From ac3215e746f039d1254728deec096f59a0488449 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Christoffer=20M=C3=BCller=20Madsen?=
 <christoffer@guava.space>
Date: Fri, 9 Jun 2017 15:00:06 +0200
Subject: [PATCH] beviser til emne 3 og 4 done

---
 beviser.lyx | 1303 +++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 1303 insertions(+)

diff --git a/beviser.lyx b/beviser.lyx
index ca49d6b..cf0d2bd 100644
--- a/beviser.lyx
+++ b/beviser.lyx
@@ -946,5 +946,1308 @@ A & =A\cdot{\rm I}_{n}\\
 Beviset er nu afsluttet.
 \end_layout
 
+\begin_layout Standard
+\begin_inset Newpage newpage
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Vektorrum, underrum og dimension
+\end_layout
+
+\begin_layout Subsection
+Definition 5.1 (Vektorrum)
+\end_layout
+
+\begin_layout Standard
+En mængde 
+\begin_inset Formula $V$
+\end_inset
+
+samt to afbildninger 
+\begin_inset Formula $\mathcal{A}$
+\end_inset
+
+ og 
+\begin_inset Formula $\mathcal{S}$
+\end_inset
+
+, der opfylder identiteterne
+\end_layout
+
+\begin_layout Enumerate
+Den 
+\emph on
+kommutative
+\emph default
+ lov
+\end_layout
+
+\begin_layout Enumerate
+Den 
+\emph on
+associative
+\emph default
+ lov
+\end_layout
+
+\begin_layout Enumerate
+Eksistens af 
+\emph on
+neutralelement
+\end_layout
+
+\begin_layout Enumerate
+Eksistens af 
+\emph on
+inverse
+\emph default
+ elementer
+\end_layout
+
+\begin_layout Enumerate
+Distributiv lov 1
+\end_layout
+
+\begin_layout Enumerate
+Distributiv lov 2
+\end_layout
+
+\begin_layout Enumerate
+Associativitet for skalarmultiplikation (
+\begin_inset Formula $\alpha\cdot(\beta\cdot\boldsymbol{u})=(\alpha\beta)\cdot\boldsymbol{u}$
+\end_inset
+
+)
+\end_layout
+
+\begin_layout Subsection
+Definition 5.7 (Underrum)
+\end_layout
+
+\begin_layout Standard
+Et underrum af et vektorrum 
+\begin_inset Formula $V$
+\end_inset
+
+ er 
+\emph on
+en delmængde 
+\emph default
+
+\begin_inset Formula $S$
+\end_inset
+
+ af vektorrummet 
+\begin_inset Formula $V$
+\end_inset
+
+, der opfylder følgende betingelser:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\boldsymbol{0}\in S$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\forall\boldsymbol{u},\boldsymbol{v}\in S:\quad\boldsymbol{u}+\boldsymbol{v}\in S$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\forall\boldsymbol{u}\in S,\:\alpha\in\mathbb{F}:\quad\alpha\cdot\boldsymbol{u}\in S$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Punkt 2 og 3 kaldes 
+\emph on
+stabilitet overfor addition
+\emph default
+ hhv.
+ 
+\emph on
+multiplikation
+\emph default
+ 
+\end_layout
+
+\begin_layout Subsection
+Definition 5.9 (Linearkombination)
+\end_layout
+
+\begin_layout Standard
+En vektor 
+\begin_inset Formula $\boldsymbol{v}$
+\end_inset
+
+ i vektorrummet 
+\begin_inset Formula $V$
+\end_inset
+
+ kaldes en 
+\emph on
+linearkombination
+\emph default
+ af 
+\begin_inset Formula $\boldsymbol{v}_{1},\boldsymbol{v}_{2},\dots,\boldsymbol{v}_{n}$
+\end_inset
+
+, hvis der eksisterer skalarer 
+\begin_inset Formula $\alpha_{1},\alpha_{2},\dots,\alpha_{n}\in\mathbb{F}$
+\end_inset
+
+, så
+\begin_inset Formula 
+\[
+\boldsymbol{v}=\alpha_{1}\cdot\boldsymbol{v}_{1}+\alpha_{2}\cdot\boldsymbol{v}_{2}+\cdots+\alpha_{n}\cdot\boldsymbol{v}_{n}
+\]
+
+\end_inset
+
+og 
+\begin_inset Formula $\boldsymbol{v}\in V$
+\end_inset
+
+.
