Initial Commit and Notes from HS24
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hs24/lineare_algebra/.DS_Store
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\chapter{Ausgleichsrechnung}
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\input{ausgleichsrechnung_least_squares.tex}
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\input{normalengleichung.tex}
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\input{qrzerlegung.tex}
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\nt{
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Nützliche Videos:
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\begin{itemize}
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\item Least squares approximation $|$ Linear Algebra $|$ Khan Academy (\url{https://www.youtube.com/watch?v=MC7l96tW8V8})
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\item A=QR Factorizations: Part 4/5 Least Squares (\url{https://www.youtube.com/watch?v=B3_vu5WcBzg})
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\end{itemize}
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}
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\newpage
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\section{Ausgleichsrechnung ("Least Sqares")}
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Nehmen wir an, dass wir eine Datenbank haben mit Messungen, welches wir in einem Graphen eingefügt haben. Nun wollen wir eine Funktion finden, welches durch die Messwerte geht. Wir können das folgende Gleichungssystem bilden:
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\begin{equation}
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\mathbf{A} \vec{x} = \vec{b}
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\label{eq:least square prev}
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\end{equation}
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Wobei die Matrix $\mathbf{A}$ die parameter von $\vec{x}$ sind und $\vec{b}$ die resultierende Gerade bildet.
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Wir schauen uns nun die folgende Grafik an:
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\includegraphics[width=\textwidth]{fig/Fig_3.png}
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Wie man anhand der Grafik sehen kann, sieht man, dass es $\vec{x}$ nicht existiert, da es keine Funktion existiert, welche durch alle Punkte geht. Was ist, wenn man trotz allem eine Funktion finden möchte?
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\\
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Wir nehmen eine neue Betrachtungsweise. Nehmen wir an, dass die Matrix $\mathbf{A}$ eine Ebene bildet. Die Gleichung $\mathbf{A} \vec{x} = \vec{b}$ bildet dann einen Vector $\vec{b}$ welches in der Ebene ist, die von $\mathbf{A}$ gebildet wird. Nehmen wir an, dass $\vec{b}$ nicht in der Ebene ist. Daraus folgt, dass es kein $\vec{x}$ gibt, welches die Gleichung bildet. Was kann man stattdessen machen?
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\\
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\\
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Man könnte ein $\vec{\hat{x}}$ wählen, welche die Gleichung zwar nicht bildet aber den Vektor $\vec{b}$ annähert. Daraus schliesst man, dass $\vec{\hat{x}}$ $\vec{b}$ am ehesten annähert, wenn die Distanz von $b'$, welches den $\vec{b'}$ in der Ebene bildet und $b$ am geringsten ist.
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\includegraphics[width=\textwidth]{fig/Fig_4.png}
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Anahand der Figur kann man erkennen, dass $\vec{\hat{x}}$ so gewählt werden sollte, dass es die Projektion von $\vec{b}$ bildet. Daraus kann man die folgende Definition ableiten.
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\dfn{Lineare Ausgleichsrechnung als lineares Gleichungssystem \cite{Gradinaru2024}}{
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Sei eine $m \times n$ Matrix $\mathbf{A}$ und der Vektor $\vec{b}$ mit $m$ Einträgen. Gesucht ist ein $\hat{x}$, so dass
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\[
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||\mathbf{A}\vec{\hat{x}} - \vec{b}|| = \text{min} ||\mathbf{A}\vec{x} - \vec{b}||
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.\]
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}
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Es gibt zwei wege um $\vec{\hat{x}}$ zu berechnen, nähmlich die Lösung mit der Normalengleichung und Lösung mit der QR-Zerlegung.
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\section{Lösung mit der Normalengleichung}
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\dfn{Normalengleichung \cite{Gradinaru2024}}{
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Die Gleichung
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\[
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\mathbf{A} ^{\text{T}} \mathbf{A} \vec{x} = \vec{b} \mathbf{A} ^{\text{T}}
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\]
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heisst Normalengleichung.
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}
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Mit dieser Funktion kann man ein $\vec{\hat{x}}$ finden, welche den Vektor $\vec{b}$ annähert. Dabei wird wie folgt vorgegangen:
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\begin{enumerate}
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\item Man multipliziert die Matrix $\mathbf{A}$ mit ihrem Transponierten
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\item Man multipliziert den Vector $\vec{b}$ mit $\mathbf{A} ^{\text{T}}$.
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\item Man gausst die Matrix $\mathbf{A}$ um nach den Parametern von $\vec{x}$ aufzulösen
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\end{enumerate}
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Wie man sieht, ist der Nachteil dieser Methode, dass man die Matrix $\mathbf{A}$ gaussen muss. Es gibt aber eine zweite Methode um $\vec{\hat{x}}$ herauszufinden.
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hs24/lineare_algebra/ausgleichsrechnung/qrzerlegung.tex
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hs24/lineare_algebra/ausgleichsrechnung/qrzerlegung.tex
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\section{Lösung mit der QR-Zerlegung}
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Eine einfachere Methode $\vec{\hat{x}}$ zu finden ist mit Hilfe der QR-Zerlegung.
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\dfn{Lösung mit der QR-Zerlegung}{
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Sei die QR-Zerlegung einer $m \times n$ Matrix $\mathbf{A}$ gegeben, so ist $\vec{\hat{x}}$ durch
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\[
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\mathbf{R} \vec{x} = \mathbf{Q} ^{\text{T}} \vec{b}
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\]
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definiert.
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}
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Dabei geht man wie bei der Normalengleichung vor, jedoch fällt das Gaussen weg, da $\mathbf{R}$ eine obere Dreiecksmatrix ist.
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hs24/lineare_algebra/determinante/anwendungen.tex
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hs24/lineare_algebra/determinante/anwendungen.tex
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\section{Anwendungen}
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Wie vorher erwähnt, wird die Determinante als ein Werkzeug für lineare Transformationen verwendet. Was wir bereits auch angeschaut haben anhand von Beispielen ist, dass det die Fläche ist welche eine $2 \times 2$ Matrix aufspannt.
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\\
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Für eine $3 \times 3$ Matrix, ist det das Volumen des Körpers, welches von den Vektoren der Matrix gebildet wird.
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\\
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Neben dem hat det eine Relevanz mit dem Kreuzprodukt:
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\[
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|(\mathbf{u} \times \mathbf{v}) \mathbf{w}| = |\text{det} \begin{bmatrix}
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u_1 & u_2 & u_3\\
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v_1 & v_2 & v_3\\
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w_1 & w_2 & w_3\\
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\end{bmatrix}|
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.\]
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hs24/lineare_algebra/determinante/definition_eigenschaften.tex
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hs24/lineare_algebra/determinante/definition_eigenschaften.tex
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\section{Definition und Eigenschaften}
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\dfn{Determinante \cite{Gradinaru2024}}{
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Die Determinante ist eine Funktion
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\[
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\text{det : } \underbrace{\mathbb{R} ^{n} \times \mathbb{R} ^{n} \times ... \times \mathbb{R} ^{n}}_{n} \rightarrow \mathbb{R}
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\]
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mit folgenden Eigenschaften:
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\begin{itemize}
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\item[D1] $\text{det } \mathbf{I}_n = 1$
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\item[D2] det wechselt das Vorzeichen, wenn zwei Zeilen oder Spalten vertauscht werden (Antisymmetrie).
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\item[D3] det ist linear in jeder Zeile und Spalte:
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\[
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\text{det} \begin{bmatrix}
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ta & tb\\
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c & d\\
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\end{bmatrix} = t \cdot \text{det} \begin{bmatrix}
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a & b\\
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c & d\\
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\end{bmatrix}
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.\]
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\[
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\text{det} \begin{bmatrix}
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a + \tilde{a} & b + \tilde{b}\\
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c & d\\
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\end{bmatrix} = \text{det} \begin{bmatrix}
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a & b\\
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c & d\\
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\end{bmatrix} + \text{det} \begin{bmatrix}
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\tilde{a} & \tilde{b}\\
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c & d\\
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\end{bmatrix}
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.\]
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\end{itemize}
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}
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Was ist genau die Determinante und was ist der Konzept dahinter?
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\\
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\\
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Die Determinante ist ein Hilfsmittel um eine lineare Transformation besser verstehen zu können, genauer um welchen Faktor die Fläche sich vergrössert. Betrachten wir es anhand einer Grafik.
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\\
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Wir wissen von D1, dass $\text{det } \mathbf{I}_2 = 1$. Versuchen wir mal dies in einer Grafik darzustellen.
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\includegraphics[width=\textwidth]{fig/Fig_5.png}
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$\mathbf{I}_2$ bildet im Koordinatensystem eine Fläche mit einer Grösse von 1. Wir nehmen nun einfachtshalber mal eine willkürliche Dreiecksmatrix $\mathbf{A}$. Diese Zeichnen wir auch in das Koordinatensystem ein.
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\includegraphics[width=\textwidth]{fig/Fig_6.png}
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Wir sehen, dass die Matrix $\mathbf{A}$ eine Fläche aufspannt mit Grösse 6. Somit ist det 6. Wir nehemen nun eine andere Matrix $\mathbf{B}$.
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\includegraphics[width=\textwidth]{fig/Fig_7.png}
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Wir sehen, dass die Matrix eine lineare Transformation ist, jedoch bleibt die Fläche gleich gross da det 1 ist. Somit würden die Flächen ihre Grösse nach der Transformation nicht verändern.
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\\
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Was ist mit negativen det. Gibt es negative det und wie soll man sich negative det sich vorstellen?
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\\
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Wir betrachten neben der $\mathbf{I}_2$ die Matrix $\mathbf{C}$.
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\includegraphics[width=\textwidth]{fig/Fig_8.png}
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Wir sehen, dass die Fläche, welche die Matrix $\mathbf{C}$ bildet in vergleich zu Matrix $\mathbf{B}$ gespiegelt ist. Daraus kann man ziehen, dass negative det eine Skalierung und eine Spiegelung sind.
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\\
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Wie sieht es aus mit det die 0 sind?
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Betrachten wir die Matrix $\mathbf{D}$.
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\includegraphics[width=\textwidth]{fig/Fig_9.png}
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Die Matrix $\mathbf{D}$ hat keine Fläche, da die Vektoren, welche die Fläche bilden übereinander sind. In manchen Fällen können Determinanten auch ein Punkt bilden.
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\\
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\\
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Da wir das Konzept von Determinanten jetzt besser verstehen. Können wir mit ein paar wichtige Rechenregel für Determinanten fortfahren. Die Rechenregel sind ausführlich im Skript von Dr. Gradinaru \cite{Gradinaru2024} beschrieben.
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\\
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\begin{itemize}
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\item Falls zwei Spalten oder Zeilen einer Matrix identisch sind, so ist die Determinante der Matrix 0.
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\item Lineare Kombinationen von Zeilen der Matrix ändert die Determinante dieser Matrix nicht.
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\item Wenn eine Matrix eine Nullzeile oder eine Nullspalte hat, so ist ihre Determinante 0.
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\item Falls $\mathbf{A}$ eine Dreiecksmatrix ist, so ist die Determinante von A das Produkt der Diagonaleinträge.
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\item Falls A singulär ist, entsteht bei der Gauss-Elimination eine Nullzeile und die Determinante ist 0.
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\item det $\mathbf{A} \mathbf{B}$ = det $\mathbf{A} \cdot$ det $\mathbf{B}$.
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\item \[
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\text{det } \mathbf{A} ^{-1} = \frac{1}{\text{det } \mathbf{A}}
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.\]
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\item $\text{det } \mathbf{A} = a_{11} \cdot \text{det } \mathbf{A} _{11} - a_{12} \cdot \text{det } \mathbf{A} _{12} + ... + a_{1n} \cdot det \mathbf{A}_{1n}$
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\end{itemize}
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\nt{
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Es gibt noch ein paar Tricks um det zu berechnen:
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\begin{itemize}
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\item det einer $3 \times 3$ Matrix lässt sich auch so rechnen:
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\begin{center}
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\includegraphics[width=0.6\textwidth]{fig/Fig_10.png}
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\end{center}
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\item Die Regel mit der Dreiecksmatrix lässt sich ein wenig erweitern. Falls man Blöcke bilden kann so lässt sich det durch das Multiplizieren der einzelnen det der Blöcke berechnen.
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\begin{center}
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\includegraphics[width=0.3\textwidth]{fig/Fig_11.png}
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\end{center}
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\end{itemize}
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}
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hs24/lineare_algebra/determinante/determinante.tex
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\chapter{Determinante}
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\input{definition_eigenschaften.tex}
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\input{anwendungen.tex}
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\nt{
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Nützliche Videos:
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\begin{itemize}
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\item The determinant $|$ Chapter 6, Essence of linear algebra \url{https://www.youtube.com/watch?v=Ip3X9LOh2dk}
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\end{itemize}
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}
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\newpage
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hs24/lineare_algebra/disclaimer.tex
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\section*{DISCLAIMER}
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Diese Notizen wurden verfasst auf Basis der Vorlesung Lineare Algebra (HS24) von V. Gradinaru und dem Skript "Lineare Algebra" von Vasile Gradinaru.
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\\
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\\
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Ich übernehme keine Haftung über mögliche Fehler in den Notizen (Es hat sicherlich ein paar drinnen, da ich teils Sätze umformuliert habe und meine Persönliche Notizen beigefügt habe!).
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\\
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\\
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Alle Grafiken wurden eigenhändig mit Manim \cite{MCD2024} generiert.
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\\
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\\
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Fehler können per Mail an \href{mailto:jirruh@ethz.ch}{jirruh@ethz.ch} gemeldet werden.
