From 44ca71b66f5c0a00807c126618e7cae36b5cc4f9 Mon Sep 17 00:00:00 2001
From: Pownie <Alexmunchhansen@gmail.com>
Date: Sat, 10 Jun 2017 11:11:16 +0200
Subject: [PATCH] Did stuff

---
 beviser.lyx | 1028 ++++++++++++++++++++++++++++++++++++++++++++++++++-
 1 file changed, 1027 insertions(+), 1 deletion(-)

diff --git a/beviser.lyx b/beviser.lyx
index 6261979..1afc185 100644
--- a/beviser.lyx
+++ b/beviser.lyx
@@ -2460,9 +2460,39 @@ Proposition 8.10(1) (Matrixrepræsentationer og koordinatvektorer)
 \end_layout
 
 \begin_layout Standard
+Lad 
+\begin_inset Formula $L:\:V\rightarrow W$
+\end_inset
+
+ betegne en lineær afbildning mellem 
+\begin_inset Formula $\mathbb{F}-vektorrum$
+\end_inset
+
+ 
+\begin_inset Formula $V$
+\end_inset
+
+ og 
+\begin_inset Formula $W$
+\end_inset
+
+ med baser hhv.
+ 
+\begin_inset Formula $\mathcal{V}$
+\end_inset
+
+ og 
+\begin_inset Formula $\mathcal{W}$
+\end_inset
+
+, så gælder:
+\end_layout
+
+\begin_layout Standard
+(1)
 \begin_inset Formula 
 \[
-\left[L(\boldsymbol{v})\right]_{\mathcal{W}}={}_{\mathcal{V}}\left[L\right]_{\mathcal{W}}\cdot\left[\boldsymbol{v}\right]_{\mathcal{V}}.
+\left[L(\boldsymbol{v})\right]_{\mathcal{W}}={}_{\mathcal{W}}\left[L\right]_{\mathcal{V}}\cdot\left[\boldsymbol{v}\right]_{\mathcal{V}}.
 \]
 
 \end_inset
@@ -2470,12 +2500,1008 @@ Proposition 8.10(1) (Matrixrepræsentationer og koordinatvektorer)
 
 \end_layout
 
+\begin_layout Standard
+(2) Hvis 
+\begin_inset Formula $A\,\in\text{Mat}_{m,n}(\mathbb{F})$
+\end_inset
+
+ opfylder relationen 
+\begin_inset Formula $\left[L(\boldsymbol{v})\right]_{\mathcal{W}}=A\cdot\left[\boldsymbol{v}\right]_{\mathcal{V}}$
+\end_inset
+
+ for alle 
+\begin_inset Formula $\boldsymbol{v}\,\in V$
+\end_inset
+
+, så er 
+\begin_inset Formula $A=_{\mathcal{W}}\left[L\right]_{\mathcal{V}}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Bevis:
+\end_layout
+
+\begin_layout Standard
+(1)
+\end_layout
+
+\begin_layout Standard
+Lad 
+\begin_inset Formula $\mathcal{V}$
+\end_inset
+
+ og 
+\begin_inset Formula $\mathcal{W}$
+\end_inset
+
+ være givet ved hhv.
+ 
+\begin_inset Formula $\mathcal{V}=(\boldsymbol{v_{1}},\boldsymbol{v_{2},\dots,v}_{n})$
+\end_inset
+
+ og 
+\begin_inset Formula $\mathcal{W}=(\boldsymbol{w_{1}},\boldsymbol{w_{2},\dots,w}_{n})$
+\end_inset
+
+.
+ Eftersom 
+\begin_inset Formula $\mathcal{V}$
+\end_inset
+
+ er en basis for 
+\begin_inset Formula $V$
+\end_inset
+
+ så kan alle elementer 
+\begin_inset Formula $\boldsymbol{v}\,\in V$
+\end_inset
+
+ beskrives som en linearkombination af basen 
+\begin_inset Formula $\mathcal{V}$
+\end_inset
+
+: 
+\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
+
+
+\end_layout
+
+\begin_layout Standard
+og specielt vil 
+\begin_inset Formula 
+\[
+L(\boldsymbol{v})=\alpha_{1}\cdot L(\boldsymbol{v})_{1}+\alpha_{2}\cdot L(\boldsymbol{v})_{2}+\cdots+\alpha_{n}\cdot L(\boldsymbol{v})_{n}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+grundet 
+\series bold
+Definition 6.1(b)
+\series default
+.