+ Dette kan oplagt også noteres som 
+\begin_inset Formula $\boldsymbol{v=\sum_{i=1}^{n}}\alpha_{i}\cdot\boldsymbol{v}_{i}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subsection
+Definition 5.11 (Span)
+\end_layout
+
+\begin_layout Standard
+Mængden af alle linearkombination af 
+\begin_inset Formula $\boldsymbol{v}_{1},\boldsymbol{v}_{2},\dots,\boldsymbol{v}_{n}$
+\end_inset
+
+ kaldes for 
+\emph on
+spannet
+\emph default
+ af elementerne 
+\begin_inset Formula $\boldsymbol{v}_{1},\boldsymbol{v}_{2},\dots,\boldsymbol{v}_{n}$
+\end_inset
+
+ og betegnes med
+\begin_inset Formula 
+\[
+{\rm Span}(\boldsymbol{v}_{1},\boldsymbol{v}_{2},\dots,\boldsymbol{v}_{n})
+\]
+
+\end_inset
+
+Se også Lemma 5.12 for et interessant resultat.
+\end_layout
+
+\begin_layout Subsection
+Definition 5.14 (Dimension)
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $V$
+\end_inset
+
+er et 
+\begin_inset Formula $\mathbb{F}$
+\end_inset
+
+-vektorrum.
+ Vi definerer da:
+\end_layout
+
+\begin_layout Itemize
+Hvis 
+\begin_inset Formula $V=\{\boldsymbol{0}\}$
+\end_inset
+
+ har 
+\begin_inset Formula $V$
+\end_inset
+
+ dimension 
+\begin_inset Formula $0$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+Hvis 
+\begin_inset Formula $V$
+\end_inset
+
+ er forskellig fra 
+\begin_inset Formula $\{\boldsymbol{0}\}$
+\end_inset
+
+ og kan udspændes af 
+\begin_inset Formula $n$
+\end_inset
+
+elementer, men ikke af færre end 
+\begin_inset Formula $n$
+\end_inset
+
+ elementer, så siger vi, at dimensionen af 
+\begin_inset Formula $V$
+\end_inset
+
+er lig 
+\begin_inset Formula $n$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+Hivs 
+\begin_inset Formula $V$
+\end_inset
+
+ ikke kan udspændes af en endelig mængde, så siges 
+\begin_inset Formula $V$
+\end_inset
+
+at have uendelig dimension.
+\end_layout
+
+\begin_layout Standard
+Dimensionen af 
+\begin_inset Formula $V$
+\end_inset
+
+betegnes med 
+\begin_inset Formula $\dim(V)$
+\end_inset
+
+.
+ Uendelig dimension beskrives ved 
+\begin_inset Formula $\dim(V)=\infty$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subsection
+Formel 5.25 (Matrixprodukter og linearkombinationer)
+\end_layout
+
+\begin_layout Standard
+\begin_inset Quotes ald
+\end_inset
+
+Ekstrem vigtig sammenhæng
+\begin_inset Quotes ard
+\end_inset
+
+ mellem linearkombinationer og matrixproduktet.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $A=(a_{ij})\in{\rm Mat}_{m,n}(\mathbb{F})$
+\end_inset
+
+.