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\chapter{Eigenwertproblem}
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hs24/lineare_algebra/graphs/__pycache__/graph.cpython-312.pyc
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391
hs24/lineare_algebra/graphs/graph.py
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hs24/lineare_algebra/graphs/graph.py
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from manim import *
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class LSgraph1(Scene):
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def construct(self):
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# self.camera.background_color = "WHITE"
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ax = Axes(
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x_range=[-0.3, 10.3, 1],
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y_range=[-0.3, 5.3, 1],
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x_length=10.6,
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axis_config={
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"include_numbers": True,
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"color": BLACK,
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},
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)
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self.camera.background_color = WHITE
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axes_lables = ax.get_axis_labels().set_color(BLACK)
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dot1 = Dot(point=ax.c2p(2, 1.3, 0), color=BLUE)
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dot2 = Dot(point=ax.c2p(3, 1.6, 0), color=BLUE)
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dot3 = Dot(point=ax.c2p(5, 2.0, 0), color=BLUE)
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dot4 = Dot(point=ax.c2p(8, 4.7, 0), color=BLUE)
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dot5 = Dot(point=ax.c2p(10, 4.2, 0), color=BLUE)
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function_graph = ax.plot(lambda x: 0.5 * x, color=BLACK)
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plot = VGroup(ax, axes_lables, dot1, dot2, dot3, dot4, dot5, function_graph)
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self.add(plot)
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class LSgraph2(ThreeDScene):
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def construct(self):
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# self.camera.background_color = "WHITE"
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ax = ThreeDAxes(
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x_range=[-0.3, 10.3, 1],
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y_range=[-0.3, 10.3, 1],
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z_range=[-0.3, 10.3, 1],
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x_length=10.6,
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axis_config={
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"include_numbers": True,
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"color": BLACK,
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},
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)
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self.camera.background_color = WHITE
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self.set_camera_orientation(
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phi=56 * DEGREES, theta=-30 * DEGREES, gamma=-7 * DEGREES, zoom=0.5
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)
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axes_lables = ax.get_axis_labels().set_color(BLACK)
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plane = Surface(
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lambda u, v: ax.c2p(u, v, u * 0),
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u_range=[2, 8],
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v_range=[2, 8],
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checkerboard_colors=[BLUE, BLUE],
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fill_opacity=0.75,
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stroke_width=0,
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)
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vector = Arrow3D(ax.c2p(3, 3, 0), ax.c2p(6, 6, 3), color=RED)
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vector1 = Arrow3D(ax.c2p(3, 3, 0), ax.c2p(6, 6, 0), color=GREEN)
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line = Line(ax.c2p(6, 6, 0), ax.c2p(6, 6, 3), color=ORANGE)
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self.add(ax, axes_lables, vector, vector1, plane, line)
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class Detgraph1(Scene):
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def construct(self):
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ax = Axes(
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x_range=[-5.3, 5.3, 1],
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y_range=[-5.3, 5.3, 1],
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x_length=7,
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y_length=7,
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||||
axis_config={
|
||||
"include_numbers": True,
|
||||
"color": BLACK,
|
||||
},
|
||||
)
|
||||
x = ax.get_x_axis()
|
||||
x.numbers.set_color(BLACK)
|
||||
y = ax.get_y_axis()
|
||||
y.numbers.set_color(BLACK)
|
||||
|
||||
self.camera.background_color = WHITE
|
||||
|
||||
axes_lables = ax.get_axis_labels().set_color(BLACK)
|
||||
|
||||
vec1 = Arrow3D(start=ax.c2p(0, 0, 0), end=ax.c2p(0, 1, 0), color=RED)
|
||||
vec2 = Arrow3D(start=ax.c2p(0, 0, 0), end=ax.c2p(1, 0, 0), color=BLUE)
|
||||
|
||||
plane = Surface(
|
||||
lambda u, v: ax.c2p(u, v, u * 0),
|
||||
u_range=[0, 1],
|
||||
v_range=[0, 1],
|
||||
checkerboard_colors=[GREEN, GREEN],
|
||||
fill_opacity=0.75,
|
||||
stroke_width=0,
|
||||
)
|
||||
|
||||
text = MathTex(
|
||||
r"\begin{bmatrix} 1 & 0\\ 0 & 1\end{bmatrix}", color=BLACK
|
||||
).move_to(ax.c2p(-2, 2, 0))
|
||||
|
||||
self.add(ax, axes_lables, vec1, vec2, plane, text)
|
||||
|
||||
|
||||
class Detgraph2(Scene):
|
||||
def construct(self):
|
||||
ax = Axes(
|
||||
x_range=[-5.3, 5.3, 1],
|
||||
y_range=[-5.3, 5.3, 1],
|
||||
x_length=7,
|
||||
y_length=7,
|
||||
axis_config={
|
||||
"include_numbers": True,
|
||||
"color": BLACK,
|
||||
},
|
||||
)
|
||||
x = ax.get_x_axis()
|
||||
x.numbers.set_color(BLACK)
|
||||
y = ax.get_y_axis()
|
||||
y.numbers.set_color(BLACK)
|
||||
self.camera.background_color = WHITE
|
||||
axes_lables = ax.get_axis_labels().set_color(BLACK)
|
||||
|
||||
vec1 = Arrow3D(start=ax.c2p(0, 0, 0), end=ax.c2p(0, 1, 0), color=RED)
|
||||
vec2 = Arrow3D(start=ax.c2p(0, 0, 0), end=ax.c2p(1, 0, 0), color=BLUE)
|
||||
plane1 = Surface(
|
||||
lambda u, v: ax.c2p(u, v, u * 0),
|
||||
u_range=[0, 1],
|
||||
v_range=[0, 1],
|
||||
checkerboard_colors=[GREEN, GREEN],
|
||||
fill_opacity=0.75,
|
||||
stroke_width=0,
|
||||
)
|
||||
|
||||
vec3 = Arrow3D(start=ax.c2p(0, 0, 0), end=ax.c2p(0, 2, 0), color=PURPLE)
|
||||
vec4 = Arrow3D(start=ax.c2p(0, 0, 0), end=ax.c2p(3, 0, 0), color=PINK)
|
||||
plane2 = Surface(
|
||||
lambda u, v: ax.c2p(u, v, u * 0),
|
||||
u_range=[0, 3],
|
||||
v_range=[0, 2],
|
||||
checkerboard_colors=[YELLOW, YELLOW],
|
||||
fill_opacity=0.75,
|
||||
stroke_width=0,
|
||||
)
|
||||
|
||||
text = MathTex(
|
||||
r"\mathbf{A} \begin{bmatrix} 3 & 0\\ 0 & 2\end{bmatrix}", color=BLACK
|
||||
).move_to(ax.c2p(-2.5, 2, 0))
|
||||
|
||||
self.add(ax, axes_lables, plane2, plane1, vec4, vec3, vec2, vec1, text)
|
||||
|
||||
|
||||
class Detgraph3(Scene):
|
||||
def construct(self):
|
||||
ax = Axes(
|
||||
x_range=[-5.3, 5.3, 1],
|
||||
y_range=[-5.3, 5.3, 1],
|
||||
x_length=7,
|
||||
y_length=7,
|
||||
axis_config={
|
||||
"include_numbers": True,
|
||||
"color": BLACK,
|
||||
},
|
||||
)
|
||||
x = ax.get_x_axis()
|
||||
x.numbers.set_color(BLACK)
|
||||
y = ax.get_y_axis()
|
||||
y.numbers.set_color(BLACK)
|
||||
self.camera.background_color = WHITE
|
||||
axes_lables = ax.get_axis_labels().set_color(BLACK)
|
||||
|
||||
vec1 = Arrow3D(start=ax.c2p(0, 0, 0), end=ax.c2p(0, 1, 0), color=RED)
|
||||
vec2 = Arrow3D(start=ax.c2p(0, 0, 0), end=ax.c2p(1, 0, 0), color=BLUE)
|
||||
plane1 = Surface(
|
||||
lambda u, v: ax.c2p(u, v, u * 0),
|
||||
u_range=[0, 1],
|
||||
v_range=[0, 1],
|
||||
checkerboard_colors=[GREEN, GREEN],
|
||||
fill_opacity=0.75,
|
||||
stroke_width=0,
|
||||
)
|
||||
|
||||
vec3 = Arrow3D(start=ax.c2p(0, 0, 0), end=ax.c2p(1, 1, 0), color=PURPLE)
|
||||
vec4 = Arrow3D(start=ax.c2p(0, 0, 0), end=ax.c2p(1, 0, 0), color=PINK)
|
||||
plane2 = Polygon(
|
||||
ax.c2p(0, 0, 0),
|
||||
ax.c2p(1, 0, 0),
|
||||
ax.c2p(2, 1, 0),
|
||||
ax.c2p(1, 1, 0),
|
||||
fill_color=YELLOW,
|
||||
fill_opacity=0.75,
|
||||
color=YELLOW,
|
||||
)
|
||||
|
||||
text = MathTex(
|
||||
r"\mathbf{B} \begin{bmatrix} 1 & 1\\ 0 & 1\end{bmatrix}", color=BLACK
|
||||
).move_to(ax.c2p(-2.5, 2, 0))
|
||||
|
||||
self.add(ax, axes_lables, plane2, plane1, vec4, vec3, vec2, vec1, text)
|
||||
|
||||
|
||||
class Detgraph4(Scene):
|
||||
def construct(self):
|
||||
ax = Axes(
|
||||
x_range=[-5.3, 5.3, 1],
|
||||
y_range=[-5.3, 5.3, 1],
|
||||
x_length=7,
|
||||
y_length=7,
|
||||
axis_config={
|
||||
"include_numbers": True,
|
||||
"color": BLACK,
|
||||
},
|
||||
)
|
||||
x = ax.get_x_axis()
|
||||
x.numbers.set_color(BLACK)
|
||||
y = ax.get_y_axis()
|
||||
y.numbers.set_color(BLACK)
|
||||
self.camera.background_color = WHITE
|
||||
axes_lables = ax.get_axis_labels().set_color(BLACK)
|
||||
|
||||
vec1 = Arrow3D(start=ax.c2p(0, 0, 0), end=ax.c2p(0, 1, 0), color=RED)
|
||||
vec2 = Arrow3D(start=ax.c2p(0, 0, 0), end=ax.c2p(1, 0, 0), color=BLUE)
|
||||
plane1 = Surface(
|
||||