+ Ligeledes, grundet egenskaberne ved koordinatvektorer beskrevet i prop
+ 8.4, så kan følgende konkluderes:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+[L(\boldsymbol{v})]_{\mathcal{W}}=\alpha_{1}\cdot[L(\boldsymbol{v})_{1}]_{\mathcal{W}}+\alpha_{2}\cdot[L(\boldsymbol{v})_{2}]_{\mathcal{W}}+\cdots+\alpha_{n}\cdot[L(\boldsymbol{v})_{n}]_{\mathcal{W}}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Hvilket kan opskrives som et produkt jf.
+ med formel 5.25
+\begin_inset Formula 
+\[
+_{\mathcal{W}}\left[L\right]_{\mathcal{V}}\cdot\begin{pmatrix}\alpha_{1}\\
+\alpha_{2}\\
+\vdots\\
+\alpha_{n}
+\end{pmatrix}={}_{\mathcal{W}}\left[L\right]_{\mathcal{V}}\cdot\left[\boldsymbol{v}\right]_{\mathcal{V}}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Hvilket viser udsagn (1).
+\end_layout
+
+\begin_layout Standard
+(2)
+\end_layout
+
+\begin_layout Standard
+Antag at en matrix 
+\begin_inset Formula $A\in{\rm Mat}_{m,n}(\mathbb{F})$
+\end_inset
+
+ opfylder egenskaben beskrevet i (2), så vil der gælde:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+[L(\boldsymbol{v}_{i})]_{\mathcal{W}}=A\cdot[\boldsymbol{v}_{i}]_{\mathcal{V}}=A\cdot\boldsymbol{e}_{i}\,\,\,\,for\,\,ethvert\,i=1,2,\dots,n
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Hvori højresiden er må være lig den i'te søjle i 
+\begin_inset Formula $A$
+\end_inset
+
+, men venstresiden er lig den i'te søjle i 
+\begin_inset Formula $_{\mathcal{W}}\left[L\right]_{\mathcal{V}}$
+\end_inset
+
+ og altså må de to være ens, som påstået i udsagn (2).
+\end_layout
+
 \begin_layout Subsection
 Lemma 8.19
 \end_layout
 
 \begin_layout Standard
+Lad 
+\begin_inset Formula $L:\:V\rightarrow W$
+\end_inset
 
+ betegne en lineær afbildning, og lad 
+\begin_inset Formula $\mathcal{V}$
+\end_inset
+
+ og 
+\begin_inset Formula $\mathcal{W}$
+\end_inset
+
+ betegne baser for hhv.
+ 
+\begin_inset Formula $V$
+\end_inset
+
+ og 
+\begin_inset Formula $W$
+\end_inset
+
+.
+ Så gælder:
+\end_layout
+
+\begin_layout Standard
+(1) Et element 
+\begin_inset Formula $\boldsymbol{v}\in V$
+\end_inset
+
+ tilhører kernen ker(
+\begin_inset Formula $L$
+\end_inset
+
+) for 
+\begin_inset Formula $L$
+\end_inset
+
+ hvis og kun hvis den tilsvarende koordinatvektor 
+\begin_inset Formula $[\boldsymbol{v}]_{\mathcal{V}}$
+\end_inset
+
+ er et element i nulrummet 
+\begin_inset Formula $N(_{\mathcal{W}}\left[L\right]_{\mathcal{V}})$
+\end_inset
+
+ for matrixrepræsentationen 
+\begin_inset Formula $_{\mathcal{W}}\left[L\right]_{\mathcal{V}}$
+\end_inset
+
+.