+ Da vil
+\begin_inset Formula 
+\begin{align*}
+A\cdot\begin{pmatrix}\alpha_{1}\\
+\alpha_{2}\\
+\vdots\\
+\alpha_{n}
+\end{pmatrix} & =\begin{pmatrix}a_{11}\alpha_{1}+a_{12}\alpha_{2}+\cdots+a_{1n}\alpha_{n}\\
+a_{21}\alpha_{1}+a_{22}\alpha_{2}+\cdots+a_{2n}\alpha_{n}\\
+\vdots\\
+a_{m1}\alpha_{1}+a_{m2}\alpha_{2}+\cdots+a_{mn}\alpha_{n}
+\end{pmatrix}\\
+ & =\alpha_{1}\begin{pmatrix}a_{11}\\
+a_{21}\\
+\vdots\\
+a_{m1}
+\end{pmatrix}+\alpha_{2}\begin{pmatrix}a_{12}\\
+a_{22}\\
+\vdots\\
+a_{m2}
+\end{pmatrix}+\cdots+\alpha_{n}\begin{pmatrix}a_{1n}\\
+a_{2n}\\
+\vdots\\
+a_{mn}
+\end{pmatrix}\\
+ & =\alpha_{1}\cdot\boldsymbol{a}_{1}+\alpha_{2}\cdot\boldsymbol{a}_{2}+\cdots+\alpha_{n}\cdot\boldsymbol{a}_{n}
+\end{align*}
+
+\end_inset
+
+Dette giver også at
+\begin_inset Formula 
+\[
+{\rm Span}(\boldsymbol{a_{1},a_{2},\dots,a_{n}})=\{A\cdot\boldsymbol{v}\mid\boldsymbol{v}\in\mathbb{F}^{n}\}.
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Proposition 5.17 (Vektorrum)
+\end_layout
+
+\begin_layout Standard
+Dimensionen af 
+\begin_inset Formula $\mathbb{F}^{n}$
+\end_inset
+
+ er 
+\begin_inset Formula $n$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Bevis
+\end_layout
+
+\begin_layout Standard
+Det er allerede bemærket at 
+\begin_inset Formula $\mathbb{F}^{n}$
+\end_inset
+
+ kan udspændes af 
+\begin_inset Formula $n$
+\end_inset
+
+elementer 
+\begin_inset Formula $(\boldsymbol{e}_{1},\boldsymbol{e}_{2},\dots,\boldsymbol{e}_{n})$
+\end_inset
+
+.
+ Det ønskes derfor kun vist at 
+\begin_inset Formula $\mathbb{F}^{n}$
+\end_inset
+
+ ikke kan udspændes af 
+\emph on
+færre end 
+\begin_inset Formula $n$
+\end_inset
+
+ elementer.
+\end_layout
+
+\begin_layout Standard
+Det antages at 
+\begin_inset Formula $\mathbb{F}^{n}$
+\end_inset
+
+ udspændes af 
+\begin_inset Formula $m$
+\end_inset
+
+ elementer 
+\begin_inset Formula $\boldsymbol{a_{1},a_{2},\dots,a_{m}}\in\mathbb{F}^{n}$
+\end_inset
+
+, dvs.
+\begin_inset Formula 
+\[
+{\rm Span}(\boldsymbol{a_{1},a_{2},\dots,a_{m}})=\mathbb{F}^{n}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Lad nu 
+\begin_inset Formula $A\in{\rm Mat}_{n,m}(\mathbb{F})$
+\end_inset
+
+ beskrive matricen, hvis 
+\begin_inset Formula $i$
+\end_inset
+
+'te søjle er givet ved 
+\begin_inset Formula $\boldsymbol{a}_{i}$
+\end_inset
+
+.
+ Da vil ligningssystemet 
+\begin_inset Formula $A\cdot\boldsymbol{x}=\boldsymbol{b}$
+\end_inset
+
+ have en løsning for ethvert 
+\begin_inset Formula $\boldsymbol{b}\in\mathbb{F}^{n}$
+\end_inset
+
+, grundet Lemma 5.16 samholdt med ovenstående.
