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%\subsection{Kombinationen}\label{vektorraum:kombination} - %Voraussetzungen für die nächsten Definitionen: - %Es seien $v_1, \dots, v_m$ Elemente eines Vektorraums $V$ über einem Skalarenkörper $\K$, - %und es sei $\alpha := (\alpha_1, \dots, \alpha_m)$ ein m-Tupel aus $\K^m$ für ein $m\in \N \backslash \{0\}$. - %\subsubsection{Linearkombination} + \subsection{Kombinationen}\label{vektorraum:kombination} + Voraussetzungen für die nächsten Definitionen: + Seien $v_1, \dots, v_m$ Elemente eines Vektorraums $V$ über einem Skalarenkörper $\K$, + und es sei $\alpha := (\alpha_1, \dots, \alpha_m)$ ein m-Tupel aus $\K^m$ für ein $m\in \N \backslash \{0\}$. + \subsubsection{Linearkombination} + Eine Linearkombination ist eine Vektroaddition, bei der jeder Vektor einer Menge $V$ zunächst mit einem Skalar $\alpha$ multipliziert wird. + \[\sum_{j=1}^{m} \alpha_j \cdot v_j \] + \subsubsection{Affinkombination} + Wenn außerdem gilt, dass die Summe aller Koeffizienten 1 ergibt, also + \[\sum_{j=1}^{m} \alpha_j = 1\] + dann spricht man von einer \emph{Affinkombination}. + \subsubsection{Konvexkombination} + Wenn darüber hinaus $\K = \R$ ist und + \[\alpha_j \in [0,1] \; \forall j, 1\le j \le m \] + gilt, spricht man von einer \emph{Konvexkombination}. + Vergleiche zum Verständnis für Konvex die Definition auf Wikipedia\footnote{\url{https://de.wikipedia.org/wiki/Konvexe_Menge}} + \vspace{\fill} \doclicenseThis{} diff --git a/README.md b/README.md index eb67842..f56d2fa 100644 --- a/README.md +++ b/README.md @@ -4,3 +4,4 @@ Hier findet sich eine kleine Sammlung von Zusammenfassungen aus meinen Kursen. Diese sind meist unvollständig und nur nach Lust und Laune erstellt, also Obacht!. [DiMa](DiMa/dima.pdf) +[MafIA 1](MafIA1/mafia.pdf)