+ 
+\end_layout
+
+\begin_layout Standard
+(2) Et element 
+\begin_inset Formula $\boldsymbol{w\in}W$
+\end_inset
+
+ tilhører billedet af 
+\begin_inset Formula $L$
+\end_inset
+
+ hvis og kun hvis den tilsvarende koordinatvektor 
+\begin_inset Formula $[\boldsymbol{w}]_{\mathcal{W}}$
+\end_inset
+
+ er et element i søjlerummet 
+\begin_inset Formula $R(_{\mathcal{W}}\left[L\right]_{\mathcal{V}})$
+\end_inset
+
+ til matrixrepræsentation 
+\begin_inset Formula $_{\mathcal{W}}\left[L\right]_{\mathcal{V}}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Bevis
+\end_layout
+
+\begin_layout Standard
+(1)
+\end_layout
+
+\begin_layout Standard
+Idet 
+\begin_inset Formula $L_{\mathcal{W}}$
+\end_inset
+
+ er en isomorfi (der findes en invers funktion, matrixrepræsentationens
+ inverse), så er 
+\begin_inset Formula $\boldsymbol{v}\in V$
+\end_inset
+
+ et element i ker(
+\begin_inset Formula $L$
+\end_inset
+
+) hvis og kun hvis
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+L_{\mathcal{W}}^{-1}(L(\boldsymbol{v}))=0
+\]
+
+\end_inset
+
+Venstresiden af dette er dog lig 
+\begin_inset Formula $[L(\boldsymbol{v})]_{\mathcal{W}}$
+\end_inset
+
+ hvilket kan skrives som:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+[L(\boldsymbol{v})]_{\mathcal{W}}={}_{\mathcal{W}}\left[L\right]_{\mathcal{V}}\cdot\left[\boldsymbol{v}\right]_{\mathcal{V}}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+hvoraf det er oplagt at koordinatvektoren 
+\begin_inset Formula $[\boldsymbol{v}]_{\mathcal{V}}$
+\end_inset
+
+ skal være i nulrummet.
+\end_layout
+
+\begin_layout Standard
+(2)
+\end_layout
+
+\begin_layout Standard
+Lad nu 
+\begin_inset Formula $\boldsymbol{w}\in W$
+\end_inset
+
+.
+ Hvis 
+\begin_inset Formula $\boldsymbol{w}$
+\end_inset
+
+ er i billedet 
+\begin_inset Formula $L(V)$
+\end_inset
+
+ så eksisterer der et 
+\begin_inset Formula $\boldsymbol{v}\in V$
+\end_inset
+
+, således at 
+\begin_inset Formula $\boldsymbol{w}=L(\boldsymbol{v}).$
+\end_inset
+
+ Dette leder til følgende sammenhæng:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+[\boldsymbol{w}]_{\mathcal{W}}=[L(\boldsymbol{v})]_{\mathcal{W}}={}_{\mathcal{W}}\left[L\right]_{\mathcal{V}}\cdot\left[\boldsymbol{v}\right]_{\mathcal{V}}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Hvilket betyder at 
+\begin_inset Formula $[\boldsymbol{w}]_{\mathcal{W}}$
+\end_inset
+
+ må være et element i søjlerummet til 
+\begin_inset Formula $_{\mathcal{W}}\left[L\right]_{\mathcal{V}}$
+\end_inset
+
+.