lambda u, v: ax.c2p(u, v, u * 0),
|
||||
u_range=[0, 1],
|
||||
v_range=[0, 1],
|
||||
checkerboard_colors=[GREEN, GREEN],
|
||||
fill_opacity=0.75,
|
||||
stroke_width=0,
|
||||
)
|
||||
|
||||
vec3 = Arrow3D(start=ax.c2p(0, 0, 0), end=ax.c2p(1, 1, 0), color=PURPLE)
|
||||
vec4 = Arrow3D(start=ax.c2p(0, 0, 0), end=ax.c2p(2, -1, 0), color=PINK)
|
||||
plane2 = Polygon(
|
||||
ax.c2p(0, 0, 0),
|
||||
ax.c2p(2, -1, 0),
|
||||
ax.c2p(3, 0, 0),
|
||||
ax.c2p(1, 1, 0),
|
||||
fill_color=YELLOW,
|
||||
fill_opacity=0.75,
|
||||
color=YELLOW,
|
||||
)
|
||||
|
||||
text = MathTex(
|
||||
r"\mathbf{C} \begin{bmatrix} 1 & 2\\ 1 & -1\end{bmatrix}", color=BLACK
|
||||
).move_to(ax.c2p(-2.5, 2, 0))
|
||||
|
||||
self.add(ax, axes_lables, plane2, plane1, vec4, vec3, vec2, vec1, text)
|
||||
|
||||
|
||||
class Detgraph5(Scene):
|
||||
def construct(self):
|
||||
ax = Axes(
|
||||
x_range=[-5.3, 5.3, 1],
|
||||
y_range=[-5.3, 5.3, 1],
|
||||
x_length=7,
|
||||
y_length=7,
|
||||
axis_config={
|
||||
"include_numbers": True,
|
||||
"color": BLACK,
|
||||
},
|
||||
)
|
||||
x = ax.get_x_axis()
|
||||
x.numbers.set_color(BLACK)
|
||||
y = ax.get_y_axis()
|
||||
y.numbers.set_color(BLACK)
|
||||
self.camera.background_color = WHITE
|
||||
axes_lables = ax.get_axis_labels().set_color(BLACK)
|
||||
|
||||
vec1 = Arrow3D(start=ax.c2p(0, 0, 0), end=ax.c2p(0, 1, 0), color=RED)
|
||||
vec2 = Arrow3D(start=ax.c2p(0, 0, 0), end=ax.c2p(1, 0, 0), color=BLUE)
|
||||
plane1 = Surface(
|
||||
lambda u, v: ax.c2p(u, v, u * 0),
|
||||
u_range=[0, 1],
|
||||
v_range=[0, 1],
|
||||
checkerboard_colors=[GREEN, GREEN],
|
||||
fill_opacity=0.75,
|
||||
stroke_width=0,
|
||||
)
|
||||
|
||||
vec3 = Arrow3D(start=ax.c2p(0, 0, 0), end=ax.c2p(4, 2, 0), color=PURPLE)
|
||||
vec4 = Arrow3D(start=ax.c2p(0, 0, 0), end=ax.c2p(2, 1, 0), color=PINK)
|
||||
|
||||
text = MathTex(
|
||||
r"\mathbf{C} \begin{bmatrix} 4 & 2\\ 2 & 1\end{bmatrix}", color=BLACK
|
||||
).move_to(ax.c2p(-2.5, 2, 0))
|
||||
|
||||
self.add(ax, axes_lables, plane1, vec3, vec4, vec2, vec1, text)
|
||||
|
||||
|
||||
class Detgraph6(Scene):
|
||||
def construct(self):
|
||||
self.camera.background_color = WHITE
|
||||
|
||||
text1 = MathTex(
|
||||
r"\begin{bmatrix} a & b & c\\ d & e & f\\ g & h & i\\ \end{bmatrix} \begin{matrix} a & b\\ d & e\\ g & h\\ \end{matrix}",
|
||||
color=BLACK,
|
||||
).scale(0.5)
|
||||
text2 = (
|
||||
MathTex(
|
||||
r"\Rightarrow a \cdot e \cdot i + b \cdot f \cdot g + c \cdot d \cdot h - g \cdot e \cdot c - h \cdot f \cdot a - i \cdot d \cdot b",
|
||||
color=BLACK,
|
||||
)
|
||||
.move_to([0, -1, 0])
|
||||
.scale(0.5)
|
||||
)
|
||||
|
||||
arrow1 = Arrow(
|
||||
[0, 0, 0],
|
||||
[1.2, -0.9, 0],
|
||||
max_tip_length_to_length_ratio=0.1,
|
||||
stroke_width=0.7,
|
||||
color=RED,
|
||||
).move_to([-0.4, -0.1, 0])
|
||||
arrow2 = Arrow(
|
||||
[0, 0, 0],
|
||||
[1.2, -0.9, 0],
|
||||
max_tip_length_to_length_ratio=0.1,
|
||||
stroke_width=0.7,
|
||||
color=RED,
|
||||
).move_to([0, -0.1, 0])
|
||||
arrow3 = Arrow(
|
||||
[0, 0, 0],
|
||||
[1.2, -0.9, 0],
|
||||
max_tip_length_to_length_ratio=0.1,
|
||||
stroke_width=0.7,
|
||||
color=RED,
|
||||
).move_to([0.4, -0.1, 0])
|
||||
|
||||
arrow4 = Arrow(
|
||||
[0, 0, 0],
|
||||
[1.2, 0.9, 0],
|
||||
max_tip_length_to_length_ratio=0.1,
|
||||
stroke_width=0.7,
|
||||
color=BLUE,
|
||||
).move_to([-0.3, -0.1, 0])
|
||||
arrow5 = Arrow(
|
||||
[0, 0, 0],
|
||||
[1.2, 0.9, 0],
|
||||
max_tip_length_to_length_ratio=0.1,
|
||||
stroke_width=0.7,
|
||||
color=BLUE,
|
||||
).move_to([0.1, -0.1, 0])
|
||||
arrow6 = Arrow(
|
||||
[0, 0, 0],
|
||||
[1.2, 0.9, 0],
|
||||
max_tip_length_to_length_ratio=0.1,
|
||||
stroke_width=0.7,
|
||||
color=BLUE,
|
||||
).move_to([0.5, -0.1, 0])
|
||||
|
||||
self.add(text1, text2, arrow1, arrow2, arrow3, arrow4, arrow5, arrow6)
|
||||
|
||||
|
||||
class Detgraph7(Scene):
|
||||
def construct(self):
|
||||
self.camera.background_color = WHITE
|
||||
|
||||
text1 = MathTex(
|
||||
r"\begin{bmatrix} a & b & 0 & 0\\ c & d & 0 & 0 \\ 0 & 0 & e & f \\ 0 & 0 & g & h \end{bmatrix}",
|
||||
color=BLACK,
|
||||
).scale(0.5)
|
||||
text2 = (
|
||||
MathTex(
|
||||
r"\Rightarrow \text{det } \begin{bmatrix} a & b \\ c & d \end{bmatrix} \cdot \text{det } \begin{bmatrix} e & f \\ g & h \end{bmatrix}",
|
||||
color=BLACK,
|
||||
)
|
||||
.move_to([0, -1, 0])
|
||||
.scale(0.5)
|
||||
)
|
||||
|
||||
box1 = Polygon([0, 0, 0], [0.6, 0, 0], [0.6, 0.6, 0], [0, 0.6, 0]).move_to(
|
||||
[-0.36, 0.3, 0]
|
||||
)
|
||||
box2 = Polygon(
|
||||
[0, 0, 0], [0.6, 0, 0], [0.6, 0.6, 0], [0, 0.6, 0], color=RED
|
||||
).move_to([0.36, -0.3, 0])
|
||||
box3 = Polygon([0, 0, 0], [0.6, 0, 0], [0.6, 0.6, 0], [0, 0.6, 0]).move_to(
|
||||
[-0.3, -1, 0]
|
||||
)
|
||||
box4 = Polygon(
|
||||
[0, 0, 0], [0.6, 0, 0], [0.6, 0.6, 0], [0, 0.6, 0], color=RED
|
||||
).move_to([1.13, -1, 0])
|
||||
|
||||
self.add(text1, text2, box1, box2, box3, box4)
|
||||
10
hs24/lineare_algebra/graphs/media/Tex/2b7ffb3c38a5a6e0.svg
Normal file
10
hs24/lineare_algebra/graphs/media/Tex/2b7ffb3c38a5a6e0.svg
Normal file
@@ -0,0 +1,10 @@
|
||||
<?xml version='1.0' encoding='UTF-8'?>
|
||||
<!-- This file was generated by dvisvgm 3.2.2 -->
|
||||
<svg version='1.1' xmlns='http://www.w3.org/2000/svg' xmlns:xlink='http://www.w3.org/1999/xlink' width='4.981339pt' height='6.420368pt' viewBox='169.364866 -10.006955 4.981339 6.420368'>
|
||||
<defs>
|
||||
<path id='g0-50' d='M1.265255-.767123L2.321295-1.793275C3.875467-3.16812 4.473225-3.706102 4.473225-4.702366C4.473225-5.838107 3.576588-6.635118 2.361146-6.635118C1.235367-6.635118 .498132-5.718555 .498132-4.83188C.498132-4.273973 .996264-4.273973 1.026152-4.273973C1.195517-4.273973 1.544209-4.393524 1.544209-4.801993C1.544209-5.061021 1.364882-5.32005 1.016189-5.32005C.936488-5.32005 .916563-5.32005 .886675-5.310087C1.115816-5.957659 1.653798-6.326276 2.231631-6.326276C3.138232-6.326276 3.566625-5.519303 3.566625-4.702366C3.566625-3.905355 3.068493-3.118306 2.520548-2.500623L.607721-.368618C.498132-.259029 .498132-.239103 .498132 0H4.194271L4.473225-1.733499H4.224159C4.174346-1.43462 4.104608-.996264 4.004981-.846824C3.935243-.767123 3.277709-.767123 3.058531-.767123H1.265255Z'/>
|
||||
</defs>
|
||||
<g id='page1'>
|
||||
<use x='169.364866' y='-3.586587' xlink:href='#g0-50'/>
|
||||
</g>
|
||||
</svg>
|
||||
|
After Width: | Height: | Size: 1.1 KiB |
@@ -0,0 +1,9 @@
|
||||
\documentclass[preview]{standalone}
|
||||
\usepackage[english]{babel}
|
||||
\usepackage{amsmath}
|
||||
\usepackage{amssymb}
|
||||
\begin{document}
|
||||
\begin{align*}
|
||||
2
|
||||
\end{align*}
|
||||
\end{document}
|
||||
10
hs24/lineare_algebra/graphs/media/Tex/31d3165490bf3404.svg
Normal file
10
hs24/lineare_algebra/graphs/media/Tex/31d3165490bf3404.svg
Normal file
@@ -0,0 +1,10 @@
|
||||
<?xml version='1.0' encoding='UTF-8'?>
|
||||
<!-- This file was generated by dvisvgm 3.2.2 -->
|
||||
<svg version='1.1' xmlns='http://www.w3.org/2000/svg' xmlns:xlink='http://www.w3.org/1999/xlink' width='4.981339pt' height='6.420368pt' viewBox='169.364866 -10.006955 4.981339 6.420368'>
|
||||
<defs>
|
||||
<path id='g0-52' d='M2.929016-1.643836V-.777086C2.929016-.418431 2.909091-.308842 2.171856-.308842H1.96264V0C2.371108-.029888 2.889166-.029888 3.307597-.029888S4.254047-.029888 4.662516 0V-.308842H4.4533C3.716065-.308842 3.696139-.418431 3.696139-.777086V-1.643836H4.692403V-1.952677H3.696139V-6.485679C3.696139-6.684932 3.696139-6.744707 3.536737-6.744707C3.447073-6.744707 3.417186-6.744707 3.337484-6.625156L.278954-1.952677V-1.643836H2.929016ZM2.988792-1.952677H.557908L2.988792-5.668742V-1.952677Z'/>
|
||||
</defs>
|
||||
<g id='page1'>
|
||||
<use x='169.364866' y='-3.586587' xlink:href='#g0-52'/>
|
||||
</g>
|
||||
</svg>
|
||||
|
After Width: | Height: | Size: 879 B |
@@ -0,0 +1,9 @@
|
||||
\documentclass[preview]{standalone}
|
||||
\usepackage[english]{babel}
|
||||
\usepackage{amsmath}
|
||||
\usepackage{amssymb}
|
||||
\begin{document}
|
||||
\begin{align*}
|
||||
4
|
||||
\end{align*}
|
||||
\end{document}
|
||||
21
hs24/lineare_algebra/graphs/media/Tex/3247af9f4399a897.svg
Normal file
21
hs24/lineare_algebra/graphs/media/Tex/3247af9f4399a897.svg
Normal file
@@ -0,0 +1,21 @@
|
||||
<?xml version='1.0' encoding='UTF-8'?>
|
||||
<!-- This file was generated by dvisvgm 3.2.2 -->
|
||||
<svg version='1.1' xmlns='http://www.w3.org/2000/svg' xmlns:xlink='http://www.w3.org/1999/xlink' width='40.7638pt' height='23.910579pt' viewBox='151.473646 -23.910576 40.7638 23.910579'>
|
||||
<defs>
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10
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x
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40
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47
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\documentclass[preview]{standalone}
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||||
\usepackage[english]{babel}
|
||||
\usepackage{amsmath}
|
||||
\usepackage{amssymb}
|
||||
\begin{document}
|
||||
\begin{align*}
|
||||
\begin{bmatrix} a & b & 0 & 0\\ c & d & 0 & 0 \\ 0 & 0 & e & f \\ 0 & 0 & g & h \end{bmatrix}
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\end{align*}
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178
hs24/lineare_algebra/letterfonts.tex
Normal file
178
hs24/lineare_algebra/letterfonts.tex
Normal file
@@ -0,0 +1,178 @@
|
||||
% Things Lie
|
||||
\newcommand{\kb}{\mathfrak b}
|
||||
\newcommand{\kg}{\mathfrak g}
|
||||
\newcommand{\kh}{\mathfrak h}
|
||||
\newcommand{\kn}{\mathfrak n}
|
||||
\newcommand{\ku}{\mathfrak u}
|
||||
\newcommand{\kz}{\mathfrak z}
|
||||
\DeclareMathOperator{\Ext}{Ext} % Ext functor
|
||||
\DeclareMathOperator{\Tor}{Tor} % Tor functor
|
||||
\newcommand{\gl}{\opname{\mathfrak{gl}}} % frak gl group
|
||||
\renewcommand{\sl}{\opname{\mathfrak{sl}}} % frak sl group chktex 6
|
||||
|
||||
% More script letters etc.
