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authorAndreas Müller <andreas.mueller@ost.ch>2021-03-12 17:27:08 +0100
committerAndreas Müller <andreas.mueller@ost.ch>2021-03-12 17:27:08 +0100
commit972eba4c4cb38eb330bd053500e627085a3f4328 (patch)
treeaf8833cc0c292addd679a7557c3f903fb3fb5ebc /vorlesungen/slides/2
parentnew slides (diff)
downloadSeminarMatrizen-972eba4c4cb38eb330bd053500e627085a3f4328.tar.gz
SeminarMatrizen-972eba4c4cb38eb330bd053500e627085a3f4328.zip
add new slides
Diffstat (limited to 'vorlesungen/slides/2')
-rw-r--r--vorlesungen/slides/2/funktionenalgebra.tex6
-rw-r--r--vorlesungen/slides/2/skalarprodukt.tex27
2 files changed, 18 insertions, 15 deletions
diff --git a/vorlesungen/slides/2/funktionenalgebra.tex b/vorlesungen/slides/2/funktionenalgebra.tex
index e3339c3..9116be4 100644
--- a/vorlesungen/slides/2/funktionenalgebra.tex
+++ b/vorlesungen/slides/2/funktionenalgebra.tex
@@ -31,12 +31,12 @@ f(x)\cdot g(x)
\begin{align*}
\|f\cdot g\|_\infty
&=
-\sup_{x\in[0,1]} f(x)g(x)
+\sup_{x\in[0,1]} |f(x)g(x)|
\\
\uncover<4->{
&\le
-\sup_{x\in[0,1]}f(x)
-\sup_{y\in[0,1]}g(y)
+\sup_{x\in[0,1]}|f(x)|
+\sup_{y\in[0,1]}|g(y)|
}
\\
\uncover<5->{
diff --git a/vorlesungen/slides/2/skalarprodukt.tex b/vorlesungen/slides/2/skalarprodukt.tex
index 2a9784f..99d8a73 100644
--- a/vorlesungen/slides/2/skalarprodukt.tex
+++ b/vorlesungen/slides/2/skalarprodukt.tex
@@ -13,7 +13,7 @@
\begin{block}{Positiv definite, symmetrische Bilinearform}
$\langle \;\,,\;\rangle\colon V\times V\to \mathbb{R}$
\begin{itemize}
-\item
+\item<2->
Bilinear:
\begin{align*}
\langle \alpha u+\beta v,w\rangle
@@ -28,18 +28,19 @@ Bilinear:
+
\beta\langle u,w\rangle
\end{align*}
-\item
+\item<3->
Symmetrisch: $\langle u,v\rangle = \langle v,u\rangle$
-\item
+\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
+\item<6->
Sesquilinear:
\begin{align*}
\langle \alpha u+\beta v,w\rangle
@@ -54,40 +55,42 @@ Sesquilinear:
+
\beta\langle u,w\rangle
\end{align*}
-\item
+\item<7->
Hermitesch: $\langle u,v\rangle = \overline{\langle v,u\rangle}$
-\item
+\item<8->
$\langle x,x\rangle >0 \quad\forall x\ne 0$
\end{itemize}
-\end{block}
+\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{block}}
\end{column}
\begin{column}{0.78\textwidth}
+\uncover<10->{%
\begin{itemize}
-\item $\|v\|_2 = \sqrt{\langle v,v\rangle} > 0\quad\forall v\ne 0$
-\item $\| \lambda v \|_2
+\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
+\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{itemize}}
\end{column}
\end{columns}
\end{frame}