\documentclass[11pt]{article}

% Pakker

\usepackage[danish]{babel}
\usepackage[latin1]{inputenc}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{pgf}       % PGF og tikz
\usepackage{tikz}
\usepackage{pgfkeys}
\usetikzlibrary{arrows}
\usetikzlibrary{positioning}
\usetikzlibrary{shapes.geometric}
\usetikzlibrary{calc}

% Kommandoer

\DeclareMathOperator{\spn}{span}

\newcommand{\treD}[3]{($#1*(1,0)+#3*(0,1)+#2*(-0.7071,-0.7071)$)}
\newcommand{\R}{\mathbf{R}}
\newcommand{\nvp}[1]{\textbf{#1}}

% Titel og forfatter

\title{Illustrationer til \\
       Niels Vigand Pedersens Lineær Algebra} 
\author{ 
        Rune Johansen             \\ 
        \texttt{rune@math.ku.dk} 
        }
\date{Version 1.0 \\
      \today}

%############################## BEGIN DOCUMENT #######################

\begin{document}

\maketitle

\listoffigures

\vspace{2 cm}

\noindent Figurerne i dette dokument skal illustrere vigtige sætninger og
eksempler fra Niels Vigand Pedersens bog 'Lineær Algebra', og det er
meningen, at de skal integreres i teksten, hvis bogen skal bruges igen
næste år. 

\begin{description}
\item[Pakker:] Figurerne er lavet med pakken \texttt{PGF}. Den
kræver, at man bruger \texttt{pdflatex}. Det giver forhåbentlig ikke
konflikter med resten af bogen.

\item[Referencer:] Hver figurtekst refererer til den sætning eller
det afsnit i bogen, som den illustrerer. I øjeblikket står
referencerne bare som en del af teksten, men de skal naturligvis
erstattes af rigtige dynamiske referencer. Alle referencer markeret
med \textbf{fed}, og i koden er det gjort med den hjemmelavede
kommando $\backslash$\texttt{nvp}, for at gøre dem lettere at finde. 

\item[Kommentarer:] Der er ikke ret mange kommentarer i koden, men
de enkelte figurer er tydeligt adskilt og navngivet.

\item[Preamble:] Preamblen indeholder så få kommandoer som
muligt. De skal så vidt muligt bevares.
\end{description}

\hspace{2 cm} - Rune


\newpage

%######### KAPITEL 1

%########### Figur: Regneregler for vektorer

\begin{figure}[htb]
\begin{center}
\begin{tikzpicture}
  [bend angle=45,
   knude/.style = {circle, inner sep = 0pt, fill=black!80},
   punkt/.style = {circle, draw, minimum size = 1pt, inner sep =
      0pt, fill=black!80},
   akse/.style = {->, thick},
   vektor/.style = {->, >=stealth', semithick},
   prikket/.style = {semithick, dotted},
   stiplet/.style = {semithick, dashed},
   stipletVektor/.style = {->, >=stealth', semithick, dashed},
   prikketVektor/.style = {->, >=stealth', semithick, dotted},
   fedVektor/.style = {->, >=stealth', very thick},
]

% Koordinatsystem
\node[knude] (left)  at (-2.5,0) {};
\node[knude] (right) at (2.5,0)  {};
\node[knude] (top)   at (0,3)  {};
\node[knude] (bot)   at (0,-2) {};
\node[knude] (O) at (0,0) {};

%\draw[help lines] (-4,-4) grid +(8,8);
\draw[stipletVektor] (left) to node[auto] {} (right);
\draw[stipletVektor] (bot)  to node[auto]      {} (top);

\node (trans) at (6,0) {};

\node[knude] (left')  at ($(trans)+(left)$) {};
\node[knude] (right') at ($(trans)+(right)$)  {};
\node[knude] (top')   at ($(trans)+(top)$)  {};
\node[knude] (bot')   at ($(trans)+(bot)$) {};
\node[knude] (O')     at ($(trans)+(O)$) {};

%\draw[help lines] (-4,-4) grid +(8,8);
\draw[stipletVektor] (left') to node[auto] {} (right');
\draw[stipletVektor] (bot')  to node[auto] {} (top');

% knuder
\node[knude] (U)   at (-2.5,2) {};
\node[knude] (V)   at (1,0.8) {};
\node[knude] (-U)  at ($-1*(U)$) {};
\node[knude] (2V)  at ($2*(V)$) {};

\node[knude] (U')   at ($(U)+(trans)$) {};
\node[knude] (V')   at ($(V)+(trans)$) {};

% Vektorer
\draw[fedVektor] (O) to node[auto] {$\underline u$} (U);
\draw[fedVektor] (O) to node[auto] {$\underline v$} (V);
\draw[fedVektor] (O') to (U');
\draw[fedVektor] (O') to (V');

\draw[vektor] (O) to node[auto] {$-\underline u$} (-U);
\draw[vektor] (O) to (2V);
\draw[vektor] (O') to ($(U)+(V)+(trans)$);

% linjer
\draw[prikket] (U') to ($(U)+(V)+(trans)$);
\draw[prikket] (V') to ($(U)+(V)+(trans)$);

% Tekst
\node[right] at (2V) {$2\underline v$};

\node[right] at (V') {$\underline v$};
\node[left]  at (U') {$\underline u$};
\node[above] at ($(U)+(V)+(trans)$) {$\underline u+\underline v$};