+ Hvis omvendt 
+\begin_inset Formula $[\boldsymbol{w}]_{\mathcal{W}}$
+\end_inset
+
+ er et element i søjlerummet til 
+\begin_inset Formula $_{\mathcal{W}}\left[L\right]_{\mathcal{V}}$
+\end_inset
+
+, så må der findes en vektor 
+\begin_inset Formula $\boldsymbol{a}\in\mathbb{F}^{n}$
+\end_inset
+
+ hvor 
+\begin_inset Formula $n$
+\end_inset
+
+ beskriver dimensionen af 
+\begin_inset Formula $V$
+\end_inset
+
+, så:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+[\boldsymbol{w}]_{\mathcal{W}}={}_{\mathcal{W}}\left[L\right]_{\mathcal{V}}\cdot\boldsymbol{a}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Hvis 
+\begin_inset Formula $\boldsymbol{v}=L_{\mathcal{V}}(\boldsymbol{a})$
+\end_inset
+
+ så:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{align*}
+[L(\boldsymbol{v})]_{\mathcal{W}} & =_{\mathcal{W}}\left[L\right]_{\mathcal{V}}\cdot\left[\boldsymbol{v}\right]_{\mathcal{V}}\\
+ & =_{\mathcal{W}}\left[L\right]_{\mathcal{V}}\cdot\boldsymbol{a}\\
+ & =[\boldsymbol{w}]_{\mathcal{W}}
+\end{align*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Idet 
+\begin_inset Formula $L_{\mathcal{W}}$
+\end_inset
+
+ er en isomorfi, så følger det at 
+\begin_inset Formula $\boldsymbol{w}=L(\boldsymbol{v})$
+\end_inset
+
+ og 
+\begin_inset Formula $\boldsymbol{w}$
+\end_inset
+
+ er derfor et element i billedet af 
+\begin_inset Formula $L$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subsection
+Lemma 8.20
+\end_layout
+
+\begin_layout Standard
+Lad 
+\begin_inset Formula $L\,:\,V\rightarrow W$
+\end_inset
+
+ betegne en lineær afbildning og lad 
+\begin_inset Formula $\mathcal{V}$
+\end_inset
+
+ og 
+\begin_inset Formula $\mathcal{W}$
+\end_inset
+
+ betegne baser for hhv.
+ 
+\begin_inset Formula $V$
+\end_inset
+
+ og 
+\begin_inset Formula $W$
+\end_inset
+
+.
+ Lad 
+\begin_inset Formula $r$
+\end_inset
+
+ betegne rangen af matrixrepræsentationen 
+\begin_inset Formula $_{\mathcal{W}}\left[L\right]_{\mathcal{V}}$
+\end_inset
+
+.
+ Så gælder:
+\end_layout
+
+\begin_layout Standard
+(1) Billedet af 
+\begin_inset Formula $L_{\mathcal{V}}(N({}_{\mathcal{W}}\left[L\right]_{\mathcal{V}}))$
+\end_inset
+
+ af nulrummet til 
+\begin_inset Formula $_{\mathcal{W}}\left[L\right]_{\mathcal{V}}$
+\end_inset
+
+ under isormorfien 
+\begin_inset Formula $L_{\mathcal{V}}$
+\end_inset
+
+ er lig kernen ker
+\begin_inset Formula $(L)$
+\end_inset
+
+.
+ Specielt inducerer 
+\begin_inset Formula $L_{\mathcal{V}}$
+\end_inset
+
+ en isomorfi:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{align*}
+N(_{\mathcal{W}}\left[L\right]_{\mathcal{V}}) & \rightarrow ker\,L\\
+ & \boldsymbol{a\mapsto}L_{\mathcal{V}}(\boldsymbol{a})
+\end{align*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Og vi har derfor: dim(ker(
+\begin_inset Formula $L$
+\end_inset
+
+))
+\begin_inset Formula $=\text{dim}(V)-r$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+(2) Billedet 
+\begin_inset Formula $L_{\mathcal{W}}(R(_{W}[L]_{\mathcal{V}}))$
+\end_inset
+
+ af søjlerummet til 
+\begin_inset Formula $_{\mathcal{W}}[L]_{\mathcal{V}}$
+\end_inset
+
+ under 
+\begin_inset Formula $L_{\mathcal{W}}$
+\end_inset
+
+ er lig billedet 
+\begin_inset Formula $L(V)$
+\end_inset
+
+.
+ Specielt inducerer 
+\begin_inset Formula $L_{\mathcal{W}}$
+\end_inset
+
+ en isomorfi
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{align*}
+R(_{\mathcal{W}}[L]_{\mathcal{V}}) & \rightarrow L(V),\\
+ & \boldsymbol{b}\mapsto L_{\mathcal{W}}(\boldsymbol{b})
+\end{align*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Og derfor har vi: 
+\begin_inset Formula $\text{dim}(L(V))=r$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Determinanter
+\end_layout
+
+\begin_layout Subsection
+Noget med definitionen på en determinant
+\end_layout
+
+\begin_layout Subsection
+Sætning 11.18
+\end_layout
+
+\begin_layout Standard
+Lad 
+\begin_inset Formula $A,\,B\in\text{Mat}_{n}(\mathbb{F})$
+\end_inset
+
+, så er 
+\begin_inset Formula 
+\[
+\text{Det}(A\cdot B)=\text{Det}(A)\cdot\text{Det}(B)
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Bevis
+\end_layout
+
+\begin_layout Standard
+Antag at 
+\begin_inset Formula $A$
+\end_inset
+
+ er singulær, altså den har ingen invers.
