geben Sie die Ergebnisser folgender Aufgaben in den daf\"ur vorgesehenen Feldern an. - \begin{enumerate} - \item [a)] Berechnen Sie die Determinante der folgenden Matrix - $$A=\left(\begin{array}{ccc} - 0.3 & 0 & 0\\ - 0.7 & 0 & 1\\ - 0 & 2 & 0 - \end{array}\right)$$ - $det(A)=$ - \item[b)] Berechnen Sie einen Eigenvektor $v\in\mathbb{R}^2$ zum Eigenwert $\lambda = 2$ der folgenden Matrix $B$: - $$B=\left(\begin{array}{cc} - 1 & -1\\ - -1 & 1 - \end{array}\right)\in\mathbb{R}^{2\times 2}$$ - \item[c)] Bestimmen Sie das charakteristische Polinom $p(\lambda) = \lambda^3+p_2\lambda^2+p_1\lambda+p_0$ der folgenden Matrix $A$ und berechnen Sie mit dessen Hilfe alle Eigenwerte $\lambda_i\in\mathbb{R}, i\in\{1,2,3\}$, der Matrix $A$: - $$A=\left(\begin{array}{ccc} - 0.9 & 0 & 0\\ - 0 & -1 & 1\\ - 0.1 & -2 & 2 - \end{array}\right)$$ - \end{enumerate} - \end{enumerate} -\end{enumerate} + Geben Sie die Ergebnisser folgender Aufgaben in den daf\"ur vorgesehenen Feldern an. + \begin{enumerate} + \item [a)] Berechnen Sie die Determinante der folgenden Matrix + $$A=\left(\begin{array}{ccc} + 0.3 & 0 & 0\\ + 0.7 & 0 & 1\\ + 0 & 2 & 0 + \end{array}\right)$$ + $det(A)=$ + \item[b)] Berechnen Sie einen Eigenvektor $v\in\mathbb{R}^2$ zum Eigenwert $\lambda = 2$ der folgenden Matrix $B$: + $$B=\left(\begin{array}{cc} + 1 & -1\\ + -1 & 1 + \end{array}\right)\in\mathbb{R}^{2\times 2}$$ + \item[c)] Bestimmen Sie das charakteristische Polinom $p(\lambda) = \lambda^3+p_2\lambda^2+p_1\lambda+p_0$ der folgenden Matrix $A$ und berechnen Sie mit dessen Hilfe alle Eigenwerte $\lambda_i\in\mathbb{R}, i\in\{1,2,3\}$, der Matrix $A$: + $$A=\left(\begin{array}{ccc} + 0.9 & 0 & 0\\ + 0 & -1 & 1\\ + 0.1 & -2 & 2 + \end{array}\right)$$ + \end{enumerate} + \end{enumerate} \end{document} \ No newline at end of file