\end{tikzpicture}
\end{center}
\caption[Regneregler (\nvp{afsnit 1.1})]{Regneregler for vektorer i $\R^2$
  (\nvp{afsnit 1.1, s. 1-2}).}   
\label{fig_regneregler}
\end{figure}

%########### Figur: Standardenhedsvektorer

\begin{figure}[htb]
\begin{center}
\begin{tikzpicture}
  [bend angle=45,
   knude/.style = {circle, inner sep = 0pt, fill=black!80},
   punkt/.style = {circle, draw, minimum size = 1pt, inner sep =
      0pt, fill=black!80},
   akse/.style = {->, thick},
   vektor/.style = {->, >=stealth', semithick},
   prikket/.style = {semithick, dotted},
   stiplet/.style = {semithick, dashed},
   stipletVektor/.style = {->, >=stealth', semithick, dashed},
   prikketVektor/.style = {->, >=stealth', semithick, dotted},
   fedVektor/.style = {->, >=stealth', very thick},
]

% Koordinatsystem er
\node[knude] (left)  at (-2,0) {};
\node[knude] (right) at (2,0)  {};
\node[knude] (top)   at (0,2)  {};
\node[knude] (bot)   at (0,-2) {};
\node[knude] (O) at (0,0) {};

%\draw[help lines] (-2,-2) grid +(4,4);
\draw[stipletVektor] (left) to node[auto] {} (right);
\draw[stipletVektor] (bot)  to node[auto]      {} (top);

\node[knude] (x') at \treD{6}{0}{0} {};
\node[knude] (y') at \treD{5}{1}{0} {};
\node[knude] (z') at \treD{5}{0}{1} {};
\node[knude] (O') at (5,0) {};

\draw[stipletVektor] ($-2*(x')+3*(O')$) to ($2*(x')-(O')$);
\draw[stipletVektor] ($-2*(y')+3*(O')$) to ($2*(y')-(O')$);
\draw[stipletVektor] ($-2*(z')+3*(O')$) to ($2*(z')-(O')$);



% knuder
\node[knude] (x)   at (1,0) {};
\node[knude] (y)   at (0,1) {};

% Vektorer
\draw[fedVektor] (O) to node[auto,swap] 
                        {$ \underline e_1$} (x);
\draw[fedVektor] (O) to node[auto] 
                        {$ \underline e_2$} (y);

\draw[fedVektor] (O') to node[very near end, above] 
                         {$ \underline e_2$} (x');
\draw[fedVektor] (O') to node[very near end, left] 
                         {$ \underline e_1$} (y');
\draw[fedVektor] (O') to node[very near end, left] 
                         {$\underline e_3$} (z');

% linjer

% Tekst

\end{tikzpicture}
\end{center}
\caption[Standardenhedsvektorer i $\R^2$ og  $\R^3$
  (\nvp{afsnit 1.1})]{Standardenhedsvektorer i $\R^2$ og  $\R^3$ (\nvp{afsnit
    1.1, s. 6}).}    
\label{fig_enhedsvektorer}
\end{figure}


%########### Figur: Linaritet

\begin{figure}[htb]
\begin{center}
\begin{tikzpicture}
  [bend angle=45,
   knude/.style = {circle, inner sep = 0pt, fill=black!80},
   punkt/.style = {circle, draw, minimum size = 1pt, inner sep =
      0pt, fill=black!80},
   akse/.style = {->, thick},
   vektor/.style = {->, >=stealth', semithick},
   prikket/.style = {semithick, dotted},
   stiplet/.style = {semithick, dashed},
   stipletVektor/.style = {->, >=stealth', semithick, dashed},
   prikketVektor/.style = {->, >=stealth', semithick, dotted},
   fedVektor/.style = {->, >=stealth', very thick},
]

% Koordinatsystem
\node[knude] (left)  at (-3,0) {};
\node[knude] (right) at (2.5,0)  {};
\node[knude] (top)   at (0,5)  {};
\node[knude] (bot)   at (0,-1) {};
\node[knude] (O) at (0,0) {};

\draw[stipletVektor] (left) to (right);
\draw[stipletVektor] (bot)  to (top);

\node (trans) at (7,0) {};

\node[knude] (left')  at ($(trans)+(left)$) {};
\node[knude] (right') at ($(trans)+(right)$)  {};
\node[knude] (top')   at ($(trans)+(top)$)  {};
\node[knude] (bot')   at ($(trans)+(bot)$) {};
\node[knude] (O')     at ($(trans)+(O)$) {};

\draw[stipletVektor] (left') to (right');
\draw[stipletVektor] (bot')  to (top');

% Knuder
\node[knude] (x) at (-1,1) {};
\node[knude] (y) at (2,1) {};

\node[knude] (fx) at ($(1,2)+(trans)$)  {};
\node[knude] (fy) at ($(-3,-1)+(trans)$) {};


% Vektorer
\draw[fedVektor] (O) to node[auto] {$\underline x$} (x);
\draw[fedVektor] (O) to node[auto,swap] {$\underline y$} (y);

\draw[vektor] (O) to ($2*(x)$);
\draw[vektor] (O) to ($(x)+(y)$);

\draw[fedVektor] (O') to node[auto,swap] {$f(\underline x)$} (fx);
\draw[fedVektor] (O') to node[auto] {$f(\underline y)$} (fy);

\draw[vektor] (O') to ($2*(fx)-(trans)$);
\draw[vektor] (O') to ($(fx)+(fy)-(trans)$);