|
||||
\newcommand{\SA}{\mathcal A}
|
||||
\newcommand{\SB}{\mathcal B}
|
||||
\newcommand{\SC}{\mathcal C}
|
||||
\newcommand{\SF}{\mathcal F}
|
||||
\newcommand{\SG}{\mathcal G}
|
||||
\newcommand{\SH}{\mathcal H}
|
||||
\newcommand{\OO}{\mathcal O}
|
||||
|
||||
\newcommand{\SCA}{\mathscr A}
|
||||
\newcommand{\SCB}{\mathscr B}
|
||||
\newcommand{\SCC}{\mathscr C}
|
||||
\newcommand{\SCD}{\mathscr D}
|
||||
\newcommand{\SCE}{\mathscr E}
|
||||
\newcommand{\SCF}{\mathscr F}
|
||||
\newcommand{\SCG}{\mathscr G}
|
||||
\newcommand{\SCH}{\mathscr H}
|
||||
|
||||
% Mathfrak primes
|
||||
\newcommand{\km}{\mathfrak m}
|
||||
\newcommand{\kp}{\mathfrak p}
|
||||
\newcommand{\kq}{\mathfrak q}
|
||||
|
||||
% number sets
|
||||
\newcommand{\RR}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{R}}{\mathbb{R}^{#1}}}}
|
||||
\newcommand{\NN}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{N}}{\mathbb{N}^{#1}}}}
|
||||
\newcommand{\ZZ}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{Z}}{\mathbb{Z}^{#1}}}}
|
||||
\newcommand{\QQ}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{Q}}{\mathbb{Q}^{#1}}}}
|
||||
\newcommand{\CC}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{C}}{\mathbb{C}^{#1}}}}
|
||||
\newcommand{\PP}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{P}}{\mathbb{P}^{#1}}}}
|
||||
\newcommand{\HH}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{H}}{\mathbb{H}^{#1}}}}
|
||||
\newcommand{\FF}[1][]{\ensuremath{\ifstrempty{#1}{\mathbb{F}}{\mathbb{F}^{#1}}}}
|
||||
% expected value
|
||||
\newcommand{\EE}{\ensuremath{\mathbb{E}}}
|
||||
\newcommand{\charin}{\text{ char }}
|
||||
\DeclareMathOperator{\sign}{sign}
|
||||
\DeclareMathOperator{\Aut}{Aut}
|
||||
\DeclareMathOperator{\Inn}{Inn}
|
||||
\DeclareMathOperator{\Syl}{Syl}
|
||||
\DeclareMathOperator{\Gal}{Gal}
|
||||
\DeclareMathOperator{\GL}{GL} % General linear group
|
||||
\DeclareMathOperator{\SL}{SL} % Special linear group
|
||||
|
||||
%---------------------------------------
|
||||
% BlackBoard Math Fonts :-
|
||||
%---------------------------------------
|
||||
|
||||
%Captital Letters
|
||||
\newcommand{\bbA}{\mathbb{A}} \newcommand{\bbB}{\mathbb{B}}
|
||||
\newcommand{\bbC}{\mathbb{C}} \newcommand{\bbD}{\mathbb{D}}
|
||||
\newcommand{\bbE}{\mathbb{E}} \newcommand{\bbF}{\mathbb{F}}
|
||||
\newcommand{\bbG}{\mathbb{G}} \newcommand{\bbH}{\mathbb{H}}
|
||||
\newcommand{\bbI}{\mathbb{I}} \newcommand{\bbJ}{\mathbb{J}}
|
||||
\newcommand{\bbK}{\mathbb{K}} \newcommand{\bbL}{\mathbb{L}}
|
||||
\newcommand{\bbM}{\mathbb{M}} \newcommand{\bbN}{\mathbb{N}}
|
||||
\newcommand{\bbO}{\mathbb{O}} \newcommand{\bbP}{\mathbb{P}}
|
||||
\newcommand{\bbQ}{\mathbb{Q}} \newcommand{\bbR}{\mathbb{R}}
|
||||
\newcommand{\bbS}{\mathbb{S}} \newcommand{\bbT}{\mathbb{T}}
|
||||
\newcommand{\bbU}{\mathbb{U}} \newcommand{\bbV}{\mathbb{V}}
|
||||
\newcommand{\bbW}{\mathbb{W}} \newcommand{\bbX}{\mathbb{X}}
|
||||
\newcommand{\bbY}{\mathbb{Y}} \newcommand{\bbZ}{\mathbb{Z}}
|
||||
|
||||
%---------------------------------------
|
||||
% MathCal Fonts :-
|
||||
%---------------------------------------
|
||||
|
||||
%Captital Letters
|
||||
\newcommand{\mcA}{\mathcal{A}} \newcommand{\mcB}{\mathcal{B}}
|
||||
\newcommand{\mcC}{\mathcal{C}} \newcommand{\mcD}{\mathcal{D}}
|
||||
\newcommand{\mcE}{\mathcal{E}} \newcommand{\mcF}{\mathcal{F}}
|
||||
\newcommand{\mcG}{\mathcal{G}} \newcommand{\mcH}{\mathcal{H}}
|
||||
\newcommand{\mcI}{\mathcal{I}} \newcommand{\mcJ}{\mathcal{J}}
|
||||
\newcommand{\mcK}{\mathcal{K}} \newcommand{\mcL}{\mathcal{L}}
|
||||
\newcommand{\mcM}{\mathcal{M}} \newcommand{\mcN}{\mathcal{N}}
|
||||
\newcommand{\mcO}{\mathcal{O}} \newcommand{\mcP}{\mathcal{P}}
|
||||
\newcommand{\mcQ}{\mathcal{Q}} \newcommand{\mcR}{\mathcal{R}}
|
||||
\newcommand{\mcS}{\mathcal{S}} \newcommand{\mcT}{\mathcal{T}}
|
||||
\newcommand{\mcU}{\mathcal{U}} \newcommand{\mcV}{\mathcal{V}}
|
||||
\newcommand{\mcW}{\mathcal{W}} \newcommand{\mcX}{\mathcal{X}}
|
||||
\newcommand{\mcY}{\mathcal{Y}} \newcommand{\mcZ}{\mathcal{Z}}
|
||||
|
||||
|
||||
%---------------------------------------
|
||||
% Bold Math Fonts :-
|
||||
%---------------------------------------
|
||||
|
||||
%Captital Letters
|
||||
\newcommand{\bmA}{\boldsymbol{A}} \newcommand{\bmB}{\boldsymbol{B}}
|
||||
\newcommand{\bmC}{\boldsymbol{C}} \newcommand{\bmD}{\boldsymbol{D}}
|
||||
\newcommand{\bmE}{\boldsymbol{E}} \newcommand{\bmF}{\boldsymbol{F}}
|
||||
\newcommand{\bmG}{\boldsymbol{G}} \newcommand{\bmH}{\boldsymbol{H}}
|
||||
\newcommand{\bmI}{\boldsymbol{I}} \newcommand{\bmJ}{\boldsymbol{J}}
|
||||
\newcommand{\bmK}{\boldsymbol{K}} \newcommand{\bmL}{\boldsymbol{L}}
|
||||
\newcommand{\bmM}{\boldsymbol{M}} \newcommand{\bmN}{\boldsymbol{N}}
|
||||
\newcommand{\bmO}{\boldsymbol{O}} \newcommand{\bmP}{\boldsymbol{P}}
|
||||
\newcommand{\bmQ}{\boldsymbol{Q}} \newcommand{\bmR}{\boldsymbol{R}}
|
||||
\newcommand{\bmS}{\boldsymbol{S}} \newcommand{\bmT}{\boldsymbol{T}}
|
||||
\newcommand{\bmU}{\boldsymbol{U}} \newcommand{\bmV}{\boldsymbol{V}}
|
||||
\newcommand{\bmW}{\boldsymbol{W}} \newcommand{\bmX}{\boldsymbol{X}}
|
||||
\newcommand{\bmY}{\boldsymbol{Y}} \newcommand{\bmZ}{\boldsymbol{Z}}
|
||||
%Small Letters
|
||||
\newcommand{\bma}{\boldsymbol{a}} \newcommand{\bmb}{\boldsymbol{b}}
|
||||
\newcommand{\bmc}{\boldsymbol{c}} \newcommand{\bmd}{\boldsymbol{d}}
|
||||
\newcommand{\bme}{\boldsymbol{e}} \newcommand{\bmf}{\boldsymbol{f}}
|
||||
\newcommand{\bmg}{\boldsymbol{g}} \newcommand{\bmh}{\boldsymbol{h}}
|
||||
\newcommand{\bmi}{\boldsymbol{i}} \newcommand{\bmj}{\boldsymbol{j}}
|
||||
\newcommand{\bmk}{\boldsymbol{k}} \newcommand{\bml}{\boldsymbol{l}}
|
||||
\newcommand{\bmm}{\boldsymbol{m}} \newcommand{\bmn}{\boldsymbol{n}}
|
||||
\newcommand{\bmo}{\boldsymbol{o}} \newcommand{\bmp}{\boldsymbol{p}}
|
||||
\newcommand{\bmq}{\boldsymbol{q}} \newcommand{\bmr}{\boldsymbol{r}}
|
||||
\newcommand{\bms}{\boldsymbol{s}} \newcommand{\bmt}{\boldsymbol{t}}
|
||||
\newcommand{\bmu}{\boldsymbol{u}} \newcommand{\bmv}{\boldsymbol{v}}
|
||||
\newcommand{\bmw}{\boldsymbol{w}} \newcommand{\bmx}{\boldsymbol{x}}
|
||||
\newcommand{\bmy}{\boldsymbol{y}} \newcommand{\bmz}{\boldsymbol{z}}
|
||||
|
||||
%---------------------------------------
|
||||
% Scr Math Fonts :-
|
||||
%---------------------------------------
|
||||
|
||||
\newcommand{\sA}{{\mathscr{A}}} \newcommand{\sB}{{\mathscr{B}}}
|
||||
\newcommand{\sC}{{\mathscr{C}}} \newcommand{\sD}{{\mathscr{D}}}
|
||||
\newcommand{\sE}{{\mathscr{E}}} \newcommand{\sF}{{\mathscr{F}}}
|
||||
\newcommand{\sG}{{\mathscr{G}}} \newcommand{\sH}{{\mathscr{H}}}
|
||||
\newcommand{\sI}{{\mathscr{I}}} \newcommand{\sJ}{{\mathscr{J}}}
|
||||
\newcommand{\sK}{{\mathscr{K}}} \newcommand{\sL}{{\mathscr{L}}}
|
||||
\newcommand{\sM}{{\mathscr{M}}} \newcommand{\sN}{{\mathscr{N}}}
|
||||
\newcommand{\sO}{{\mathscr{O}}} \newcommand{\sP}{{\mathscr{P}}}
|
||||
\newcommand{\sQ}{{\mathscr{Q}}} \newcommand{\sR}{{\mathscr{R}}}
|
||||
\newcommand{\sS}{{\mathscr{S}}} \newcommand{\sT}{{\mathscr{T}}}
|
||||
\newcommand{\sU}{{\mathscr{U}}} \newcommand{\sV}{{\mathscr{V}}}
|
||||
\newcommand{\sW}{{\mathscr{W}}} \newcommand{\sX}{{\mathscr{X}}}
|
||||
\newcommand{\sY}{{\mathscr{Y}}} \newcommand{\sZ}{{\mathscr{Z}}}
|
||||
|
||||
|
||||
%---------------------------------------
|
||||
% Math Fraktur Font
|
||||
%---------------------------------------
|
||||
|
||||
%Captital Letters
|
||||
\newcommand{\mfA}{\mathfrak{A}} \newcommand{\mfB}{\mathfrak{B}}
|
||||
\newcommand{\mfC}{\mathfrak{C}} \newcommand{\mfD}{\mathfrak{D}}
|
||||
\newcommand{\mfE}{\mathfrak{E}} \newcommand{\mfF}{\mathfrak{F}}
|
||||
\newcommand{\mfG}{\mathfrak{G}} \newcommand{\mfH}{\mathfrak{H}}
|
||||
\newcommand{\mfI}{\mathfrak{I}} \newcommand{\mfJ}{\mathfrak{J}}
|
||||
\newcommand{\mfK}{\mathfrak{K}} \newcommand{\mfL}{\mathfrak{L}}
|
||||
\newcommand{\mfM}{\mathfrak{M}} \newcommand{\mfN}{\mathfrak{N}}
|
||||
\newcommand{\mfO}{\mathfrak{O}} \newcommand{\mfP}{\mathfrak{P}}
|
||||
\newcommand{\mfQ}{\mathfrak{Q}} \newcommand{\mfR}{\mathfrak{R}}
|
||||
\newcommand{\mfS}{\mathfrak{S}} \newcommand{\mfT}{\mathfrak{T}}
|
||||
\newcommand{\mfU}{\mathfrak{U}} \newcommand{\mfV}{\mathfrak{V}}
|
||||
\newcommand{\mfW}{\mathfrak{W}} \newcommand{\mfX}{\mathfrak{X}}
|
||||
\newcommand{\mfY}{\mathfrak{Y}} \newcommand{\mfZ}{\mathfrak{Z}}
|
||||
%Small Letters
|
||||
\newcommand{\mfa}{\mathfrak{a}} \newcommand{\mfb}{\mathfrak{b}}
|
||||
\newcommand{\mfc}{\mathfrak{c}} \newcommand{\mfd}{\mathfrak{d}}
|
||||
\newcommand{\mfe}{\mathfrak{e}} \newcommand{\mff}{\mathfrak{f}}
|
||||
\newcommand{\mfg}{\mathfrak{g}} \newcommand{\mfh}{\mathfrak{h}}
|
||||
\newcommand{\mfi}{\mathfrak{i}} \newcommand{\mfj}{\mathfrak{j}}
|
||||
\newcommand{\mfk}{\mathfrak{k}} \newcommand{\mfl}{\mathfrak{l}}
|
||||
\newcommand{\mfm}{\mathfrak{m}} \newcommand{\mfn}{\mathfrak{n}}
|
||||
\newcommand{\mfo}{\mathfrak{o}} \newcommand{\mfp}{\mathfrak{p}}
|
||||
\newcommand{\mfq}{\mathfrak{q}} \newcommand{\mfr}{\mathfrak{r}}
|
||||
\newcommand{\mfs}{\mathfrak{s}} \newcommand{\mft}{\mathfrak{t}}
|
||||
\newcommand{\mfu}{\mathfrak{u}} \newcommand{\mfv}{\mathfrak{v}}
|
||||
\newcommand{\mfw}{\mathfrak{w}} \newcommand{\mfx}{\mathfrak{x}}
|
||||
\newcommand{\mfy}{\mathfrak{y}} \newcommand{\mfz}{\mathfrak{z}}
|
||||
@@ -0,0 +1,3 @@
|
||||
\chapter{Lineare Abbildungen}
|
||||
|
||||
\newpage
|
||||
@@ -0,0 +1,3 @@
|
||||
\chapter{Lineare Gleichungssysteme}
|
||||
|
||||
\input{vektoren_matrizen.tex}
|
||||
@@ -0,0 +1,287 @@
|
||||
\section{Vektoren, Matrizen}
|
||||
|
||||
\subsection{2 Dimensionen}
|
||||
|
||||
Im Zweidimensionalen bestehen Vektoren aus zwei Komponenten, welche in einem Koordinatensystem dargestellt werden kann.
|
||||
|
||||
\begin{figure}[h]
|
||||
\begin{minipage}{0.7\linewidth}
|
||||
\includegraphics[width=\linewidth]{fig/Fig_1.png}
|
||||
\caption{Zweidimensionaler Vektor}
|
||||
\label{fig:vek1}
|
||||
\end{minipage}
|
||||
\begin{minipage}{0.3\linewidth}
|
||||
\[
|
||||
\vec{x} =
|
||||
\begin{bmatrix}
|
||||
x_1 \\
|
||||
x_2 \\
|
||||
\end{bmatrix}
|
||||
\in \mathbb{R} ^2
|
||||
.\]
|
||||
\end{minipage}
|
||||
\end{figure}
|
||||
|
||||
Die Vektoren $\vec{e}_1$ und $\vec{e}_2$ aus Grafik \ref{fig:vek1} sind die Einheitsvektoren und können einen Vektor als Gleichung darstellen. Des Weiteren werden die Einheitsvektoren für die kanonische Basis wichtig sein.
|
||||
|
||||
\[
|
||||
\vec{e}_1 =
|
||||
\begin{bmatrix}
|
||||
1 \\
|
||||
0 \\
|
||||
\end{bmatrix},
|
||||
\vec{e}_2 =
|
||||
\begin{bmatrix}
|
||||
0 \\
|
||||
1 \\
|
||||
\end{bmatrix}
|
||||
.\]
|
||||
|
||||
\[
|
||||
\vec{x} =
|
||||
\begin{bmatrix}
|
||||
x_1 \\
|
||||
x_2 \\
|
||||
\end{bmatrix} =
|
||||
\begin{bmatrix}
|
||||
x_1 \\
|
||||
0 \\
|
||||
\end{bmatrix} +
|
||||
\begin{bmatrix}
|
||||
0 \\
|
||||
x_2 \\
|
||||
\end{bmatrix} =
|
||||
x_1 \cdot \begin{bmatrix}
|
||||
1 \\
|
||||
0 \\
|
||||
\end{bmatrix} + x_2 \cdot
|
||||
\begin{bmatrix}
|
||||
0 \\
|
||||
1 \\
|
||||
\end{bmatrix} = x_1 \cdot \vec{e}_1 + x_2 \cdot \vec{e}_2
|
||||
.\]
|
||||
|
||||
Unser Beispiel bestand aus reellen Zahlen $\mathbb{R}$ und befand sich im $\mathbb{R}^2$. Wir können dies jedoch erweitern zu den komplexen Zahlen $\mathbb{C}$. Somit können die Komponenten nun aus Komplexen Zahlen bestehen. Des Weiteren können wir die Dimension erweitern ins Dreidimensionale.
|
||||
|
||||
\subsection{3 Dimensionen}
|
||||
|
||||
Im Vergleich zum Zweidimensionalen bestehen Vektoren im Dreidimensionalen aus drei Komponenten, welche auch in einem Koordinatensystem dargestellt werden kann.