+ Der påstås at dette betyder at 
+\begin_inset Formula $A\cdot B$
+\end_inset
+
+ er singulær, da 
+\begin_inset Formula $B\cdot(A\cdot B)^{-1}$
+\end_inset
+
+ ellers ville være en invers til 
+\begin_inset Formula $A$
+\end_inset
+
+, eftersom
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+A\cdot(B\cdot(A\cdot B)^{-1})=(A\cdot B)\cdot(A\cdot B)^{-1}={\rm {\rm I}}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Hvilket er umuligt, da 
+\begin_inset Formula $A$
+\end_inset
+
+ per antagelse er singulær.
+ Eftersom 
+\begin_inset Formula $A$
+\end_inset
+
+ er singulær, må 
+\begin_inset Formula $\text{Det}(A)=0$
+\end_inset
+
+, da dette, jf.
+ prop 11.17, betyder at 
+\begin_inset Formula $A$
+\end_inset
+
+ ikke er invertibel.
+ Derfor gælder følgende:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+\text{Det}(A\cdot B)=\text{Det}(A)=\boldsymbol{0}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Hvilket opfylder det oprindelige, da alt ganget med 0, vil give 0 og det
+ derfor ikke gør nogen forskel, hvad B er.
+ Desuden er produktet af 
+\begin_inset Formula $A\cdot B$
+\end_inset
+
+ også en singulær kvadratisk matrice og dermed er 
+\begin_inset Formula $\text{Det}(A\cdot B)=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Antag så at 
+\begin_inset Formula $A$
+\end_inset
+
+ er invertibel og dermed rækkeækvivalent med identitetsmatricen.
+ Dette betyder at den opdelte matrix 
+\begin_inset Formula $(A\,\vline\,A\cdot B)$
+\end_inset
+
+ er rækkeækvivalent med 
+\begin_inset Formula $({\rm I}\,\vline\,C)$
+\end_inset
+
+ jf.
+ prop 4.6, for en passende matric 
+\begin_inset Formula $C$
+\end_inset
+
+, i dette tilfælde noget der er rækkeækvivalent med 
+\begin_inset Formula $A\cdot B$
+\end_inset
+
+.
+ 
+\end_layout
+
+\begin_layout Standard
+Dette resultat betyder, at der jf.
+ Lemma 11.16 eksisterer en skalar 
+\begin_inset Formula $\alpha\in\mathbb{F}$
+\end_inset
+
+, så
+\begin_inset Formula 
+\[
+\text{Det}(A)=\alpha\cdot\text{Det}({\rm I})
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+og dermed også
+\begin_inset Formula 
+\[
+\text{Det}(A\cdot B)=\alpha\cdot\text{Det}(C)
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Men jf.
+ prop 11.17, så implicerer ovenstående at 
+\begin_inset Formula $\text{Det}(A)=\alpha$
+\end_inset
+
+ og hvis dette indsættes i sidstnævnte formel, så opnås følgende:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+\text{Det}(A\cdot B)=\text{Det}(A)\cdot\text{Det}(C)
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Hvori C er lig
+\begin_inset Formula 
+\[
+C=A^{-1}\cdot(A\cdot B)=B
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+jf.
+ prop 4.12, der siger at da 
+\begin_inset Formula $A\cdot B$
+\end_inset
+
+ er rækkeækvivalent med 
+\begin_inset Formula $C$
+\end_inset
+
+, så gælder ovenstående formel.
+\end_layout
+
+\begin_layout Subsection
+Proposition 11.30
+\end_layout
+
+\begin_layout Standard
+Lad 
+\begin_inset Formula $A\in\text{Mat}_{n}(\mathbb{F})$
+\end_inset
+
+.