% Tekst
\node[above] at ($(x)+(y)$) {$\underline x + \underline y$};
\node[above] at ($2*(x)$)   {$2 \underline x$};

\draw[bend left, ->] (2.5,4) to node[auto] {$f$} ($(trans)+(-2.5,4)$);

\node[above] at ($(fx)+(fy)-(trans)$) 
                {$f(\underline x + \underline y) 
                      = f(\underline x) + f(\underline y) $};
\node[above] at ($2*(fx)-(trans)+(-0.7,0)$)   
                {$f(2 \underline x) = 2 f(\underline x)$};


% Linjer
\draw[prikket] (x) to ($(x)+(y)$) to (y);
\draw[prikket] (fx) to ($(fx)+(fy)-(trans)$) to (fy);


\end{tikzpicture}
\end{center}
\caption[Lineær afbildning (\nvp{1.3.9})]{Sætning
  \nvp{1.3.9}. En lineær afbildning $f \colon \R^2 \to \R^2$ opfylder
  at $f(\lambda \underline x) = \lambda f(\underline x)$ og
  $f(\underline x + \underline y) = f(\underline x) + f(\underline y)$
for alle $\underline x, \underline y \in \R^2$ og $\lambda \in \R$.}   
\label{fig_lin_afb}
\end{figure}




%######### KAPITEL 3

%######### Figur: Determinant 2D

\begin{figure}[htb]
\begin{center}
\begin{tikzpicture}
  [bend angle=45,
   knude/.style = {circle, inner sep = 0pt, fill=black!80},
   punkt/.style = {circle, draw, minimum size = 1pt, inner sep =
      0pt, fill=black!80},
   akse/.style = {->, thick},
   vektor/.style = {->, >=stealth', semithick},
   prikket/.style = {semithick, dotted},
   stiplet/.style = {semithick, dashed},
   stipletVektor/.style = {->, >=stealth', semithick, dashed},
   prikketVektor/.style = {->, >=stealth', semithick, dotted},
   fedVektor/.style = {->, >=stealth', very thick},
]

% Koordinatsystem
\node[knude] (x) at (1,0) {};
\node[knude] (y) at (0,1)  {};
\node[knude] (O) at (0,0) {};

\draw[stipletVektor] ($-1*(x)$) to node[auto]  {} ($4*(x)$);
\draw[stipletVektor] ($-1*(y)$)  to node[auto] {} ($4*(y)$);

% knuder
\node[knude] (v) at ($3*(x)+1*(y)$) {};
\node[knude] (u) at ($1*(x)+2*(y)$) {};

% Skravering
\fill[black!20] (0,0) -- (1,2) -- (4,3) -- (3,1) -- (0,0) -- cycle;

% Vektorer
\draw[fedVektor] (O) to (u);
\draw[fedVektor] (O) to (v);


% linjer
\draw[prikket, very thick] (v) to ($(u)+(v)$);
\draw[prikket, very thick] (u) to ($(u)+(v)$);

% Tekst
\node[above] at (u) {$\left( \begin{array}{r}
                        1 \\
                        2  
                  \end{array} \right)$};
\node[right] at (v) {$\left( \begin{array}{r}
                        3 \\
                        1  
                  \end{array} \right)$};
\node at (-2,2) {$\underline{ \underline A} = 
                  \left( \begin{array}{r r}
                     1 & 3 \\
                     2 & 1 
                  \end{array} \right)$};
\node at ($1/2*(u)+1/2*(v)$) {$|\det \underline{ \underline A}|$};


\end{tikzpicture}
\end{center}
\caption[Determinant af $2 \times 2$-matrix (\nvp{afsnit
    3.1})]{Den numeriske værdi af determinanten af en  $2 \times
  2$-matrix (\nvp{afsnit 3.1}) 
    er arealet af 
    parallelogrammet udspændt af vektorerne, som udgør søjlerne i
    matricen. Tilsvarende resultater gælder i højere 
    dimensioner.} 
\label{fig_determinant_2d}
\end{figure}


%######### Figur: determinant 3d

\newpage

\begin{figure}[htb]
\begin{center}
\begin{tikzpicture}
  [bend angle=45,
   knude/.style = {circle, inner sep = 0pt, fill=black!80},
   punkt/.style = {circle, draw, minimum size = 1pt, inner sep =
      0pt, fill=black!80},
   akse/.style = {->, thick},
   vektor/.style = {->, >=stealth', semithick},
   prikket/.style = {semithick, dotted},
   stiplet/.style = {semithick, dashed},
   stipletVektor/.style = {->, >=stealth', semithick, dashed},
   prikketVektor/.style = {->, >=stealth', semithick, dotted},
   fedVektor/.style = {->, >=stealth', very thick},
]

% Koordinatsystemer
\node[knude] (x) at \treD{1}{0}{0} {};
\node[knude] (y) at \treD{0}{1}{0} {};
\node[knude] (z) at \treD{0}{0}{1} {};
\node[knude] (O) at (0,0) {};

\draw[stipletVektor] ($-3*(x)$) to ($3*(x)$);
\draw[stipletVektor] ($-3*(y)$) to ($3*(y)$);
\draw[stipletVektor] ($-3*(z)$) to ($3*(z)$);

% knuder
\node[knude] (A1) at \treD{1.5}{-1}{0} {};
\node[knude] (A2) at \treD{1}{1}{-1} {};
\node[knude] (A3) at \treD{-1}{1}{3/2} {};

