From 3222965c31602df51b3b80c3699b1fd82607361a Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Christoffer=20M=C3=BCller=20Madsen?=
 <christoffer@guava.space>
Date: Sat, 10 Jun 2017 12:23:23 +0200
Subject: [PATCH] halvvejs med emne 7

---
 beviser.lyx | 149 +++++++++++++++++++++++++++++++++++++++++++++++++++-
 1 file changed, 147 insertions(+), 2 deletions(-)

diff --git a/beviser.lyx b/beviser.lyx
index 1afc185..bbc85fb 100644
--- a/beviser.lyx
+++ b/beviser.lyx
@@ -3065,6 +3065,13 @@ Og derfor har vi:
 \end_inset
 
 
+\end_layout
+
+\begin_layout Standard
+\begin_inset Newpage newpage
+\end_inset
+
+
 \end_layout
 
 \begin_layout Section
@@ -3353,8 +3360,8 @@ Husk at hver indgang
 
  et tal og ikke en matrice.
  Ovenstående formel beskriver jf.
- prop 11.26 også determinaten af matricen, der kan fremkomme ved at udskifte
- den 
+ Proposition 11.26 også determinaten af matricen, der kan fremkomme ved at
+ udskifte den 
 \begin_inset Formula $j$
 \end_inset
 
@@ -3504,5 +3511,143 @@ Disse kan relateres til prop 11.30 og dermed bruges til at beskrive henholdsvis
  udvikling af række og søjle.
 \end_layout
 
+\begin_layout Section
+Indre produkt
+\end_layout
+
+\begin_layout Subsection
+Definition 9.1
+\end_layout
+
+\begin_layout Standard
+Afbildningen
+\begin_inset Formula 
+\[
+\left\langle \cdot,\cdot\right\rangle :V\times V\rightarrow\mathbb{K}
+\]
+
+\end_inset
+
+benævnes som det 
+\emph on
+indre produkt
+\emph default
+ hvis der for alle 
+\begin_inset Formula $\boldsymbol{u},\boldsymbol{v},\boldsymbol{w}\in V$
+\end_inset
+
+ og skalarer 
+\begin_inset Formula $\alpha,\beta\in\mathbb{K}$
+\end_inset
+
+ gælder at:
+\end_layout
+
+\begin_layout Enumerate
+Skalaren 
+\begin_inset Formula $\left\langle \boldsymbol{v},\boldsymbol{v}\right\rangle $
+\end_inset
+
+ er et reelt tal, der er større end eller lig med nul.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\left\langle \boldsymbol{v},\boldsymbol{v}\right\rangle =0\implies\boldsymbol{v}=\boldsymbol{0}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\left\langle \boldsymbol{v},\boldsymbol{w}\right\rangle =\overline{\left\langle \boldsymbol{w},\boldsymbol{v}\right\rangle }$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\left\langle \alpha\cdot\boldsymbol{u}+\beta\cdot\boldsymbol{v},\boldsymbol{w}\right\rangle =\alpha\cdot\left\langle \boldsymbol{u},\boldsymbol{w}\right\rangle +\beta\cdot\left\langle \boldsymbol{v},\boldsymbol{w}\right\rangle $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Bemærkning 9.4 (Naiv definition af komplekst skalarprodukt)
+\end_layout
+
+\begin_layout Subsection
+Definition 9.5 (Norm)
+\end_layout
+
+\begin_layout Standard
+
+\emph on
+Normen
+\emph default
+ af et element 
+\begin_inset Formula $\boldsymbol{v}$
+\end_inset
+
+ i et indre produkt rum 
+\begin_inset Formula $V$
+\end_inset
+
+ defineres som
+\begin_inset Formula 
+\[
+\left\Vert \boldsymbol{v}\right\Vert =\sqrt{\left\langle \boldsymbol{v},\boldsymbol{v}\right\rangle }\in\mathbb{R}_{\geq0}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Definition 9.7 (Ortogonalitet)
+\end_layout
+
+\begin_layout Standard
+To elementer 
+\begin_inset Formula $\boldsymbol{v}$
+\end_inset
+
+ og 
+\begin_inset Formula $\boldsymbol{w}$
+\end_inset
+
+ i et indre produkt rum kaldes 
+\emph on
+ortogonale
+\emph default
+ hvis
+\begin_inset Formula 
+\[
+\left\langle \boldsymbol{v},\boldsymbol{w}\right\rangle =0
+\]
+
+\end_inset
+
+dette skrives også som
+\begin_inset Formula 
+\[
+\boldsymbol{v}\perp\boldsymbol{w}.
+\]
+
+\end_inset
+
+Denne betingelse er oplagt symmetrisk (betingelse 3 (c) i definitionen af
+ indre produkt) således at
+\begin_inset Formula 
+\[
+\boldsymbol{v}\perp\boldsymbol{w}\iff\boldsymbol{w}\perp\boldsymbol{v}
+\]
+
+\end_inset
+
+
+\end_layout
+
 \end_body
 \end_document