+\end_layout
+
+\begin_layout Standard
+Vi lader nu 
+\begin_inset Formula $\boldsymbol{e}_{i}\in\mathbb{F}^{n}$
+\end_inset
+
+ betegne den 
+\begin_inset Formula $i$
+\end_inset
+
+'te søjle i 
+\begin_inset Formula $I_{n}$
+\end_inset
+
+ og 
+\begin_inset Formula $\boldsymbol{b}_{i}\in\mathbb{F}^{m}$
+\end_inset
+
+ for 
+\begin_inset Formula $i=[1,n]$
+\end_inset
+
+ være en løsning til ligningssystemet 
+\begin_inset Formula $A\cdot\boldsymbol{x}=\boldsymbol{e}_{i}$
+\end_inset
+
+.
+ Lad endvidere 
+\begin_inset Formula $B\in{\rm Mat}_{m,n}(\mathbb{F})$
+\end_inset
+
+ betegne matricen hvis 
+\begin_inset Formula $i$
+\end_inset
+
+'te søjle er 
+\begin_inset Formula $\boldsymbol{b}_{i}$
+\end_inset
+
+ for 
+\begin_inset Formula $i=[1,n]$
+\end_inset
+
+.
+ Da gælder der at
+\begin_inset Formula 
+\begin{align*}
+A\cdot B & =\left(A\cdot\boldsymbol{b}_{1}\mid A\cdot\boldsymbol{b}_{2}\mid\cdots\mid A\cdot\boldsymbol{b}_{n}\right)\\
+ & =\left(\boldsymbol{e}_{1}\mid\boldsymbol{e}_{2}\mid\cdots\mid\boldsymbol{e}_{n}\right)\\
+ & =I_{n}.
+\end{align*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Det homogene ligningssystem 
+\begin_inset Formula $B\cdot\boldsymbol{x}=\boldsymbol{0}$
+\end_inset
+
+ har kun har nulvektoren som løsning, da 
+\begin_inset Formula $B$
+\end_inset
+
+ er invertibel samholdt med Lemma 4.4 og 4.6.
+ Proposition 1.14 implicerer da at 
+\begin_inset Formula $m\geq n$
+\end_inset
+
+, hvilket fuldender beviset.
+\end_layout
+
+\begin_layout Subsubsection
+Hjælpesætning - Lemma 5.16
+\end_layout
+
+\begin_layout Standard
+Et lineært ligningssystem 
+\begin_inset Formula $A\cdot\boldsymbol{x}=\boldsymbol{b}$
+\end_inset
+
+ har en løsning hvis og kun hvis 
+\begin_inset Formula $\boldsymbol{b}\in R(A)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Bevis
+\end_layout
+
+\begin_layout Standard
+Det at 
+\begin_inset Formula $\boldsymbol{b}$
+\end_inset
+
+ er i søjlerummet for 
+\begin_inset Formula $A$
+\end_inset
+
+ betyder at 
+\begin_inset Formula $\boldsymbol{b}$
+\end_inset
+
+ er på formen 
+\begin_inset Formula $A\cdot\boldsymbol{v}$
+\end_inset
+
+ for et passende 
+\begin_inset Formula $\boldsymbol{v}\in\mathbb{F}^{n}$
+\end_inset
+
+.
+ Dette er netop betingelsen for at 
+\begin_inset Formula $A\cdot\boldsymbol{x}=\boldsymbol{b}$
+\end_inset
+
+ har en løsning.
+\end_layout
+
+\begin_layout Subsection
+Lemma 5.12 (Hvis tid?)
+\end_layout
+
+\begin_layout Standard
+Mængden 
+\begin_inset Formula ${\rm Span}(\boldsymbol{v}_{1},\boldsymbol{v}_{2},\dots,\boldsymbol{v}_{n})$
+\end_inset
+
+ udgør et underrum i 
+\begin_inset Formula $V$
+\end_inset
+
+ indeholdende alle elementerne 
+\begin_inset Formula $\boldsymbol{v}_{1},\boldsymbol{v}_{2},\dots,\boldsymbol{v}_{n}$
+\end_inset
+
+.