\node[knude] (A12) at ($(A1)+(A2)$) {};
\node[knude] (A13) at ($(A1)+(A3)$) {};

\node[knude] (A23) at ($(A2)+(A3)$) {};

\node[knude] (A123) at ($(A1)+(A2)+(A3)$) {};


% Skravering
\filldraw[black!10] ($(A1)$) to ($(A12)$) to ($(A2)$) to ($(A23)$) to ($(A3)$) to ($(A13)$) to ($(A1)$);
%\draw[stipletVektor] (O) to ($3*(x)$); 

% Vektorer
\draw[vektor] (O) to (A1);
\draw[vektor] (O) to (A2);
\draw[vektor] (O) to (A3);

% linjer
\draw[prikket, ultra thick] (A1) to (A12) to (A123);
\draw[prikket, ultra thick] (A1) to (A13) to (A123);

\draw[prikket, ultra thick] (A2) to (A12);
\draw[prikket, ultra thick] (A2) to (A23) to (A123);

\draw[prikket, ultra thick] (A3) to (A23);
\draw[prikket, ultra thick] (A3) to (A13);

% Tekst

\end{tikzpicture}
\end{center}
\caption[Determinant af $3 \times 3$-matrix (\nvp{afsnit 3.2})]{Den
  numeriske værdi af determinanten af en  $3 \times 
  3$-matrix (\nvp{afsnit 3.2}) 
    er volumenet af det parallelepipedum som 
    udspændes af vektorerne, der udgør søjlerne i
    matricen. Sammenlign med figur
    \ref{fig_determinant_2d}. Tilsvarende resultater gælder i højere
      dimensioner. }    
\label{fig_determinant_3d}
\end{figure}



%######### KAPITEL 4

%######### Figur: Polynomier

\begin{figure}[htb]
\begin{center}
\begin{tikzpicture}
  [bend angle=45,
   knude/.style = {circle, inner sep = 0pt, fill=black!80},
   punkt/.style = {circle, draw, minimum size = 1pt, inner sep =
      0pt, fill=black!80},
   akse/.style = {->, thick},
   vektor/.style = {->, >=stealth', semithick},
   prikket/.style = {semithick, dotted},
   stiplet/.style = {semithick, dashed},
   stipletVektor/.style = {->, >=stealth', semithick, dashed},
   prikketVektor/.style = {->, >=stealth', semithick, dotted},
   fedVektor/.style = {->, >=stealth', very thick},
]

% Koordinatsystem
\node[knude] (left)  at (-2.5,0) {};
\node[knude] (right) at (2.5,0)  {};
\node[knude] (top)   at (0,4)  {};
\node[knude] (bot)   at (0,-1.5) {};
\node[knude] (O) at (0,0) {};

%\draw[help lines] (-4,-4) grid +(8,8);
\draw[akse] (left) to node[near end, below ] {$x$} (right);
\draw[akse] (bot)  to node[auto]      {} (top);

\node (trans) at (6,0) {};

\node[knude] (left')  at ($(trans)+(left)$) {};
\node[knude] (right') at ($(trans)+(right)$)  {};
\node[knude] (top')   at ($(trans)+(top)$)  {};
\node[knude] (bot')   at ($(trans)+(bot)$) {};
\node[knude] (O')     at ($(trans)+(O)$) {};

%\draw[help lines] (-4,-4) grid +(8,8);
\draw[akse] (left') to node[near end, below ] {$x$} (right');
\draw[akse] (bot')  to node[auto]      {} (top');



% Funktioner
\draw plot[domain=-sqrt(5):sqrt(5)] (\x,\x^2-1) node[right] {$f$};
\draw[dotted] plot[domain=-sqrt(5):sqrt(5)] (\x,1/2*\x^2-1/2) 
      node[right] {$\frac{1}{2} f$};

\draw plot[domain=-sqrt(5)+6:sqrt(5)+6] (\x,{(\x-6)^2-1}) node[right]
      {$f$};
\draw[dashed] plot[domain=-2.5+6:2+6]
            (\x,{(\x-6)+2}) node[below] {$g$}; 
\draw[dotted] plot[domain=-2+6:1.3+6]
            (\x,{(\x-6)^2-1+(\x-6)+2}) node[left] {$f+g$}; 



% Tekst
\node (tekst) at (-2.5,4.5) {\textbf{(A)}};
\node (tekst') at ($(trans)+(tekst)$) {\textbf{(B)}};


\end{tikzpicture}
\end{center}
\caption[Polynomier (\nvp{4.1.4})]{Eksempel \nvp{4.1.4}. \textbf{(A)}
  Polynomierne  $f(x) = x^2 -1$ og $(\frac{1}{2} f)(x) = \frac{1}{2}
  f(x)  =  \frac{1}{2}(x^2 -1)$. \textbf{(B)}
  Polynomierne  $f(x) = x^2 -1, g(x) = x+2$ og $(f+g)(x) = f(x)+g(x)
  = x^2+x+1$.}   
\label{fig_polynomier}
\end{figure}



