diff options
Diffstat (limited to 'vorlesungen/slides/2')
| -rw-r--r-- | vorlesungen/slides/2/Makefile.inc | 1 | ||||
| -rw-r--r-- | vorlesungen/slides/2/chapter.tex | 1 | ||||
| -rw-r--r-- | vorlesungen/slides/2/polarformel.tex | 113 |
3 files changed, 115 insertions, 0 deletions
diff --git a/vorlesungen/slides/2/Makefile.inc b/vorlesungen/slides/2/Makefile.inc index 2eb3ce9..7c4dfb8 100644 --- a/vorlesungen/slides/2/Makefile.inc +++ b/vorlesungen/slides/2/Makefile.inc @@ -8,5 +8,6 @@ chapter2 = \ ../slides/2/norm.tex \ ../slides/2/skalarprodukt.tex \ ../slides/2/cauchyschwarz.tex \ + ../slides/2/polarformel.tex \ ../slides/2/chapter.tex diff --git a/vorlesungen/slides/2/chapter.tex b/vorlesungen/slides/2/chapter.tex index 4c86f39..7b968d1 100644 --- a/vorlesungen/slides/2/chapter.tex +++ b/vorlesungen/slides/2/chapter.tex @@ -6,3 +6,4 @@ \folie{2/norm.tex} \folie{2/skalarprodukt.tex} \folie{2/cauchyschwarz.tex} +\folie{2/polarformel.tex} diff --git a/vorlesungen/slides/2/polarformel.tex b/vorlesungen/slides/2/polarformel.tex new file mode 100644 index 0000000..ebdbf81 --- /dev/null +++ b/vorlesungen/slides/2/polarformel.tex @@ -0,0 +1,113 @@ +% +% polarformel.tex +% +% (c) 2021 Prod Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkcolor}{rgb}{0,0.6,0} +\def\yone{-2.1} +\def\ytwo{-3.55} +\def\ythree{-5.0} +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Polarformel} +\vspace{-5pt} +\begin{block}{Aufgabe} +$\langle x,y\rangle$ aus Werten von $\|\cdot\|_2$ rekonstruieren: + +\end{block} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\node at (0,0) {$ +\begin{aligned} +\uncover<2->{ +\|x+ty\|_2^2 +&= +\|x\|_2^2 ++t\langle x,y\rangle ++\overline{t}\langle y,x\rangle ++ \|y\|_2^2} +\\ +\uncover<3->{ +&= +\|x\|_2^2 ++t\langle x,y\rangle ++\overline{t\langle x,y\rangle} ++ \|y\|_2^2} +\\ +\uncover<4->{ +&= +\|x\|_2^2 ++2\operatorname{Re}(t\langle x,y\rangle) ++ \|y\|_2^2} +\end{aligned}$}; + +\uncover<5->{ + \draw[->] (-1,-0.9) -- (-3.3,{\yone+0.25}); + \node at (-3.5,\yone) {$ + \|x\pm y\|_2^2 + = + \|x\|_2^2 + \pm2\operatorname{Re}\langle x,y\rangle + + + \|y\|_2^2 + $}; +} + +\uncover<8->{ + \draw[->] (1,-0.9) -- (3.3,{\yone+0.25}); + \node at (3.5,\yone) {$ + \|x\pm iy\|_2^2 + = + \|x\|_2^2 + \pm2i\operatorname{Im}\langle x,y\rangle + + + \|y\|_2^2 + $}; +} + +\uncover<6->{ + \draw[->] (-3.5,{\yone-0.2}) -- (-3.5,{\ytwo+0.2}); + \node at (-3.5,\ytwo) {$\operatorname{Re}\langle x,y\rangle + = + \frac12\bigl( + \|x+y\|_2^2-\|x-y\|_2^2 + \bigr) + $}; +} + +\uncover<9->{ + \draw[->] (3.5,{\yone-0.2}) -- (3.5,{\ytwo+0.2}); + \node at (3.5,\ytwo) {$ + \operatorname{Im}\langle x,y\rangle + = + \frac1{2i}\bigl( + \|x+iy\|_2^2-\|x-iy\|_2^2 + \bigr) + $}; +} + +\uncover<7->{ + \draw[->] (-3.3,{\ytwo-0.25}) -- (-1.5,{\ythree+0.25}); + \node at (0,\ythree) {$ + \langle x,y\rangle + = + \frac12\bigl( + \|x+y\|_2^2-\|x-y\|_2^2 + \uncover<10->{ + + + \|x+iy\|_2^2-\|x-iy\|_2^2 + } + \bigr)$}; +} + +\uncover<10->{ + \draw[->] (3.3,{\ytwo-0.25}) -- (1.5,{\ythree+0.25}); +} + +\end{tikzpicture} +\end{center} +\end{frame} +\egroup |