+ Ethvert underrum af 
+\begin_inset Formula $V$
+\end_inset
+
+ indeholdende 
+\begin_inset Formula $\boldsymbol{v}_{1},\boldsymbol{v}_{2},\dots,\boldsymbol{v}_{n}$
+\end_inset
+
+ vil indeholde 
+\begin_inset Formula ${\rm Span}(\boldsymbol{v}_{1},\boldsymbol{v}_{2},\dots,\boldsymbol{v}_{n})$
+\end_inset
+
+ som delmængde.
+\end_layout
+
+\begin_layout Paragraph
+Meget uformel begrundelse
+\end_layout
+
+\begin_layout Standard
+Dette er på grund af spannets og underrummets definition.
+ Underrum er stabile overfor netop addition og skalarmultiplikation, hvilket
+ er det linearkombinationer benytter sig af.
+\end_layout
+
+\begin_layout Subsection
+Noter
+\end_layout
+
+\begin_layout Standard
+Skriv måske mere til Lemma 5.12
+\end_layout
+
+\begin_layout Standard
+\begin_inset Newpage newpage
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Basis for vektorrum
+\end_layout
+
+\begin_layout Subsection
+Definition 7.1 (
+\emph on
+Udspænde
+\emph default
+, 
+\emph on
+lineær
+\emph default
+ 
+\emph on
+uafhængighed
+\emph default
+ og 
+\emph on
+basis
+\emph default
+)
+\end_layout
+
+\begin_layout Standard
+For en samling af elementer 
+\begin_inset Formula $\mathcal{V}=(\boldsymbol{v}_{1},\boldsymbol{v}_{2},\dots,\boldsymbol{v}_{n})$
+\end_inset
+
+ i et 
+\begin_inset Formula $\mathbb{F}$
+\end_inset
+
+-vektorrum 
+\begin_inset Formula $V$
+\end_inset
+
+defineres:
+\end_layout
+
+\begin_layout Itemize
+Udspænding af 
+\begin_inset Formula $V$
+\end_inset
+
+: 
+\begin_inset Formula ${\rm Span}(\boldsymbol{v}_{1},\boldsymbol{v}_{2},\dots,\boldsymbol{v}_{n})=V$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+Lineær uafhængighed: 
+\begin_inset Formula $\begin{pmatrix}\alpha_{1}\\
+\alpha_{2}\\
+\vdots\\
+\alpha_{n}
+\end{pmatrix}\cdot\begin{pmatrix}\vdots & \vdots &  & \vdots\\
+\boldsymbol{v}_{1} & \boldsymbol{v}_{2} & \cdots & \boldsymbol{v}_{n}\\
+\vdots & \vdots &  & \vdots
+\end{pmatrix}=\boldsymbol{0}\iff\alpha_{i}=0\:{\rm for}\:i=1,2,\dots,n$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+Basis for 
+\begin_inset Formula $V$
+\end_inset
+
+ såfremt 
+\begin_inset Formula $\mathcal{V}$
+\end_inset
+
+ udspænder 
+\begin_inset Formula $V$
+\end_inset
+
+ samt er lineært uafhængig.
