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-rw-r--r--vorlesungen/slides/2/Makefile.inc1
-rw-r--r--vorlesungen/slides/2/chapter.tex1
-rw-r--r--vorlesungen/slides/2/polarformel.tex113
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