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diff --git a/vorlesungen/slides/2/skalarprodukt.tex b/vorlesungen/slides/2/skalarprodukt.tex new file mode 100644 index 0000000..99d8a73 --- /dev/null +++ b/vorlesungen/slides/2/skalarprodukt.tex @@ -0,0 +1,96 @@ +% +% skalarprodukt.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Skalarprodukt} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Positiv definite, symmetrische Bilinearform} +$\langle \;\,,\;\rangle\colon V\times V\to \mathbb{R}$ +\begin{itemize} +\item<2-> +Bilinear: +\begin{align*} +\langle \alpha u+\beta v,w\rangle +&= +\alpha\langle u,w\rangle ++ +\beta\langle v,w\rangle +\\ +\langle u,\alpha v+\beta w\rangle +&= +\alpha\langle u,v\rangle ++ +\beta\langle u,w\rangle +\end{align*} +\item<3-> +Symmetrisch: $\langle u,v\rangle = \langle v,u\rangle$ +\item<4-> +$\langle x,x\rangle >0 \quad\forall x\ne 0$ +\end{itemize} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<5->{% +\begin{block}{Positive definite, hermitesche Sesquilinearform} +$\langle \;\,,\;\rangle\colon V\times V\to \mathbb{C}$ +\begin{itemize} +\item<6-> +Sesquilinear: +\begin{align*} +\langle \alpha u+\beta v,w\rangle +&= +\overline{\alpha}\langle u,w\rangle ++ +\overline{\beta}\langle v,w\rangle +\\ +\langle u,\alpha v+\beta w\rangle +&= +\alpha\langle u,v\rangle ++ +\beta\langle u,w\rangle +\end{align*} +\item<7-> +Hermitesch: $\langle u,v\rangle = \overline{\langle v,u\rangle}$ +\item<8-> +$\langle x,x\rangle >0 \quad\forall x\ne 0$ +\end{itemize} +\end{block}} +\end{column} +\end{columns} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.28\textwidth} +\uncover<9->{% +\begin{block}{$2$-Norm} +$\|v\|_2^2 = \langle v,v\rangle$ +\\ +$\|v\|_2 = \sqrt{\langle v,v\rangle}$ +\end{block}} +\end{column} +\begin{column}{0.78\textwidth} +\uncover<10->{% +\begin{itemize} +\item<11-> $\|v\|_2 = \sqrt{\langle v,v\rangle} > 0\quad\forall v\ne 0$ +\item<12-> $\| \lambda v \|_2 += +\sqrt{\langle \lambda v,\lambda v\rangle\mathstrut} += +\sqrt{\overline{\lambda}\lambda\langle v,v\rangle} += +|\lambda|\cdot \|v\|_2$ +\item<13-> +\raisebox{-8pt}{ +$\begin{aligned} +\|u+v\|_2^2 &= \|u\|_2^2 + 2{\color{red}\operatorname{Re}\langle u,v\rangle} + \|v\|_2^2 +\\ +(\|u\|_2+\|v\|_2)^2 &= \|u\|_2^2 + 2{\color{red}\|u\|_2\|v\|_2} + \|v\|_2^2 +\end{aligned}$} +\end{itemize}} +\end{column} +\end{columns} +\end{frame} |