%############# Figur: Baser

\begin{figure}[htb]
\begin{center}
\begin{tikzpicture}
  [bend angle=45,
   knude/.style = {circle, inner sep = 0pt, fill=black!80},
   punkt/.style = {circle, draw, minimum size = 1pt, inner sep =
      0pt, fill=black!80},
   akse/.style = {->, thick},
   vektor/.style = {->, >=stealth', semithick},
   prikket/.style = {semithick, dotted},
   stiplet/.style = {semithick, dashed},
   stipletVektor/.style = {->, >=stealth', semithick, dashed},
   prikketVektor/.style = {->, >=stealth', semithick, dotted},
   fedVektor/.style = {->, >=stealth', very thick},
]

% Koordinatsystemer
\node[knude] (left)  at (-4,0) {};
\node[knude] (right) at (1,0)  {};
\node[knude] (top)   at (0,3)  {};
\node[knude] (bot)   at (0,-2) {};
\node[knude] (O) at (0,0) {};

\draw[akse] (left) to node[auto,swap] {} (right);
\draw[akse] (bot)  to node[auto]      {} (top);

\node (trans) at (6,0) {};

\node[knude] (left')  at ($(trans)+(-3,0)$) {};
\node[knude] (right') at ($(trans)+(2,0)$)  {};
\node[knude] (top')   at ($(trans)+(0,3)$)  {};
\node[knude] (bot')   at ($(trans)+(0,-2)$) {};
\node[knude] (O')     at ($(trans)+(O)$) {};

\draw[akse] (left') to node[auto,swap] {} (right');
\draw[akse] (bot')  to node[auto]      {} (top');


% Knuder
\node[knude] (A1)  at (-1,3) {};
\node[knude] (A2)  at (-4,2) {};
\node[knude] (A1') at ($(trans)+(1,1)$) {};
\node[knude] (A2') at ($(trans)+(-1,1)$) {};
\node[knude] (X)   at (-3,-1) {};
\node[knude] (X')   at ($(trans)+(X)$) {};

% Vektorer
\draw[fedVektor] (O) to node[very near end, left] {$\underline a_1$} (A1);
\draw[fedVektor] (O) to node[auto] {$\underline a_2$} (A2);
\draw[fedVektor] (O') to (A1');
\draw[fedVektor] (O') to (A2');

\draw[vektor] (O)  to node[auto] {$\underline x$} (X);
\draw[vektor] (O') to node[auto] {$\underline x$} (X');

\draw[stipletVektor] (O') to node[auto] {$-2 \underline{\tilde a}_1$} 
                                         ($(O')-2*(1,1)$);

% Linjer
\draw[prikket] (X) to (A2);
\draw[prikket] (A1) to (A2);
\draw[prikket] (X') to ($(O')-2*(1,1)$);
\draw[prikket] (X') to (A2');

% Tekst
\node[above] at (A1') {\quad $\underline{ \tilde a}_1$};
\node[above] at (A2') {$\underline{ \tilde a}_2$};

\node[below] at (X) {$\underline x = -\underline a_1 + \underline a_2$};
\node[left] at (X') 
   {$\underline x = -2\underline{\tilde a}_1 + \underline{\tilde a}_2$}; 
\end{tikzpicture}
\end{center}
\caption[Baser (\nvp{4.3.3})]{Definition \nvp{4.3.3}. Vektorerne
  $\underline a_1$ og 
  $\underline a_2$ udgør en basis for $\R^2$. Det samme gør vektorerne
  $\underline{\tilde a}_1$ og $\underline{\tilde a}_2$. Enhver vektor
  $\underline x \in \R^2 $ kan skrives på præcis en måde som en
  linearkombination af $\underline a_1$ og $\underline a_2$ og på
  præcis en måde som en linearkombination af $\underline{\tilde a}_1$
  og $\underline{\tilde a}_2$.} 
\label{fig_baser}
\end{figure}



%######### Figur: Underrum 2D

\begin{figure}[htb]
\begin{center}
\begin{tikzpicture}
  [bend angle=45,
   knude/.style = {circle, inner sep = 0pt, fill=black!80},
   punkt/.style = {circle, draw, minimum size = 1pt, inner sep =
      0pt, fill=black!80},
   akse/.style = {->, thick},
   vektor/.style = {->, >=stealth', semithick},
   prikket/.style = {semithick, dotted},
   stiplet/.style = {semithick, dashed},
   stipletVektor/.style = {->, >=stealth', semithick, dashed},
   prikketVektor/.style = {->, >=stealth', semithick, dotted},
   fedVektor/.style = {->, >=stealth', very thick},
]

% Koordinatsystem
\node[knude] (x) at (1,0) {};
\node[knude] (y) at (0,1)  {};
\node[knude] (O) at (0,0) {};

\draw[stipletVektor] ($-3*(x)$) to node[auto]  {} ($3*(x)$);
\draw[stipletVektor] ($-2*(y)$)  to node[auto] {} ($3*(y)$);

% knuder
\node[knude] (v) at ($2*(x)-1*(y)$){};

% Vektorer
\draw[fedVektor] (O) to node[auto,swap] {$\underline v$} (v);

% linjer
\draw[prikket] ($-1.5*(v)$) to node[near start, above] {$U$} ($1.5*(v)$);


% Tekst


\end{tikzpicture}
\end{center}
\caption[Underrum af $\R^2$ (\nvp{4.4.1})]
  {Definition \nvp{4.4.1}. For hver vektor $\underline v \in \R^2$ er
    $U = \{ t \underline v \mid t \in \R \}$ et underrum. Alle underrum
    af $\R^2$ (undtagen det trivielle underrum $\R^2$) har denne
    form.}  
\label{fig_underrum_2d}
\end{figure}