+ Så er
+\begin_inset Formula 
+\[
+\text{adj}(A)\cdot A=\text{Det}(A)\cdot{\rm I}=A\cdot\text{adj}(A)
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+hvori 
+\begin_inset Formula ${\rm I}\in\text{Mat}_{n}(\mathbb{F})$
+\end_inset
+
+ betegner identitetsmatricen.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Bevis
+\end_layout
+
+\begin_layout Standard
+Jf.
+ deinitionen på matrixproduktet, så kan den 
+\begin_inset Formula $(i,j)$
+\end_inset
+
+'te indgang i produktet 
+\begin_inset Formula $A\cdot\text{adj}(A)$
+\end_inset
+
+ beskrives som
+\begin_inset Formula 
+\[
+\sum_{r=1}^{n}a_{i,r}A_{j,r}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Husk at hver indgang 
+\begin_inset Formula $(i,j)$
+\end_inset
+
+ i 
+\begin_inset Formula $\text{adj}(A)$
+\end_inset
+
+ består af kofaktorer og derfor er 
+\begin_inset Formula $A_{i,j}$
+\end_inset
+
+ et tal og ikke en matrice.
+ Ovenstående formel beskriver jf.
+ prop 11.26 også determinaten af matricen, der kan fremkomme ved at udskifte
+ den 
+\begin_inset Formula $j$
+\end_inset
+
+'te række i 
+\begin_inset Formula $A$
+\end_inset
+
+ med den 
+\begin_inset Formula $i$
+\end_inset
+
+'te række i 
+\begin_inset Formula $A$
+\end_inset
+
+.
+ Såfremt 
+\begin_inset Formula $i\neq j$
+\end_inset
+
+, så er determinanten lig 0, jf.
+ Lemma 11.13, da der isåfald vil være to ens rækker, men hvis 
+\begin_inset Formula $i=j$
+\end_inset
+
+, så er determinanten lig 
+\begin_inset Formula $\text{Det}(A)$
+\end_inset
+
+.
+ Derfor gælder identiteten
+\begin_inset Formula 
+\[
+\text{Det}(A)\cdot{\rm I}=A\cdot\text{adj}(A)
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Den resterende del af den oprindelige proposition, følger ved at anvende
+ ovenstående på matricen 
+\begin_inset Formula $A^{T}$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+\text{Det}(A^{T})\cdot{\rm I}=A^{T}\cdot\text{adj}(A^{T})=A^{T}\cdot\text{adj}(A)^{T}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Hvor det sidste lighedstegn følger af Lemma 11.29.
+ Dermed jf.
+ Lemma 11.20, vil:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+\text{Det}(A)\cdot{\rm I}=A^{T}\cdot\text{adj}(A)^{T}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+hvilket implicerer
+\begin_inset Formula 
+\[
+\text{adj}(A)\cdot A=(A^{T}\cdot\text{adj}(A)^{T})^{T}=(\text{Det}(A)\cdot{\rm I)^{T}=\text{Det}(A)\cdot{\rm I}}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+hvori det sidste lighedstegn følger, da 
+\begin_inset Formula $\text{Det}(A)\cdot{\rm I}$
+\end_inset
+
+ er diagonal.
+ Hermed er beviset afsluttet.
+\end_layout
+
+\begin_layout Subsection
+Korollar 13.32
+\end_layout
+
+\begin_layout Standard
+Lad 
+\begin_inset Formula $A\in\text{Mat}_{n}(\mathbb{F})$
+\end_inset
+
+ med 
+\begin_inset Formula $n>1$
+\end_inset
+
+.
+ For 
+\begin_inset Formula $i\le i\le n$
+\end_inset
+
+ gælder
+\begin_inset Formula 
+\[
+\text{Det}(A)=\sum_{j=1}^{n}a_{i,j}\cdot A_{i,j}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Tilsvarende gælder der, for 
+\begin_inset Formula $1\leq j\leq n$
+\end_inset
+
+, at
+\begin_inset Formula 
+\[
+\text{\text{Det}(A)=\sum_{i=1}^{n}a_{i,j}\cdot A_{i,j}}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Disse kan relateres til prop 11.30 og dermed bruges til at beskrive henholdsvis
+ udvikling af række og søjle.
 \end_layout
 
 \end_body