+\end_layout
+
+\begin_layout Subsection
+Lemma 7.2 (Relation af samlinger til afbildningsbegreber)
+\end_layout
+
+\begin_layout Standard
+For en samling af elementer 
+\begin_inset Formula $\mathcal{V}=(\boldsymbol{v}_{1},\boldsymbol{v}_{2},\dots,\boldsymbol{v}_{n})$
+\end_inset
+
+ i et 
+\begin_inset Formula $\mathbb{F}$
+\end_inset
+
+-vektorrum 
+\begin_inset Formula $V$
+\end_inset
+
+gælder det at
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\mathcal{V}$
+\end_inset
+
+udspænder 
+\begin_inset Formula $V$
+\end_inset
+
+ hvis og kun hvis 
+\begin_inset Formula $L_{\mathcal{V}}$
+\end_inset
+
+ er surjektiv
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\mathcal{V}$
+\end_inset
+
+ er lineært uafhængig hvis og kun hvis 
+\begin_inset Formula $L_{\mathcal{V}}$
+\end_inset
+
+ er injektiv
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\mathcal{V}$
+\end_inset
+
+ er en basis for 
+\begin_inset Formula $V$
+\end_inset
+
+ hvis og kun hvis 
+\begin_inset Formula $L_{\mathcal{V}}$
+\end_inset
+
+ er en isomorfi
+\end_layout
+
+\begin_layout Standard
+Dette giver endvidere at 
+\begin_inset Formula $\mathcal{V}$
+\end_inset
+
+ er en basis for 
+\begin_inset Formula $V$
+\end_inset
+
+ netop når ethvert element i 
+\begin_inset Formula $V$
+\end_inset
+
+ 
+\emph on
+på entydig vis
+\emph default
+ er en linearkombination af vektorerne i 
+\begin_inset Formula $\mathcal{V}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Paragraph
+Bevis
+\end_layout
+
+\begin_layout Standard
+(1): Billedet af 
+\begin_inset Formula $L_{\mathcal{V}}$
+\end_inset
+
+ er lig 
+\begin_inset Formula ${\rm Span}(\mathcal{V})$
+\end_inset
+
+.
+ 
+\begin_inset Formula $L_{\mathcal{V}}$
+\end_inset
+
+ er surjektiv hvis og kun hvis 
+\begin_inset Formula ${\rm Span}(\mathcal{V})=V$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+(2): 
+\begin_inset Formula $L_{\mathcal{V}}$
+\end_inset
+
+ er injektiv hvis og kun hvis at kernen for 
+\begin_inset Formula $L_{\mathcal{V}}$
+\end_inset
+
+ er lig 
+\begin_inset Formula $\{\boldsymbol{0}\}$
+\end_inset
+
+ jf.
+ Sætning 6.14.
+ Vektoren 
+\begin_inset Formula $(\alpha_{1},\alpha_{2},\dots,\alpha_{n})^{T}$
+\end_inset
+
+ er et element i kernen for 
+\begin_inset Formula $L_{\mathcal{V}}$
+\end_inset
+
+ hvis og kun hvis identiteten 
+\begin_inset Formula $\alpha_{1}\cdot\boldsymbol{v}_{1}+\alpha_{2}\cdot\boldsymbol{v}_{2}+\cdots+\alpha_{n}\cdot\boldsymbol{v}_{n}=0$
+\end_inset
+
+ er opfyldt.
+\end_layout
+
+\begin_layout Standard
+For at 
+\begin_inset Formula $L_{\mathcal{V}}$
+\end_inset
+
+ kan være injektiv må identiteten kun være opfyldt hvis alle 
+\begin_inset Formula $\alpha_{i}=0$
+\end_inset
+
+, hvilket netop er definitionen på, at 
+\begin_inset Formula $\mathcal{V}$
+\end_inset
+
+ er lineært uafhængig.
+\end_layout
+
+\begin_layout Standard
+(3): Følger af (1) og (2), da isomorfi kræver samtidig surjektivitet og
+ injektivitet ligesom basis kræver samtidig udspænding (surjektivitet) og
+ lineær uafhængighed (injektivitet).
+\end_layout
+
+\begin_layout Subsection
+Sætning 7.12 (
+\emph on
+Udtynding
+\emph default
+ og 
+\emph on
+udvidelse
+\emph default
+)
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $V$
+\end_inset
+
+er et vektorrum af endelig dimension 
+\begin_inset Formula $n>0$
+\end_inset
+
+ og 
+\begin_inset Formula $\mathcal{W}=(\boldsymbol{v}_{1},\boldsymbol{v}_{2},\dots,\boldsymbol{v}_{m})$
+\end_inset
+
+ er en samling af 
+\begin_inset Formula $m$
+\end_inset
+
+ elementer i vektorrummet 
+\begin_inset Formula $V$
+\end_inset
+
+.