%######### Figur: Underrum 3d

\newpage

\begin{figure}[htb]
\begin{center}
\begin{tikzpicture}
  [bend angle=45,
   knude/.style = {circle, inner sep = 0pt, fill=black!80},
   punkt/.style = {circle, draw, minimum size = 1pt, inner sep =
      0pt, fill=black!80},
   akse/.style = {->, thick},
   vektor/.style = {->, >=stealth', semithick},
   prikket/.style = {semithick, dotted},
   stiplet/.style = {semithick, dashed},
   stipletVektor/.style = {->, >=stealth', semithick, dashed},
   prikketVektor/.style = {->, >=stealth', semithick, dotted},
   fedVektor/.style = {->, >=stealth', very thick},
]

% Koordinatsystemer
\node[knude] (x) at \treD{1}{0}{0} {};
\node[knude] (y) at \treD{0}{1}{0} {};
\node[knude] (z) at \treD{0}{0}{1} {};
\node[knude] (O') at (0,0) {};

\draw[stipletVektor] ($-2*(x)$) to ($2*(x)$);
%\draw[stipletVektor] ($-2*(y)$) to ($3*(y)$);
%\draw[stipletVektor] ($-2*(z)$) to ($2*(z)$);

\node (trans) at (-5,0) {};
\node[knude] (O') at (trans) {};

\draw[stipletVektor] ($-2*(x)+(trans)$) to ($2*(x)+(trans)$);
\draw[stipletVektor] ($-2*(y)+(trans)$) to ($2*(y)+(trans)$);
\draw[stipletVektor] ($-2*(z)+(trans)$) to ($2*(z)+(trans)$);


% knuder
\node[knude] (u) at \treD{0}{1}{-2/3} {};
\node[knude] (v) at \treD{1}{0}{1/3} {};

\node[knude] (u2) at \treD{0}{1}{0} {};
\node[knude] (v1) at \treD{1}{0}{0} {};

\node[knude] (left)  at ($-1*(v)$) {};
\node[knude] (right) at ($1*(v)$)  {};
\node[knude] (bot)   at ($-1*(u)$) {};
\node[knude] (top)  at ($1*(u)$)   {};

\node[knude] (w)     at \treD{1}{-1/2}{1/2} {};
\node[knude] (wproj) at \treD{1}{-1/2}{0}   {};
\node[knude] (w1)     at \treD{1}{0}{0} {};  
\node[knude] (w2)     at \treD{0}{-1/2}{0} {};

\node[knude] (w') at ($(w)+(trans)$) {};  
\node[knude] (wproj') at ($(wproj)+(trans)$) {};  
\node[knude] (w1') at ($(w1)+(trans)$) {};  
\node[knude] (w2') at ($(w2)+(trans)$) {};  


% Skravering
\fill[magenta!40] ($(right)+(bot)$) -- ($(right)+(top)$) -- 
                ($(left)+(top)$) -- ($(left)+(bot)$) -- ($(right)+(bot)$) -- cycle;

% Vektorer
\draw[fedVektor] (O) to (u);
\draw[fedVektor] (O) to (v);

\draw[stiplet] (O) to ($-2*(x)$);
\draw[stipletVektor] (O) to ($2*(y)$);
\draw[stipletVektor] (O) to ($2*(z)$);

\draw[fedVektor] (O') to  (w');


% linjer
\draw[prikket] (u) to (u2);
\draw[prikket] (v) to (v1);
\draw[prikket] ($-1*(v)$) to ($-1*(v1)$);

\draw[prikket] ($-1*(v)$) to (O);
\draw[prikket] ($-1*(u)$) to (O);


\draw[prikket] ($(right)+(bot)$) to ($(right)+(top)$);
\draw[prikket] ($(right)+(bot)$) to ($(left)+(bot)$);
\draw[prikket] ($(left)+(top)$) to ($(right)+(top)$);
\draw[prikket] ($(left)+(bot)$) to ($(left)+(top)$);

\draw[prikket, ultra thick] ($-1.5*(w)+(trans)$) to ($1.5*(w)+(trans)$);

\draw[prikket] (w') to (wproj');
\draw[prikket] (wproj') to (w1');
\draw[prikket] (wproj') to (w2');

\draw[prikket] (w') to (wproj');
\draw[prikket] (wproj') to (w1');
\draw[prikket] (wproj') to (w2');

\draw[prikket] ($-1*(w)+(trans)$) to ($-1*(wproj)+(trans)$);
\draw[prikket] ($-1*(w1)+(trans)$) to ($-1*(wproj)+(trans)$);
\draw[prikket] ($-1*(w2)+(trans)$) to ($-1*(wproj)+(trans)$);



% Tekst
\node[right] at (v) {$\underline u$};
\node[below] at (u) {$\underline v$};
\node[left] at ($(left)+(bot)$) {$U$};

\node[below right] at (w') {$\underline w$};
\node[below] at ($-1.5*(w)+(trans)$) {$V$};

\end{tikzpicture}
\end{center}
\caption[Underrum af $\R^3$ (\nvp{4.4.1})]{Definition \nvp{4.4.1}. 
  For hver vektor $w \in \R^3$ er $V = \{ t \underline w \mid t \in \R
  \}$ et underrum af $\R^3$. 
  For
  hvert par af vektorer $\underline u, \underline v \in \R^3$ er
    $U = \{ s \underline u + t \underline v \mid s,t \in \R \}$ et
  underrum af $\R^3$. Alle underrum af $\R^3$ (undtagen det trivielle
  underrum $\R^3$) har en af disse to former.}    
\label{fig_underrum_3d}
\end{figure}