|
||||
|
||||
\begin{figure}[h]
|
||||
\begin{minipage}{0.7\linewidth}
|
||||
\includegraphics[width=\textwidth]{fig/Fig_2.png}
|
||||
\caption{Dreidimensionaler Vektor}
|
||||
\label{fig:vek2}
|
||||
\end{minipage}
|
||||
\begin{minipage}{0.3\linewidth}
|
||||
\[
|
||||
\vec{x} = \begin{bmatrix}
|
||||
x_1 \\
|
||||
x_2 \\
|
||||
x_3 \\
|
||||
\end{bmatrix}
|
||||
.\]
|
||||
\end{minipage}
|
||||
\end{figure}
|
||||
|
||||
Wie in Grafik \ref{fig:vek1} sind die Vektoren $\vec{e_1}, \vec{e_2}$ und $\vec{e_3}$ aus Grafik \ref{fig:vek2} Einheitsvektoren, welche dreidimensionale Vektoren als Gleichung darstellen können.
|
||||
|
||||
\[
|
||||
\vec{e_1} = \begin{bmatrix}
|
||||
1 \\
|
||||
0 \\
|
||||
0 \\
|
||||
\end{bmatrix}, \vec{e_2} = \begin{bmatrix}
|
||||
0 \\
|
||||
1 \\
|
||||
0 \\
|
||||
\end{bmatrix}, \vec{e_3} = \begin{bmatrix}
|
||||
0 \\
|
||||
0 \\
|
||||
1 \\
|
||||
\end{bmatrix}
|
||||
.\]
|
||||
|
||||
|
||||
\[
|
||||
\vec{x} = \begin{bmatrix}
|
||||
x_1 \\
|
||||
x_2 \\
|
||||
x_3 \\
|
||||
\end{bmatrix} = x_1 \cdot \begin{bmatrix}
|
||||
1 \\
|
||||
0 \\
|
||||
0 \\
|
||||
\end{bmatrix} + x_2 \cdot \begin{bmatrix}
|
||||
0 \\
|
||||
1 \\
|
||||
0 \\
|
||||
\end{bmatrix} + x_3 \cdot \begin{bmatrix}
|
||||
0 \\
|
||||
0 \\
|
||||
1 \\
|
||||
\end{bmatrix}
|
||||
.\]
|
||||
|
||||
\nt{
|
||||
Die Gleichungen, welche die Vektoren darstellen, nennt man auch eine lineare Kombination von Vektoren.
|
||||
}
|
||||
|
||||
\subsection{Lineare Kombination}
|
||||
|
||||
\dfn{lineare Kombination \cite{Gradinaru2024}}{
|
||||
Die lineare Kombination von Vektoren $\vec{v_1}, ... , \vec{v_n}$ ist:
|
||||
|
||||
\[
|
||||
c_1 \cdot \vec{v_1} + c_2 \cdot \vec{v_2} + ... + c_n \vec{v_n}
|
||||
.\]
|
||||
|
||||
wobei die $c_1, c_2, ... , c_n$ Skalare in $\mathbb{R}$ oder $\mathbb{C}$ sind.
|
||||
}
|
||||
|
||||
Die lineare Kombination ist eine Summe von Termen, wobei jeder Term ein gestreckter/gestauchter Vektor (d.h. die Multiplikation eines Skalars mit einem Vektor) ist. \cite{Gradinaru2024}
|
||||
|
||||
\subsection{$n$ Dimensionen}
|
||||
|
||||
Vektoren können natürlich mehr als zwei oder drei Komponenten haben. Wir können es auf $n$ Komponenten erweitern und deshalb auf $n$ Dimensionen. Dies bedeutet, dass wir unser Koordinatensystem auf $n$ Dimensionen erweitern. Somit hat der zugehörige Vektor $\vec{x} \in \mathbb{R} ^{n}$ die Komponenten $x_1, x_2, ... , x_{n}$.
|
||||
|
||||
\[
|
||||
\vec{x} = \begin{bmatrix}
|
||||
x_1 \\
|
||||
x_2 \\
|
||||
\vdots \\
|
||||
x_{n} \\
|
||||
\end{bmatrix}
|
||||
.\]
|
||||
|
||||
Vektoren in $n$-dimensionalen Räumen können auch als eine lineare Kombination dargestellt werden.
|
||||
|
||||
\[
|
||||
\vec{x} = \begin{bmatrix}
|
||||
x_1 \\
|
||||
x_2 \\
|
||||
\vdots \\
|
||||
x_{n} \\
|
||||
\end{bmatrix} = x_1 \cdot \vec{e_1} + x_2 \cdot \vec{e_2} + ... + x_{n} \cdot \vec{e_n}
|
||||
.\]
|
||||
|
||||
\dfn{Standardbasis \cite{Gradinaru2024}}{
|
||||
Sie Standardbasis (oder auch die kanonische Basis) in diesem $n$-dimensionalen Koordinatensystem besteht aus den folgenden $n$ Vektoren:
|
||||
|
||||
\[
|
||||
\vec{e_1} = \begin{bmatrix}
|
||||
1 \\
|
||||
0 \\
|
||||
\vdots \\
|
||||
0 \\
|
||||
\end{bmatrix}, \vec{e_2} = \begin{bmatrix}
|
||||
0 \\
|
||||
1 \\
|
||||
\vdots \\
|
||||
0 \\
|
||||
\end{bmatrix}, ... , \vec{e_j} = \begin{bmatrix}
|
||||
0 \\
|
||||
\vdots \\
|
||||
1 \\
|
||||
\vdots \\
|
||||
0 \\
|
||||
\end{bmatrix}, ... , \vec{e_n} = \begin{bmatrix}
|
||||
0 \\
|
||||
0 \\
|
||||
\vdots \\
|
||||
1 \\
|
||||
\end{bmatrix}
|
||||
.\]
|
||||
}
|
||||
|
||||
\subsection{Anwendung der linearen Kombination: Die Superposition von Feldern}
|
||||
|
||||
Die lineare Kombination wir vor allem bei der Superposition bzw. der Überlagerung von Kräften verwendet.
|
||||
|
||||
Nehmen wir an, dass 3 Vektoren $\in \mathbb{R} ^{3}$ gegeben sind.
|
||||
|
||||
\[
|
||||
\vec{a_1} = \begin{bmatrix}
|
||||
a_{11} \\
|
||||
a_{21} \\
|
||||
a_{31} \\
|
||||
\end{bmatrix}, \vec{a_2} = \begin{bmatrix}
|
||||
a_{12} \\
|
||||
a_{22} \\
|
||||
a_{32} \\
|
||||
\end{bmatrix}, \vec{a_3} = \begin{bmatrix}
|
||||
a_{13} \\
|
||||
a_{23} \\
|
||||
a_{33} \\
|
||||
\end{bmatrix}
|
||||
.\]
|
||||
|
||||
Jetzt wollen wir die Skalare herausfinden, welche den resultierenden Vektor $\vec{b}$ durch eine lineare Kombination mit den Vektoren $\vec{a_1}, \vec{a_2}, \vec{a_3}$ darstellen.
|
||||
|
||||
\[
|
||||
\vec{b} = x_1 \cdot \vec{a_1} + x_2 \cdot \vec{a_2} + x_3 \cdot \vec{a_3} \text{ mit } x_1, x_2, x_3 \in \mathbb{R}
|
||||
.\]
|
||||
|
||||
Daraus können wir nun ein Gleichungssystem bilden.
|
||||
|
||||
\[
|
||||
\left\{
|
||||
\begin{array}{lr}
|
||||
a_{11} \cdot x_1 + a_{12} \cdot x_2 + a_{13} \cdot x_3 = b_1 \\
|
||||
a_{21} \cdot x_1 + a_{22} \cdot x_2 + a_{23} \cdot x_3 = b_2 \\
|
||||
a_{31} \cdot x_1 + a_{32} \cdot x_2 + a_{33} \cdot x_3 = b_3 \\
|
||||
\end{array}
|
||||
\right.
|
||||
.\]
|
||||
|
||||
Das Gleichungssystem stellen wir nun wie folgt mit Matrizen und Vektoren dar.
|
||||
|
||||
\[
|
||||
\begin{bmatrix}
|
||||
a_{11} & a_{12} & a_{13} \\
|
||||
a_{21} & a_{22} & a_{23} \\
|
||||
a_{31} & a_{32} & a_{33} \\
|
||||
\end{bmatrix} \cdot \begin{bmatrix}
|
||||
x_1 \\
|
||||
x_2 \\
|
||||
x_3 \\
|
||||
\end{bmatrix} = \begin{bmatrix}
|
||||
b_1 \\
|
||||
b_2 \\
|
||||
b_3 \\
|
||||
\end{bmatrix}
|
||||
.\]
|
||||
|
||||
Die Vektoren $\vec{a_1}, \vec{a_2}, \vec{a_3}$ werden nebeneinander in einer Matrix geschrieben. Die Skalare $x_1, x_2, x_3$ werden als Vektor neben der Matrix geschrieben. Vereinfacht kann man auch $\vec{A} \cdot \vec{x} = \vec{b}$ schreiben.
|
||||
|
||||
\dfn{Lineares Gleichungssystem (LGS) \cite{Gradinaru2024}}{
|
||||
Ein lineares Gleichungssystem (LGS) wird kurz geschrieben als:
|
||||
|
||||
\[
|
||||
\vec{A} \cdot \vec{x} = \vec{b}
|
||||
,\]
|
||||
|
||||
wobei $\vec{A}$ die Koeffizientenmatrix, $\vec{x}$ die Unbekannte und $\vec{b}$ die rechte Seite ist.
|
||||
}
|
||||
|
||||
\nt{
|
||||
Bei der Rechenoperation $\vec{A} \cdot \vec{x}$ handelt es sich um eine Matrix Vektor Multiplikation.
|
||||
}
|
||||
|
||||
Für lineare Gleichungssysteme gilt:
|
||||
|
||||
\[
|
||||
\vec{A} \cdot \vec{x} = \vec{b} \Longleftrightarrow \vec{x} = \vec{A}^{-1} \cdot \vec{b}
|
||||
.\]
|
||||
|
||||
Die Inverse sowie die Multiplikation von Matrizen wird zu einem späteren Zeitpunkt besprochen.
|
||||
\\
|
||||
\\
|
||||
Wichtig zu erwähnen ist, dass es nicht immer eine Inverse einer Matrix gibt, dies bedeutet aber nicht, dass das Gleichungssystem nicht lösbar ist. Um herauszufinden ob ein LGS lösbar ist, führen wir die Kompabilitätsbedingung (KB) ein. Dabei gilt folgendes:
|
||||
|
||||
\begin{itemize}
|
||||
\item falls $b_1 + b_2 + b_3 \neq 0$, dann gibt es keine Lösung;
|
||||
\item falls $b_1 + b_2 + b_3 = 0$, dann gibt es unendlich viele Lösungen.
|
||||
\end{itemize}
|
||||
|
||||
Falls $b_1 + b_2 + b_3 = 0$ gilt, so kann $x_3$ beliebig gewählt werden.