+ Da er to ting mulige:
+\end_layout
+
+\begin_layout Enumerate
+Hvis 
+\begin_inset Formula $\mathcal{W}$
+\end_inset
+
+ udspænder 
+\begin_inset Formula $V$
+\end_inset
+
+, så er 
+\begin_inset Formula $n\leq m$
+\end_inset
+
+ og 
+\begin_inset Formula $\mathcal{W}$
+\end_inset
+
+ kan 
+\emph on
+udtyndes
+\emph default
+ til en basis for 
+\begin_inset Formula $V$
+\end_inset
+
+.
+ Det vil sige at nogle af vektorerne i 
+\begin_inset Formula $\mathcal{W}$
+\end_inset
+
+ kan 
+\emph on
+fjernes
+\emph default
+ for at gøre 
+\begin_inset Formula $\mathcal{W}$
+\end_inset
+
+ til en basis for 
+\begin_inset Formula $V$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Hvis 
+\begin_inset Formula $\mathcal{W}$
+\end_inset
+
+ er lineært uafhængig, så er 
+\begin_inset Formula $m\leq n$
+\end_inset
+
+ og 
+\begin_inset Formula $\mathcal{W}$
+\end_inset
+
+ kan 
+\emph on
+udvides
+\emph default
+ til en basis for 
+\begin_inset Formula $V$
+\end_inset
+
+.
+ Det vil sige at der kan 
+\emph on
+tilføjes
+\emph default
+ vektorerne fra 
+\begin_inset Formula $V$
+\end_inset
+
+ til 
+\begin_inset Formula $\mathcal{W}$
+\end_inset
+
+ for at gøre 
+\begin_inset Formula $\mathcal{W}$
+\end_inset
+
+ til en basis for 
+\begin_inset Formula $V$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Paragraph
+Bevis
+\end_layout
+
+\begin_layout Standard
+Beviset foregår i to dele.
+ Først vises udtynding og derefter udvidelse.
+\end_layout
+
+\begin_layout Subparagraph
+(1)
+\end_layout
+
+\begin_layout Standard
+Der argumenteres genmem induktion i 
+\begin_inset Formula $m>0$
+\end_inset
+
+.
+ Hvis 
+\begin_inset Formula $m=1$
+\end_inset
+
+, så vil 
+\begin_inset Formula $\mathcal{W}=(\boldsymbol{v}_{1})$
+\end_inset
+
+ pr.
+ antagelse udspænde 
+\begin_inset Formula $V,$
+\end_inset
+
+ og da 
+\begin_inset Formula $V$
+\end_inset
+
+ ikke er nulrummet (da 
+\begin_inset Formula $n>0$
+\end_inset
+
+) vil 
+\begin_inset Formula $\mathcal{W}$
+\end_inset
+
+ være lineært uafhængig (jf.
+ Eks 7.6(A)).
+ Derfor vil 
+\begin_inset Formula $\mathcal{W}$
+\end_inset
+
+ i dette tilfælde være en basis for 
+\begin_inset Formula $V$
+\end_inset
+
+.
+ Proposition 
+\begin_inset Formula $7.8$
+\end_inset
+
+ giver at 
+\begin_inset Formula $n$
+\end_inset
+
+ derved er lig 
+\begin_inset Formula $1$
+\end_inset
+
+.
+ Derfor kan det anvendes at 
+\begin_inset Formula $\mathcal{V}=\mathcal{W}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Det antages nu at 
+\begin_inset Formula $m>1$
+\end_inset
+
+ og at udsagnet er vist i tilfældet 
+\begin_inset Formula $m-1$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Hvis 
+\begin_inset Formula $\mathcal{W}$
+\end_inset
+
+ er lineært 
+\emph on
+uafhængig
+\emph default
+, så er 
+\begin_inset Formula $W$
+\end_inset
+
+ en basis og dermed er 
+\begin_inset Formula $n=m$
+\end_inset
+
+ ifølge Proposition 
+\begin_inset Formula $7.8$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Hvis 
+\begin_inset Formula $\mathcal{W}$
+\end_inset
+
+ er lineært 
+\emph on
+afhængig
+\emph default
+, så eksisterer der jf.