%######### Figur: Stiliseret diagram af kernen

\begin{figure}[htb]
\begin{center}
\begin{tikzpicture}
  [bend angle=45,
   knude/.style = {circle, inner sep = 0pt, fill=black!80},
   punkt/.style = {circle, draw, minimum size = 1pt, inner sep =
      0pt, fill=black!80},
   akse/.style = {->, thick},
   vektor/.style = {->, >=stealth', semithick},
   prikket/.style = {semithick, dotted},
   stiplet/.style = {semithick, dashed},
   stipletVektor/.style = {->, >=stealth', semithick, dashed},
   prikketVektor/.style = {->, >=stealth', semithick, dotted},
   fedVektor/.style = {->, >=stealth', very thick},
]

% Cirkler
\node [circle,draw, fill=black!10] (ker) at (-2.2,0) [minimum size=50pt]
      {}; 
\node [circle,draw] (U) at (-2.5,0) [minimum size=80pt] {};
\node [circle,draw] (V) at (2.5,0) [minimum size=80pt] {};

% knuder
\node[punkt] (o) at (V) {};
\node[punkt] (x) at ($(ker)+(0.3,-0.4)$ ) {};


% vektorer
\draw[bend left, ->, semithick] (-1,1) to node[auto] {$f$} (1,1);
\draw[->, shorten <= 3pt, semithick] (x) to
($1/2*(x)+1/2*(o)$);
\draw[shorten >= 3pt, semithick] ($1/2*(x)+1/2*(o)$) to (o);

% linjer
\draw[prikket] (tangent cs:node=ker,point={(o)},solution=1) -- (o) 
            -- (tangent cs:node=ker,point={(o)},solution=2) -- cycle; 


% Tekst
\node at ($(-35 pt,35 pt)+(U)$) {$U$};
\node at ($(35 pt,35 pt)+(V)$) {$V$};
\node[above] at (ker) {$\ker f$};
\node[above] at (x) {$\underline x$};
\node[right] at (o) {$\underline o$};


\end{tikzpicture}
\end{center}
\caption[Kerne (\nvp{4.4.6})]
  {Stiliseret diagram af kernen (definition \nvp{4.4.6}) for en lineær
    afbildning $f \colon U \to V$.
    %$\ker F = \{\underline x \in U \mid 
    %             f(\underline x) = \underline 0 \}$  
  }  
\label{fig_kerne}
\end{figure}




%######### Figur: Stiliseret diagram af billedet

\begin{figure}[htb]
\begin{center}
\begin{tikzpicture}
  [bend angle=45,
   knude/.style = {circle, inner sep = 0pt, fill=black!80},
   punkt/.style = {circle, draw, minimum size = 1pt, inner sep =
      0pt, fill=black!80},
   akse/.style = {->, thick},
   vektor/.style = {->, >=stealth', semithick},
   prikket/.style = {semithick, dotted},
   stiplet/.style = {semithick, dashed},
   stipletVektor/.style = {->, >=stealth', semithick, dashed},
   prikketVektor/.style = {->, >=stealth', semithick, dotted},
   fedVektor/.style = {->, >=stealth', very thick},
]

% funktion




% Cirkler
\node [circle,draw, fill=black!10] (im) at (2.2,0) [minimum size=50pt]
      {}; 
\node [circle,draw] (U) at (-2.5,0) [minimum size=80pt] {};
\node [circle,draw] (V) at (2.5,0) [minimum size=80pt] {};

% knuder
%\node[punkt] (o) at (V) {};
\node[punkt] (x) at ($(U)+(0.5,-0.8)$ ) {};
\node[punkt] (fx) at ($(im)+(-0.3,-0.4)$ ) {};


% vektorer
\draw[bend left, ->, semithick] (-1,1.5) to node[auto] {$f$} (1,1.5);
\draw[->, shorten <= 3pt, semithick] (x) to
($1/2*(x)+1/2*(fx)$);
\draw[shorten >= 3pt, semithick] ($1/2*(x)+1/2*(fx)$) to (fx);

% linjer
\draw[prikket] (tangent cs:node=U,point={(2.2,25 pt)},solution=1) 
               to (2.2,25 pt); 
\draw[prikket] (tangent cs:node=U,point={(2.2,-25 pt)},solution=2) 
               to (2.2,-25 pt); 


% Tekst
\node at ($(-35 pt,35 pt)+(U)$) {$U$};
\node at ($(35 pt,35 pt)+(V)$) {$V$};
\node[above] at (im) {$f(U)$};
\node[above] at (x) {$\underline x$};
\node[right] at (fx) {$f(\underline x)$};


\end{tikzpicture}
\end{center}
\caption[Billede (\nvp{4.4.7})]
  {Stiliseret diagram af billedet (sætning \nvp{4.4.7}) for en lineær
    afbildning $f \colon U \to V$.
    %$\ker F = \{\underline x \in U \mid 
    %             f(\underline x) = \underline 0 \}$  
  }  
\label{fig_billede}
\end{figure}