|
||||
3
hs24/lineare_algebra/lineare_raeume/lineare_raeume.tex
Normal file
3
hs24/lineare_algebra/lineare_raeume/lineare_raeume.tex
Normal file
@@ -0,0 +1,3 @@
|
||||
\chapter{Lineare Räume}
|
||||
|
||||
\newpage
|
||||
88
hs24/lineare_algebra/macros.tex
Normal file
88
hs24/lineare_algebra/macros.tex
Normal file
@@ -0,0 +1,88 @@
|
||||
%From M275 "Topology" at SJSU
|
||||
\newcommand{\id}{\mathrm{id}}
|
||||
\newcommand{\taking}[1]{\xrightarrow{#1}}
|
||||
\newcommand{\inv}{^{-1}}
|
||||
|
||||
%From M170 "Introduction to Graph Theory" at SJSU
|
||||
\DeclareMathOperator{\diam}{diam}
|
||||
\DeclareMathOperator{\ord}{ord}
|
||||
\newcommand{\defeq}{\overset{\mathrm{def}}{=}}
|
||||
|
||||
%From the USAMO .tex files
|
||||
\newcommand{\ts}{\textsuperscript}
|
||||
\newcommand{\dg}{^\circ}
|
||||
\newcommand{\ii}{\item}
|
||||
|
||||
% % From Math 55 and Math 145 at Harvard
|
||||
% \newenvironment{subproof}[1][Proof]{%
|
||||
% \begin{proof}[#1] \renewcommand{\qedsymbol}{$\blacksquare$}}%
|
||||
% {\end{proof}}
|
||||
|
||||
\newcommand{\liff}{\leftrightarrow}
|
||||
\newcommand{\lthen}{\rightarrow}
|
||||
\newcommand{\opname}{\operatorname}
|
||||
\newcommand{\surjto}{\twoheadrightarrow}
|
||||
\newcommand{\injto}{\hookrightarrow}
|
||||
\newcommand{\On}{\mathrm{On}} % ordinals
|
||||
\DeclareMathOperator{\img}{im} % Image
|
||||
\DeclareMathOperator{\Img}{Im} % Image
|
||||
\DeclareMathOperator{\coker}{coker} % Cokernel
|
||||
\DeclareMathOperator{\Coker}{Coker} % Cokernel
|
||||
\DeclareMathOperator{\Ker}{Ker} % Kernel
|
||||
\DeclareMathOperator{\rank}{rank}
|
||||
\DeclareMathOperator{\Spec}{Spec} % spectrum
|
||||
\DeclareMathOperator{\Tr}{Tr} % trace
|
||||
\DeclareMathOperator{\pr}{pr} % projection
|
||||
\DeclareMathOperator{\ext}{ext} % extension
|
||||
\DeclareMathOperator{\pred}{pred} % predecessor
|
||||
\DeclareMathOperator{\dom}{dom} % domain
|
||||
\DeclareMathOperator{\ran}{ran} % range
|
||||
\DeclareMathOperator{\Hom}{Hom} % homomorphism
|
||||
\DeclareMathOperator{\Mor}{Mor} % morphisms
|
||||
\DeclareMathOperator{\End}{End} % endomorphism
|
||||
|
||||
\newcommand{\eps}{\epsilon}
|
||||
\newcommand{\veps}{\varepsilon}
|
||||
\newcommand{\ol}{\overline}
|
||||
\newcommand{\ul}{\underline}
|
||||
\newcommand{\wt}{\widetilde}
|
||||
\newcommand{\wh}{\widehat}
|
||||
\newcommand{\vocab}[1]{\textbf{\color{blue} #1}}
|
||||
\providecommand{\half}{\frac{1}{2}}
|
||||
\newcommand{\dang}{\measuredangle} %% Directed angle
|
||||
\newcommand{\ray}[1]{\overrightarrow{#1}}
|
||||
\newcommand{\seg}[1]{\overline{#1}}
|
||||
\newcommand{\arc}[1]{\wideparen{#1}}
|
||||
\DeclareMathOperator{\cis}{cis}
|
||||
\DeclareMathOperator*{\lcm}{lcm}
|
||||
\DeclareMathOperator*{\argmin}{arg min}
|
||||
\DeclareMathOperator*{\argmax}{arg max}
|
||||
\newcommand{\cycsum}{\sum_{\mathrm{cyc}}}
|
||||
\newcommand{\symsum}{\sum_{\mathrm{sym}}}
|
||||
\newcommand{\cycprod}{\prod_{\mathrm{cyc}}}
|
||||
\newcommand{\symprod}{\prod_{\mathrm{sym}}}
|
||||
\newcommand{\Qed}{\begin{flushright}\qed\end{flushright}}
|
||||
\newcommand{\parinn}{\setlength{\parindent}{1cm}}
|
||||
\newcommand{\parinf}{\setlength{\parindent}{0cm}}
|
||||
% \newcommand{\norm}{\|\cdot\|}
|
||||
\newcommand{\inorm}{\norm_{\infty}}
|
||||
\newcommand{\opensets}{\{V_{\alpha}\}_{\alpha\in I}}
|
||||
\newcommand{\oset}{V_{\alpha}}
|
||||
\newcommand{\opset}[1]{V_{\alpha_{#1}}}
|
||||
\newcommand{\lub}{\text{lub}}
|
||||
\newcommand{\del}[2]{\frac{\partial #1}{\partial #2}}
|
||||
\newcommand{\Del}[3]{\frac{\partial^{#1} #2}{\partial^{#1} #3}}
|
||||
\newcommand{\deld}[2]{\dfrac{\partial #1}{\partial #2}}
|
||||
\newcommand{\Deld}[3]{\dfrac{\partial^{#1} #2}{\partial^{#1} #3}}
|
||||
\newcommand{\lm}{\lambda}
|
||||
\newcommand{\uin}{\mathbin{\rotatebox[origin=c]{90}{$\in$}}}
|
||||
\newcommand{\usubset}{\mathbin{\rotatebox[origin=c]{90}{$\subset$}}}
|
||||
\newcommand{\lt}{\left}
|
||||
\newcommand{\rt}{\right}
|
||||
\newcommand{\bs}[1]{\boldsymbol{#1}}
|
||||
\newcommand{\exs}{\exists}
|
||||
\newcommand{\st}{\strut}
|
||||
\newcommand{\dps}[1]{\displaystyle{#1}}
|
||||
|
||||
\newcommand{\sol}{\setlength{\parindent}{0cm}\textbf{\textit{Solution:}}\setlength{\parindent}{1cm} }
|
||||
\newcommand{\solve}[1]{\setlength{\parindent}{0cm}\textbf{\textit{Solution: }}\setlength{\parindent}{1cm}#1 \Qed}
|
||||
@@ -0,0 +1,2 @@
|
||||
\section{Die adjugierte Abbildung}
|
||||
|
||||
@@ -0,0 +1,2 @@
|
||||
\section{Darstellungssatz von Riesz}
|
||||
|
||||
@@ -0,0 +1,2 @@
|
||||
\section{Gram-Schmidt-Algorithmus}
|
||||
|
||||
@@ -0,0 +1,7 @@
|
||||
\chapter{Norm und Skalarprodukt in linearen Räumen}
|
||||
|
||||
%\input{gram_schmidt_algorithmus.tex}
|
||||
% \input{darstellungssatz_von_riesz.tex}
|
||||
% \input{adjugierte_abbildung.tex}
|
||||
|
||||
\newpage
|
||||
8
hs24/lineare_algebra/notes.tex
Normal file
8
hs24/lineare_algebra/notes.tex
Normal file
@@ -0,0 +1,8 @@
|
||||
\import{./lineare_gleichungssysteme}{lineare_gleichungssysteme.tex}
|
||||
\import{./lineare_raeume}{lineare_raeume.tex}
|
||||
\import{./lineare_abbildungen}{lineare_abbildungen.tex}
|
||||
\import{./norm_und_skalarprodukt_in_linearen_raeumen}{norm_und_skalarprodukt_in_linearen_raeumen.tex}
|
||||
\import{./ausgleichsrechnung}{ausgleichsrechnung.tex}
|
||||
\import{./determinante}{determinante.tex}
|
||||
\import{./eigenwertproblem}{eigenwertproblem.tex}
|
||||
\import{./singulaerwertzerlegung}{singulaerwertzerlegung.tex}
|
||||
11
hs24/lineare_algebra/notizen_lineare_algebra_ruh_jirayu.bbl
Normal file
11
hs24/lineare_algebra/notizen_lineare_algebra_ruh_jirayu.bbl
Normal file
@@ -0,0 +1,11 @@
|
||||
\begin{thebibliography}{}
|
||||
|
||||
\bibitem[Gradinaru, 2024]{Gradinaru2024}
|
||||
Gradinaru, Prof.~Dr., V. (2024).
|
||||
\newblock {Lineare Algebra}.
|
||||
|
||||
\bibitem[{The Manim Community Developers}, 2024]{MCD2024}
|
||||
{The Manim Community Developers} (2024).
|
||||
\newblock {Manim - Mathematical Animation Framework}.
|
||||
|
||||
\end{thebibliography}
|
||||
BIN
hs24/lineare_algebra/notizen_lineare_algebra_ruh_jirayu.pdf
Normal file
BIN
hs24/lineare_algebra/notizen_lineare_algebra_ruh_jirayu.pdf
Normal file
Binary file not shown.
43
hs24/lineare_algebra/notizen_lineare_algebra_ruh_jirayu.tex
Normal file
43
hs24/lineare_algebra/notizen_lineare_algebra_ruh_jirayu.tex
Normal file
@@ -0,0 +1,43 @@
|
||||
\documentclass{report}
|
||||
|
||||
\def\papertitle{Lineare Algebra}
|
||||
|
||||
\def\theorytitle{Satz}
|
||||
\def\corollarytitle{Korollar}
|
||||
\def\proposaltitle{Vorschlag}
|
||||
\def\claimtitle{Behauptung}
|
||||
\def\exercisetitle{Aufgabe}
|
||||
\def\exampletitle{Beispiel}
|
||||
\def\questiontitle{Frage}
|
||||
\def\wrongctitle{Falscher Konzept}
|
||||
|
||||
\input{preamble}
|
||||
\input{macros}
|
||||
\input{letterfonts}
|
||||
|
||||
\title{\huge{\papertitle}}
|
||||
\author{\huge{Jirayu Ruh}}
|
||||
\date{}
|
||||
|
||||
\begin{document}
|
||||
|
||||
|
||||
\maketitle
|
||||
\newpage% or \cleardoublepage
|
||||
% \pdfbookmark[<level>]{<title>}{<dest>}
|
||||
\pdfbookmark[section]{\contentsname}{toc}
|
||||
\tableofcontents
|
||||
\pagebreak
|
||||
|
||||
\input{disclaimer.tex}
|
||||
\input{notes.tex}
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% BIBLIOGRAPHY
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
\addcontentsline{toc}{chapter}{References}
|
||||
\bibliographystyle{apalike}
|
||||
\bibliography{sources}
|
||||
|
||||
\end{document}
|
||||
778
hs24/lineare_algebra/preamble.tex
Normal file
778
hs24/lineare_algebra/preamble.tex
Normal file
@@ -0,0 +1,778 @@
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% PACKAGE IMPORTS
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
\usepackage[ngerman]{babel}
|
||||
\usepackage[tmargin=2cm,rmargin=1in,lmargin=1in,margin=0.85in,bmargin=2cm,footskip=.2in]{geometry}
|
||||
\usepackage{amsmath,amsfonts,amsthm,amssymb,mathtools}
|
||||
\usepackage[varbb]{newpxmath}
|
||||
\usepackage{xfrac}
|
||||
\usepackage[makeroom]{cancel}
|
||||
\usepackage{mathtools}
|
||||
\usepackage{bookmark}
|
||||
\usepackage{enumitem}
|
||||
\usepackage{hyperref,theoremref}
|
||||
\hypersetup{
|
||||
pdftitle={Assignment},
|
||||
colorlinks=true, linkcolor=doc!90,
|
||||
bookmarksnumbered=true,
|
||||
bookmarksopen=true
|
||||
}
|
||||
\usepackage[most,many,breakable]{tcolorbox}
|
||||
\usepackage{xcolor}
|
||||
\usepackage{varwidth}
|
||||
\usepackage{varwidth}
|
||||
\usepackage{etoolbox}
|
||||
%\usepackage{authblk}
|
||||
\usepackage{nameref}
|
||||
\usepackage{multicol,array}
|
||||
\usepackage{tikz-cd}
|
||||
\usepackage[ruled,vlined,linesnumbered]{algorithm2e}
|
||||
\usepackage{comment} % enables the use of multi-line comments (\ifx \fi)
|
||||
\usepackage{import}
|
||||
\usepackage{xifthen}
|
||||
\usepackage{pdfpages}
|
||||
\usepackage{transparent}
|
||||
\usepackage{caption}
|
||||
|
||||
\newcommand\mycommfont[1]{\footnotesize\ttfamily\textcolor{blue}{#1}}
|
||||
\SetCommentSty{mycommfont}
|
||||
\newcommand{\incfig}[1]{%
|
||||
\def\svgwidth{\columnwidth}
|
||||
\import{./figures/}{#1.pdf_tex}
|
||||
}
|
||||
|
||||
\usepackage{tikzsymbols}
|
||||
\renewcommand\qedsymbol{$\Laughey$}
|
||||
|
||||
|
||||
%\usepackage{import}
|
||||
%\usepackage{xifthen}
|
||||
%\usepackage{pdfpages}
|
||||
%\usepackage{transparent}
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% SELF MADE COLORS
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
|
||||
|
||||
\definecolor{myg}{RGB}{56, 140, 70}
|
||||
\definecolor{myb}{RGB}{45, 111, 177}
|
||||
\definecolor{myr}{RGB}{199, 68, 64}
|
||||
\definecolor{mytheorembg}{HTML}{F2F2F9}
|
||||
\definecolor{mytheoremfr}{HTML}{00007B}
|
||||
\definecolor{mylenmabg}{HTML}{FFFAF8}
|
||||
\definecolor{mylenmafr}{HTML}{983b0f}
|
||||
\definecolor{mypropbg}{HTML}{f2fbfc}
|
||||
\definecolor{mypropfr}{HTML}{191971}
|
||||
\definecolor{myexamplebg}{HTML}{F2FBF8}
|
||||
\definecolor{myexamplefr}{HTML}{88D6D1}
|
||||
\definecolor{myexampleti}{HTML}{2A7F7F}
|
||||
\definecolor{mydefinitbg}{HTML}{E5E5FF}
|
||||
\definecolor{mydefinitfr}{HTML}{3F3FA3}
|
||||
\definecolor{notesgreen}{RGB}{0,162,0}
|
||||
\definecolor{myp}{RGB}{197, 92, 212}
|
||||
\definecolor{mygr}{HTML}{2C3338}
|
||||
\definecolor{myred}{RGB}{127,0,0}
|
||||
\definecolor{myyellow}{RGB}{169,121,69}
|
||||
\definecolor{myexercisebg}{HTML}{F2FBF8}
|
||||
\definecolor{myexercisefg}{HTML}{88D6D1}
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% TCOLORBOX SETUPS
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
\setlength{\parindent}{0cm}
|
||||
%================================
|
||||
% THEOREM BOX
|
||||
%================================
|
||||
|
||||
\tcbuselibrary{theorems,skins,hooks}
|
||||
\newtcbtheorem[number within=section]{Theorem}{\theorytitle}
|
||||
{%
|
||||
enhanced,
|
||||
breakable,
|
||||
colback = mytheorembg,
|
||||
frame hidden,
|
||||
boxrule = 0sp,
|
||||
borderline west = {2pt}{0pt}{mytheoremfr},
|
||||
sharp corners,
|
||||
detach title,
|
||||
before upper = \tcbtitle\par\smallskip,
|
||||