+ Lemma 7.7(1) et 
+\begin_inset Formula $i$
+\end_inset
+
+, 
+\begin_inset Formula $1\leq i\leq m$
+\end_inset
+
+, så kan en ny samling 
+\begin_inset Formula $\mathcal{W}^{\prime}$
+\end_inset
+
+ skabes ud fra 
+\begin_inset Formula $\mathcal{W}$
+\end_inset
+
+ ved fjernelse af vektoren 
+\begin_inset Formula $\boldsymbol{v}_{i}$
+\end_inset
+
+ ift.
+ 
+\begin_inset Formula $\mathcal{W}$
+\end_inset
+
+ således at 
+\begin_inset Formula $\mathcal{W}^{\prime}$
+\end_inset
+
+ udspænder 
+\begin_inset Formula $V$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Pr.
+ induktion kan det nye 
+\begin_inset Formula $\mathcal{W}^{\prime}$
+\end_inset
+
+ udtyndes til en basis for 
+\begin_inset Formula $V$
+\end_inset
+
+, og denne basis vil også være en udtynding af 
+\begin_inset Formula $\mathcal{W}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subparagraph
+(2)
+\end_layout
+
+\begin_layout Standard
+At 
+\begin_inset Formula $m\leq n$
+\end_inset
+
+ følger af Lemma 7.10.
+\end_layout
+
+\begin_layout Standard
+Der argumenteres ved induktion i tallet 
+\begin_inset Formula $n-m\geq0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Hvis 
+\begin_inset Formula $n-m=0$
+\end_inset
+
+, så er 
+\begin_inset Formula $\mathcal{W}$
+\end_inset
+
+ en basis for 
+\begin_inset Formula $V$
+\end_inset
+
+ jf.
+ Proposition 7.11.
+\end_layout
+
+\begin_layout Standard
+Det antages nu at 
+\begin_inset Formula $n-m>0$
+\end_inset
+
+ og at udsagnet er vist i tilfældet 
+\begin_inset Formula $(n-m)-1$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Da 
+\begin_inset Formula $m\neq n$
+\end_inset
+
+ er 
+\begin_inset Formula $\mathcal{W}$
+\end_inset
+
+ ikke en basis for 
+\begin_inset Formula $V$
+\end_inset
+
+ jf.
+ Proposition 
+\begin_inset Formula $7.8$
+\end_inset
+
+ (størrelse af basis skal være lig dimension af rum).
+ 
+\begin_inset Formula $\mathcal{W}$
+\end_inset
+
+ kan da ikke udspænde 
+\begin_inset Formula $V$
+\end_inset
+
+.
+ Det må derfor være muligt at vælge et element 
+\begin_inset Formula $\boldsymbol{v}^{\prime}$
+\end_inset
+
+ i 
+\begin_inset Formula $V$
+\end_inset
+
+ som ikke er indeholdet i 
+\begin_inset Formula ${\rm Span}(\mathcal{W})$
+\end_inset
+
+.
+ Pr.
+ Lemma 7.7(2) vil 
+\begin_inset Formula $\mathcal{W}^{\prime}=\mathcal{W}+\boldsymbol{v}^{\prime}$
+\end_inset
+
+ være lineært uafhængig.
+\end_layout
+
+\begin_layout Standard
+Pr.
+ induktion så kan 
+\begin_inset Formula $\mathcal{W}^{\prime}$
+\end_inset
+
+ nu udvides til en basis for 
+\begin_inset Formula $V$
+\end_inset
+
+.
+ En sådan udvidelse vil samtidig være en udvidelse af det oprindelige 
+\begin_inset Formula $\mathcal{W}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Noter
+\end_layout
+
+\begin_layout Standard
+Overvej at droppe Lemma 7.2 fra dispositionen.
+\end_layout
+
 \end_body
 \end_document