coltitle = mytheoremfr,
|
||||
fonttitle = \bfseries\sffamily,
|
||||
description font = \mdseries,
|
||||
separator sign none,
|
||||
segmentation style={solid, mytheoremfr},
|
||||
}
|
||||
{th}
|
||||
|
||||
\tcbuselibrary{theorems,skins,hooks}
|
||||
\newtcbtheorem[number within=chapter]{theorem}{\theorytitle}
|
||||
{%
|
||||
enhanced,
|
||||
breakable,
|
||||
colback = mytheorembg,
|
||||
frame hidden,
|
||||
boxrule = 0sp,
|
||||
borderline west = {2pt}{0pt}{mytheoremfr},
|
||||
sharp corners,
|
||||
detach title,
|
||||
before upper = \tcbtitle\par\smallskip,
|
||||
coltitle = mytheoremfr,
|
||||
fonttitle = \bfseries\sffamily,
|
||||
description font = \mdseries,
|
||||
separator sign none,
|
||||
segmentation style={solid, mytheoremfr},
|
||||
}
|
||||
{th}
|
||||
|
||||
|
||||
\tcbuselibrary{theorems,skins,hooks}
|
||||
\newtcolorbox{Theoremcon}
|
||||
{%
|
||||
enhanced
|
||||
,breakable
|
||||
,colback = mytheorembg
|
||||
,frame hidden
|
||||
,boxrule = 0sp
|
||||
,borderline west = {2pt}{0pt}{mytheoremfr}
|
||||
,sharp corners
|
||||
,description font = \mdseries
|
||||
,separator sign none
|
||||
}
|
||||
|
||||
%================================
|
||||
% Corollery
|
||||
%================================
|
||||
\tcbuselibrary{theorems,skins,hooks}
|
||||
\newtcbtheorem[number within=section]{Corollary}{\corollarytitle}
|
||||
{%
|
||||
enhanced
|
||||
,breakable
|
||||
,colback = myp!10
|
||||
,frame hidden
|
||||
,boxrule = 0sp
|
||||
,borderline west = {2pt}{0pt}{myp!85!black}
|
||||
,sharp corners
|
||||
,detach title
|
||||
,before upper = \tcbtitle\par\smallskip
|
||||
,coltitle = myp!85!black
|
||||
,fonttitle = \bfseries\sffamily
|
||||
,description font = \mdseries
|
||||
,separator sign none
|
||||
,segmentation style={solid, myp!85!black}
|
||||
}
|
||||
{th}
|
||||
\tcbuselibrary{theorems,skins,hooks}
|
||||
\newtcbtheorem[number within=chapter]{corollary}{\corollarytitle}
|
||||
{%
|
||||
enhanced
|
||||
,breakable
|
||||
,colback = myp!10
|
||||
,frame hidden
|
||||
,boxrule = 0sp
|
||||
,borderline west = {2pt}{0pt}{myp!85!black}
|
||||
,sharp corners
|
||||
,detach title
|
||||
,before upper = \tcbtitle\par\smallskip
|
||||
,coltitle = myp!85!black
|
||||
,fonttitle = \bfseries\sffamily
|
||||
,description font = \mdseries
|
||||
,separator sign none
|
||||
,segmentation style={solid, myp!85!black}
|
||||
}
|
||||
{th}
|
||||
|
||||
|
||||
%================================
|
||||
% LENMA
|
||||
%================================
|
||||
|
||||
\tcbuselibrary{theorems,skins,hooks}
|
||||
\newtcbtheorem[number within=section]{Lenma}{Lenma}
|
||||
{%
|
||||
enhanced,
|
||||
breakable,
|
||||
colback = mylenmabg,
|
||||
frame hidden,
|
||||
boxrule = 0sp,
|
||||
borderline west = {2pt}{0pt}{mylenmafr},
|
||||
sharp corners,
|
||||
detach title,
|
||||
before upper = \tcbtitle\par\smallskip,
|
||||
coltitle = mylenmafr,
|
||||
fonttitle = \bfseries\sffamily,
|
||||
description font = \mdseries,
|
||||
separator sign none,
|
||||
segmentation style={solid, mylenmafr},
|
||||
}
|
||||
{th}
|
||||
|
||||
\tcbuselibrary{theorems,skins,hooks}
|
||||
\newtcbtheorem[number within=chapter]{lenma}{Lenma}
|
||||
{%
|
||||
enhanced,
|
||||
breakable,
|
||||
colback = mylenmabg,
|
||||
frame hidden,
|
||||
boxrule = 0sp,
|
||||
borderline west = {2pt}{0pt}{mylenmafr},
|
||||
sharp corners,
|
||||
detach title,
|
||||
before upper = \tcbtitle\par\smallskip,
|
||||
coltitle = mylenmafr,
|
||||
fonttitle = \bfseries\sffamily,
|
||||
description font = \mdseries,
|
||||
separator sign none,
|
||||
segmentation style={solid, mylenmafr},
|
||||
}
|
||||
{th}
|
||||
|
||||
|
||||
%================================
|
||||
% PROPOSITION
|
||||
%================================
|
||||
|
||||
\tcbuselibrary{theorems,skins,hooks}
|
||||
\newtcbtheorem[number within=section]{Prop}{\proposaltitle}
|
||||
{%
|
||||
enhanced,
|
||||
breakable,
|
||||
colback = mypropbg,
|
||||
frame hidden,
|
||||
boxrule = 0sp,
|
||||
borderline west = {2pt}{0pt}{mypropfr},
|
||||
sharp corners,
|
||||
detach title,
|
||||
before upper = \tcbtitle\par\smallskip,
|
||||
coltitle = mypropfr,
|
||||
fonttitle = \bfseries\sffamily,
|
||||
description font = \mdseries,
|
||||
separator sign none,
|
||||
segmentation style={solid, mypropfr},
|
||||
}
|
||||
{th}
|
||||
|
||||
\tcbuselibrary{theorems,skins,hooks}
|
||||
\newtcbtheorem[number within=chapter]{prop}{\proposaltitle}
|
||||
{%
|
||||
enhanced,
|
||||
breakable,
|
||||
colback = mypropbg,
|
||||
frame hidden,
|
||||
boxrule = 0sp,
|
||||
borderline west = {2pt}{0pt}{mypropfr},
|
||||
sharp corners,
|
||||
detach title,
|
||||
before upper = \tcbtitle\par\smallskip,
|
||||
coltitle = mypropfr,
|
||||
fonttitle = \bfseries\sffamily,
|
||||
description font = \mdseries,
|
||||
separator sign none,
|
||||
segmentation style={solid, mypropfr},
|
||||
}
|
||||
{th}
|
||||
|
||||
|
||||
%================================
|
||||
% CLAIM
|
||||
%================================
|
||||
|
||||
\tcbuselibrary{theorems,skins,hooks}
|
||||
\newtcbtheorem[number within=section]{claim}{\claimtitle}
|
||||
{%
|
||||
enhanced
|
||||
,breakable
|
||||
,colback = myg!10
|
||||
,frame hidden
|
||||
,boxrule = 0sp
|
||||
,borderline west = {2pt}{0pt}{myg}
|
||||
,sharp corners
|
||||
,detach title
|
||||
,before upper = \tcbtitle\par\smallskip
|
||||
,coltitle = myg!85!black
|
||||
,fonttitle = \bfseries\sffamily
|
||||
,description font = \mdseries
|
||||
,separator sign none
|
||||
,segmentation style={solid, myg!85!black}
|
||||
}
|
||||
{th}
|
||||
|
||||
|
||||
|
||||
%================================
|
||||
% Exercise
|
||||
%================================
|
||||
|
||||
\tcbuselibrary{theorems,skins,hooks}
|
||||
\newtcbtheorem[number within=section]{Exercise}{\exercisetitle}
|
||||
{%
|
||||
enhanced,
|
||||
breakable,
|
||||
colback = myexercisebg,
|
||||
frame hidden,
|
||||
boxrule = 0sp,
|
||||
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|
||||
sharp corners,
|
||||
detach title,
|
||||
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|
||||
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|
||||
fonttitle = \bfseries\sffamily,
|
||||
description font = \mdseries,
|
||||
separator sign none,
|
||||
segmentation style={solid, myexercisefg},
|
||||
}
|
||||
{th}
|
||||
|
||||
\tcbuselibrary{theorems,skins,hooks}
|
||||
\newtcbtheorem[number within=chapter]{exercise}{\exercisetitle}
|
||||
{%
|
||||
enhanced,
|
||||
breakable,
|
||||
colback = myexercisebg,
|
||||
frame hidden,
|
||||
boxrule = 0sp,
|
||||
borderline west = {2pt}{0pt}{myexercisefg},
|
||||
sharp corners,
|
||||
detach title,
|
||||
before upper = \tcbtitle\par\smallskip,
|
||||
coltitle = myexercisefg,
|
||||
fonttitle = \bfseries\sffamily,
|
||||
description font = \mdseries,
|
||||
separator sign none,
|
||||
segmentation style={solid, myexercisefg},
|
||||
}
|
||||
{th}
|
||||
|
||||
%================================
|
||||
% EXAMPLE BOX
|
||||
%================================
|
||||
|
||||
\newtcbtheorem[number within=section]{Example}{\exampletitle}
|
||||
{%
|
||||
colback = myexamplebg
|
||||
,breakable
|
||||
,colframe = myexamplefr
|
||||
,coltitle = myexampleti
|
||||
,boxrule = 1pt
|
||||
,sharp corners
|
||||
,detach title
|
||||
,before upper=\tcbtitle\par\smallskip
|
||||
,fonttitle = \bfseries
|
||||
,description font = \mdseries
|
||||
,separator sign none
|
||||
,description delimiters parenthesis
|
||||
}
|
||||
{ex}
|
||||
|
||||
\newtcbtheorem[number within=chapter]{example}{\exampletitle}
|
||||
{%
|
||||
colback = myexamplebg
|
||||
,breakable
|
||||
,colframe = myexamplefr
|
||||
,coltitle = myexampleti
|
||||
,boxrule = 1pt
|
||||
,sharp corners
|
||||
,detach title
|
||||
,before upper=\tcbtitle\par\smallskip
|
||||
,fonttitle = \bfseries
|
||||
,description font = \mdseries
|
||||
,separator sign none
|
||||
,description delimiters parenthesis
|
||||
}
|
||||
{ex}
|
||||
|
||||
%================================
|
||||
% DEFINITION BOX
|
||||
%================================
|
||||
|
||||
\newtcbtheorem[number within=section]{Definition}{Definition}{enhanced,
|
||||
before skip=2mm,after skip=2mm, colback=red!5,colframe=red!80!black,boxrule=0.5mm,
|
||||
attach boxed title to top left={xshift=1cm,yshift*=1mm-\tcboxedtitleheight}, varwidth boxed title*=-3cm,
|
||||
boxed title style={frame code={
|
||||
\path[fill=tcbcolback]
|
||||
([yshift=-1mm,xshift=-1mm]frame.north west)
|
||||
arc[start angle=0,end angle=180,radius=1mm]
|
||||
([yshift=-1mm,xshift=1mm]frame.north east)
|
||||
arc[start angle=180,end angle=0,radius=1mm];
|
||||
\path[left color=tcbcolback!60!black,right color=tcbcolback!60!black,
|
||||
middle color=tcbcolback!80!black]
|
||||
([xshift=-2mm]frame.north west) -- ([xshift=2mm]frame.north east)
|
||||
[rounded corners=1mm]-- ([xshift=1mm,yshift=-1mm]frame.north east)
|
||||
-- (frame.south east) -- (frame.south west)
|
||||
-- ([xshift=-1mm,yshift=-1mm]frame.north west)
|
||||
[sharp corners]-- cycle;
|
||||
},interior engine=empty,
|
||||
},
|
||||
fonttitle=\bfseries,
|
||||
title={#2},#1}{def}
|
||||
\newtcbtheorem[number within=chapter]{definition}{Definition}{enhanced,
|
||||
before skip=2mm,after skip=2mm, colback=red!5,colframe=red!80!black,boxrule=0.5mm,
|
||||
attach boxed title to top left={xshift=1cm,yshift*=1mm-\tcboxedtitleheight}, varwidth boxed title*=-3cm,
|
||||
boxed title style={frame code={
|
||||
\path[fill=tcbcolback]
|
||||
([yshift=-1mm,xshift=-1mm]frame.north west)
|
||||
arc[start angle=0,end angle=180,radius=1mm]
|
||||
([yshift=-1mm,xshift=1mm]frame.north east)
|
||||
arc[start angle=180,end angle=0,radius=1mm];
|
||||
\path[left color=tcbcolback!60!black,right color=tcbcolback!60!black,
|
||||
middle color=tcbcolback!80!black]
|
||||
([xshift=-2mm]frame.north west) -- ([xshift=2mm]frame.north east)
|
||||
[rounded corners=1mm]-- ([xshift=1mm,yshift=-1mm]frame.north east)
|
||||
-- (frame.south east) -- (frame.south west)
|
||||
-- ([xshift=-1mm,yshift=-1mm]frame.north west)
|
||||
[sharp corners]-- cycle;
|
||||
},interior engine=empty,
|
||||
},
|
||||
fonttitle=\bfseries,
|
||||
title={#2},#1}{def}
|
||||
|
||||
|
||||
|
||||
%================================
|
||||
% Solution BOX
|
||||
%================================
|
||||
|
||||
\makeatletter
|
||||
\newtcbtheorem{question}{\questiontitle}{enhanced,
|
||||
breakable,
|
||||
colback=white,
|
||||
colframe=myb!80!black,
|
||||
attach boxed title to top left={yshift*=-\tcboxedtitleheight},
|
||||
fonttitle=\bfseries,
|
||||
title={#2},
|
||||
boxed title size=title,
|
||||
boxed title style={%
|
||||
sharp corners,
|
||||
rounded corners=northwest,
|
||||
colback=tcbcolframe,
|
||||
boxrule=0pt,
|
||||
},
|
||||
underlay boxed title={%
|
||||
\path[fill=tcbcolframe] (title.south west)--(title.south east)
|
||||
to[out=0, in=180] ([xshift=5mm]title.east)--
|
||||
(title.center-|frame.east)
|
||||
[rounded corners=\kvtcb@arc] |-
|
||||
(frame.north) -| cycle;
|
||||
},
|
||||
#1
|
||||
}{def}
|
||||
\makeatother
|
||||
|
||||
%================================
|
||||
% SOLUTION BOX
|
||||
%================================
|
||||
|
||||
\makeatletter
|
||||
\newtcolorbox{solution}{enhanced,
|
||||
breakable,
|
||||
colback=white,
|
||||
colframe=myg!80!black,
|
||||
attach boxed title to top left={yshift*=-\tcboxedtitleheight},
|
||||
title=Solution,
|
||||
boxed title size=title,
|
||||
boxed title style={%
|
||||
sharp corners,
|
||||
rounded corners=northwest,
|
||||
colback=tcbcolframe,
|
||||
boxrule=0pt,
|
||||
},
|
||||
underlay boxed title={%
|
||||
\path[fill=tcbcolframe] (title.south west)--(title.south east)
|
||||
to[out=0, in=180] ([xshift=5mm]title.east)--
|
||||
(title.center-|frame.east)
|
||||
[rounded corners=\kvtcb@arc] |-
|
||||
(frame.north) -| cycle;
|
||||
},
|
||||
}
|
||||
\makeatother
|
||||
|
||||
%================================
|
||||
% Question BOX
|
||||
%================================
|
||||
|
||||
\makeatletter
|
||||
\newtcbtheorem{qstion}{\questiontitle}{enhanced,
|
||||
breakable,
|
||||
colback=white,
|
||||
colframe=mygr,
|
||||
attach boxed title to top left={yshift*=-\tcboxedtitleheight},
|
||||
fonttitle=\bfseries,
|
||||
title={#2},
|
||||
boxed title size=title,
|
||||
boxed title style={%
|
||||
sharp corners,
|
||||
rounded corners=northwest,
|
||||
colback=tcbcolframe,
|
||||
boxrule=0pt,
|
||||
},
|
||||
underlay boxed title={%
|
||||
\path[fill=tcbcolframe] (title.south west)--(title.south east)
|
||||
to[out=0, in=180] ([xshift=5mm]title.east)--
|
||||
(title.center-|frame.east)
|
||||
[rounded corners=\kvtcb@arc] |-
|
||||
(frame.north) -| cycle;
|
||||
},
|
||||
#1
|
||||
}{def}
|
||||
\makeatother
|
||||
|
||||
\newtcbtheorem[number within=chapter]{wconc}{\wrongctitle}{
|
||||
breakable,
|
||||
enhanced,
|
||||
colback=white,
|
||||
colframe=myr,
|
||||
arc=0pt,
|
||||
outer arc=0pt,
|
||||
fonttitle=\bfseries\sffamily\large,
|
||||
colbacktitle=myr,
|
||||
attach boxed title to top left={},
|
||||
boxed title style={
|
||||
enhanced,
|
||||
skin=enhancedfirst jigsaw,
|
||||
arc=3pt,
|
||||
bottom=0pt,
|
||||
interior style={fill=myr}
|
||||
},
|
||||
#1
|
||||
}{def}
|
||||
|
||||
|
||||
|
||||
%================================
|
||||
% NOTE BOX
|
||||
%================================
|
||||
|
||||
\usetikzlibrary{arrows,calc,shadows.blur}
|
||||
\tcbuselibrary{skins}
|
||||
\newtcolorbox{note}[1][]{%
|
||||
enhanced jigsaw,
|
||||
colback=gray!20!white,%
|
||||
colframe=gray!80!black,
|
||||
size=small,
|
||||
boxrule=1pt,
|
||||
title=\textbf{Bemerkung:-},
|
||||
halign title=flush center,
|
||||
coltitle=black,
|
||||
breakable,
|
||||
drop shadow=black!50!white,
|
||||
attach boxed title to top left={xshift=1cm,yshift=-\tcboxedtitleheight/2,yshifttext=-\tcboxedtitleheight/2},
|
||||
minipage boxed title=2.5cm,
|
||||
boxed title style={%
|
||||
colback=white,
|
||||
size=fbox,
|
||||
boxrule=1pt,
|
||||
boxsep=2pt,
|
||||
underlay={%
|
||||
\coordinate (dotA) at ($(interior.west) + (-0.5pt,0)$);
|
||||
\coordinate (dotB) at ($(interior.east) + (0.5pt,0)$);
|
||||
\begin{scope}
|
||||
\clip (interior.north west) rectangle ([xshift=3ex]interior.east);
|
||||
\filldraw [white, blur shadow={shadow opacity=60, shadow yshift=-.75ex}, rounded corners=2pt] (interior.north west) rectangle (interior.south east);
|
||||
\end{scope}
|
||||
\begin{scope}[gray!80!black]
|
||||
\fill (dotA) circle (2pt);
|
||||
\fill (dotB) circle (2pt);
|
||||
\end{scope}
|
||||
},
|
||||
},
|
||||
#1,
|
||||
}
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% SELF MADE COMMANDS
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
|
||||
\newcommand{\thm}[2]{\begin{Theorem}{#1}{}#2\end{Theorem}}
|
||||
\newcommand{\cor}[2]{\begin{Corollary}{#1}{}#2\end{Corollary}}
|
||||
\newcommand{\mlenma}[2]{\begin{Lenma}{#1}{}#2\end{Lenma}}
|
||||
\newcommand{\mprop}[2]{\begin{Prop}{#1}{}#2\end{Prop}}
|
||||
\newcommand{\clm}[3]{\begin{Claim}{#1}{#2}#3\end{Claim}}
|
||||
\newcommand{\wc}[2]{\begin{wconc}{#1}{}\setlength{\parindent}{1cm}#2\end{wconc}}
|
||||
\newcommand{\thmcon}[1]{\begin{Theoremcon}{#1}\end{Theoremcon}}
|
||||
\newcommand{\ex}[2]{\begin{Example}{#1}{}#2\end{Example}}
|
||||
\newcommand{\dfn}[2]{\begin{Definition}[colbacktitle=red!75!black]{#1}{}#2\end{Definition}}
|
||||
\newcommand{\dfnc}[2]{\begin{definition}[colbacktitle=red!75!black]{#1}{}#2\end{definition}}
|
||||
\newcommand{\qs}[2]{\begin{question}{#1}{}#2\end{question}}
|
||||
\newcommand{\pf}[2]{\begin{myproof}[#1]#2\end{myproof}}
|
||||
\newcommand{\nt}[1]{\begin{note}#1\end{note}}
|
||||
|
||||
\newcommand*\circled[1]{\tikz[baseline=(char.base)]{
|
||||
Wrong Concept \node[shape=circle,draw,inner sep=1pt] (char) {#1};}}
|
||||
\newcommand\getcurrentref[1]{%
|
||||
\ifnumequal{\value{#1}}{0}
|
||||
{??}
|
||||
{\the\value{#1}}%
|
||||
}
|
||||
\newcommand{\getCurrentSectionNumber}{\getcurrentref{section}}
|
||||
\newenvironment{myproof}[1][\proofname]{%
|
||||
\proof[\bfseries #1: ]%
|
||||
}{\endproof}
|
||||
|
||||
\newcommand{\mclm}[2]{\begin{myclaim}[#1]#2\end{myclaim}}
|
||||
\newenvironment{myclaim}[1][\claimname]{\proof[\bfseries #1: ]}{}
|
||||
|
||||
\newcounter{mylabelcounter}
|
||||
|
||||
\makeatletter
|
||||
\newcommand{\setword}[2]{%
|
||||
\phantomsection
|
||||
#1\def\@currentlabel{\unexpanded{#1}}\label{#2}%
|
||||
}
|
||||
\makeatother
|
||||
|
||||
|
||||
|
||||
|
||||
\tikzset{
|
||||
symbol/.style={
|
||||
draw=none,
|
||||
every to/.append style={
|
||||
edge node={node [sloped, allow upside down, auto=false]{$#1$}}}
|
||||
}
|
||||
}
|
||||
|
||||
|
||||
% deliminators
|
||||
\DeclarePairedDelimiter{\abs}{\lvert}{\rvert}
|
||||
\DeclarePairedDelimiter{\norm}{\lVert}{\rVert}
|
||||
|
||||
\DeclarePairedDelimiter{\ceil}{\lceil}{\rceil}
|
||||
\DeclarePairedDelimiter{\floor}{\lfloor}{\rfloor}
|
||||
\DeclarePairedDelimiter{\round}{\lfloor}{\rceil}
|
||||
|
||||
\newsavebox\diffdbox
|
||||
\newcommand{\slantedromand}{{\mathpalette\makesl{d}}}
|
||||
\newcommand{\makesl}[2]{%
|
||||
\begingroup
|
||||
\sbox{\diffdbox}{$\mathsurround=0pt#1\mathrm{#2}$}%
|
||||
\pdfsave
|
||||
\pdfsetmatrix{1 0 0.2 1}%
|
||||
\rlap{\usebox{\diffdbox}}%
|
||||
\pdfrestore
|
||||
\hskip\wd\diffdbox
|
||||
\endgroup
|
||||
}
|
||||
\newcommand{\dd}[1][]{\ensuremath{\mathop{}\!\ifstrempty{#1}{%
|
||||
\slantedromand\@ifnextchar^{\hspace{0.2ex}}{\hspace{0.1ex}}}%
|
||||
{\slantedromand\hspace{0.2ex}^{#1}}}}
|
||||
\ProvideDocumentCommand\dv{o m g}{%
|
||||
\ensuremath{%
|
||||
\IfValueTF{#3}{%
|
||||
\IfNoValueTF{#1}{%
|
||||
\frac{\dd #2}{\dd #3}%
|
||||
}{%
|
||||
\frac{\dd^{#1} #2}{\dd #3^{#1}}%
|
||||
}%
|
||||
}{%
|
||||
\IfNoValueTF{#1}{%
|
||||
\frac{\dd}{\dd #2}%
|
||||
}{%
|
||||
\frac{\dd^{#1}}{\dd #2^{#1}}%
|
||||
}%
|
||||
}%
|
||||
}%
|
||||
}
|
||||
\providecommand*{\pdv}[3][]{\frac{\partial^{#1}#2}{\partial#3^{#1}}}
|
||||
% - others
|
||||
\DeclareMathOperator{\Lap}{\mathcal{L}}
|
||||
\DeclareMathOperator{\Var}{Var} % varience
|
||||
\DeclareMathOperator{\Cov}{Cov} % covarience
|
||||
\DeclareMathOperator{\E}{E} % expected
|
||||
|
||||
% Since the amsthm package isn't loaded
|
||||
|
||||
% I prefer the slanted \leq
|
||||
\let\oldleq\leq % save them in case they're every wanted
|
||||
\let\oldgeq\geq
|
||||
\renewcommand{\leq}{\leqslant}
|
||||
\renewcommand{\geq}{\geqslant}
|
||||
|
||||
% % redefine matrix env to allow for alignment, use r as default
|
||||
% \renewcommand*\env@matrix[1][r]{\hskip -\arraycolsep
|
||||
% \let\@ifnextchar\new@ifnextchar
|
||||
% \array{*\c@MaxMatrixCols #1}}
|
||||
|
||||
|
||||
%\usepackage{framed}
|
||||
%\usepackage{titletoc}
|
||||
%\usepackage{etoolbox}
|
||||
%\usepackage{lmodern}
|
||||
|
||||
|
||||
%\patchcmd{\tableofcontents}{\contentsname}{\sffamily\contentsname}{}{}
|
||||
|
||||
%\renewenvironment{leftbar}
|
||||
%{\def\FrameCommand{\hspace{6em}%
|
||||
% {\color{myyellow}\vrule width 2pt depth 6pt}\hspace{1em}}%
|
||||
% \MakeFramed{\parshape 1 0cm \dimexpr\textwidth-6em\relax\FrameRestore}\vskip2pt%
|
||||
%}
|
||||
%{\endMakeFramed}
|
||||
|
||||
%\titlecontents{chapter}
|
||||
%[0em]{\vspace*{2\baselineskip}}
|
||||
%{\parbox{4.5em}{%
|
||||
% \hfill\Huge\sffamily\bfseries\color{myred}\thecontentspage}%
|
||||
% \vspace*{-2.3\baselineskip}\leftbar\textsc{\small\chaptername~\thecontentslabel}\\\sffamily}
|
||||
%{}{\endleftbar}
|
||||
%\titlecontents{section}
|
||||
%[8.4em]
|
||||
%{\sffamily\contentslabel{3em}}{}{}
|
||||
%{\hspace{0.5em}\nobreak\itshape\color{myred}\contentspage}
|
||||
%\titlecontents{subsection}
|
||||
%[8.4em]
|
||||
%{\sffamily\contentslabel{3em}}{}{}
|
||||
%{\hspace{0.5em}\nobreak\itshape\color{myred}\contentspage}
|
||||
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% TABLE OF CONTENTS
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
\usepackage{tikz}
|
||||
\definecolor{doc}{RGB}{0,60,110}
|
||||
\usepackage{titletoc}
|
||||
\contentsmargin{0cm}
|
||||
\titlecontents{chapter}[3.7pc]
|
||||
{\addvspace{30pt}%
|
||||
\begin{tikzpicture}[remember picture, overlay]%
|
||||
\draw[fill=doc!60,draw=doc!60] (-7,-.1) rectangle (-0.9,.5);%
|
||||
\pgftext[left,x=-3.5cm,y=0.2cm]{\color{white}\Large\sc\bfseries Chapter\ \thecontentslabel};%
|
||||
\end{tikzpicture}\color{doc!60}\large\sc\bfseries}%
|
||||
{}
|
||||
{}
|
||||
{\;\titlerule\;\large\sc\bfseries Page \thecontentspage
|
||||
\begin{tikzpicture}[remember picture, overlay]
|
||||
\draw[fill=doc!60,draw=doc!60] (2pt,0) rectangle (4,0.1pt);
|
||||
\end{tikzpicture}}%
|
||||
\titlecontents{section}[3.7pc]
|
||||
{\addvspace{2pt}}
|
||||
{\contentslabel[\thecontentslabel]{2pc}}
|
||||
{}
|
||||
{\hfill\small \thecontentspage}
|
||||
[]
|
||||
\titlecontents*{subsection}[3.7pc]
|
||||
{\addvspace{-1pt}\small}
|
||||
{}
|
||||
{}
|
||||
{\ --- \small\thecontentspage}
|
||||
[ \textbullet\ ][]
|
||||
|
||||
\makeatletter
|
||||
\renewcommand{\tableofcontents}{%
|
||||
\chapter*{%
|
||||
\vspace*{-20\p@}%
|
||||
\begin{tikzpicture}[remember picture, overlay]%
|
||||
\pgftext[right,x=15cm,y=0.2cm]{\color{doc!60}\Huge\sc\bfseries \contentsname};%
|
||||
\draw[fill=doc!60,draw=doc!60] (13,-.75) rectangle (20,1);%
|
||||
\clip (13,-.75) rectangle (20,1);
|
||||
\pgftext[right,x=15cm,y=0.2cm]{\color{white}\Huge\sc\bfseries \contentsname};%
|
||||
\end{tikzpicture}}%
|
||||
\@starttoc{toc}}
|
||||
\makeatother
|
||||
@@ -0,0 +1 @@
|
||||
\chapter{Singulärwertzerlegung}
|
||||
16
hs24/lineare_algebra/sources.bib
Normal file
16
hs24/lineare_algebra/sources.bib
Normal file
@@ -0,0 +1,16 @@
|
||||
@misc{Gradinaru2024,
|
||||
author = {Gradinaru, Prof. Dr. ,Vasile},
|
||||
month = nov,
|
||||
title = {{Lineare Algebra}},
|
||||
year = {2024},
|
||||
}
|
||||
|
||||
@software{MCD2024,
|
||||
author = {{The Manim Community Developers}},
|
||||
license = {MIT},
|
||||
month = apr,
|
||||
title = {{Manim - Mathematical Animation Framework}},
|
||||
url = {https://www.manim.community/},
|
||||
version = {v0.18.1},
|
||||
year = {2024},
|
||||
}
|
||||
Reference in New Issue
Block a user