%############# Figur: Kerne og billede

\begin{figure}[htb]
\begin{center}
\begin{tikzpicture}
  [bend angle=45,
   knude/.style = {circle, inner sep = 0pt, fill=black!80},
   punkt/.style = {circle, draw, minimum size = 1pt, inner sep =
      0pt, fill=black!80},
   akse/.style = {->, thick},
   vektor/.style = {->, >=stealth', semithick},
   prikket/.style = {semithick, dotted},
   stiplet/.style = {semithick, dashed},
   stipletVektor/.style = {->, >=stealth', semithick, dashed},
   prikketVektor/.style = {->, >=stealth', semithick, dotted},
   fedVektor/.style = {->, >=stealth', very thick},
]

% Koordinatsystemer
\node[knude] (left)  at (-2.5,0) {};
\node[knude] (right) at (2.5,0)  {};
\node[knude] (top)   at (0,4.5)  {};
\node[knude] (bot)   at (0,-3) {};
\node[knude] (O) at (0,0) {};

\draw[stipletVektor] (left) to node[auto,swap] {} (right);
\draw[stipletVektor] (bot)  to node[auto]      {} (top);

\node (trans) at (7,0) {};

\node[knude] (left')  at ($(trans)+(left)$) {};
\node[knude] (right') at ($(trans)+(right)$)  {};
\node[knude] (top')   at ($(trans)+(top)$)  {};
\node[knude] (bot')   at ($(trans)+(bot)$) {};
\node[knude] (O')     at ($(trans)+(O)$) {};

\draw[stipletVektor] (left') to node[auto,swap] {} (right');
\draw[stipletVektor] (bot')  to node[auto]      {} (top');

% Knuder
\node[knude] (e1) at (1,0) {};
\node[knude] (e2) at (0,1) {};

\node[knude] (fe1) at ($(trans)+(1,-2)$) {};
\node[knude] (fe2) at ($(trans)+(-2,4)$) {};


% Vektorer
\draw[fedVektor] (O) to (e1);
\draw[fedVektor] (O) to (e2);

\draw[fedVektor] (O') to node[auto] {$f(\underline e_1)$} (fe1);
\draw[fedVektor] (O') to node[auto] {$f(\underline e_2)$} (fe2);

\draw[bend left, ->, semithick] (2.5,2) to node[auto] {$f$} ($(trans)+(-2.5,2)$); 

% Linjer
\draw[prikket] ($(trans)+1.5*(1,-2)$) to ($(trans)-2.25*(1,-2)$);
\draw[prikket] ($-1.25*(2,1)$) to node[very near start, below right ]
     {$\ker f$} ($1.25*(2,1)$);


% Tekst
\node[above right] at (e1) {$\underline e_1$};
\node[above left] at (e2) {$\underline e_2$};
\node[left] at ($(trans)+1.5*(1,2)$) {$f(\R^2)$};


\node[below] at ($(0,-5)$) 
                      {$\ker f = \left\{ t \left( \begin{array}{r}
                                                   2 \\
                                                   1
                                          \end{array} \right) \mid t \in
                          \R \right\}$};


\node[below] at ($(trans)+(0,-5)$) 
                      {$f(\R^2) = \left\{ t \left( \begin{array}{r}
                                                   1 \\
                                                  -2
                                          \end{array} \right) \mid t \in
                          \R \right\}$};


\end{tikzpicture}
\end{center}
\caption[Kerne og billede (\nvp{4.4.6} og \nvp{4.4.7})]{Kerne og
  billede (\nvp{4.4.6} og \nvp{4.4.7}) for den lineære afbildning $f
  \colon \R^2 \to \R^2$ give ved matricen 
  $\underline{ \underline A} = \left( \begin{array}{r r}
       1 & -2 \\
      -2 &  4
  \end{array} \right)$.
  Se også figur \ref{fig_kerne} og \ref{fig_billede}.} 
\label{fig_billede_og_kerne}
\end{figure}



%############ Figur: Span

\begin{figure}[htb]
\begin{center}
\begin{tikzpicture}
  [bend angle=45,
   knude/.style = {circle, inner sep = 0pt, fill=black!80},
   punkt/.style = {circle, draw, minimum size = 1pt, inner sep =
      0pt, fill=black!80},
   akse/.style = {->, thick},
   vektor/.style = {->, >=stealth', semithick},
   prikket/.style = {semithick, dotted},
   stiplet/.style = {semithick, dashed},
   stipletVektor/.style = {->, >=stealth', semithick, dashed},
   prikketVektor/.style = {->, >=stealth', semithick, dotted},
   fedVektor/.style = {->, >=stealth', very thick},
]

% Koordinatsystem er
\node[knude] (x) at \treD{1}{0}{0} {};
\node[knude] (y) at \treD{0}{1}{0} {};
\node[knude] (z) at \treD{0}{0}{1} {};
\node[knude] (O') at (0,0) {};

\draw[stipletVektor] (O) to ($5*(x)$);
\draw[stipletVektor] ($-5*(y)$) to (O);
\draw[stipletVektor] ($-5*(z)$) to (O);

% knuder
\node[knude] (u) at \treD{0}{3}{-2} {};
\node[knude] (v) at \treD{3}{0}{1} {};

\node[knude] (u2) at \treD{0}{3}{0} {};
\node[knude] (v1) at \treD{3}{0}{0} {};

\node[knude] (left)  at ($-1*(v)$) {};
\node[knude] (right) at ($1*(v)$) {};
\node[knude] (bot)   at ($-1*(u)$) {};
\node[knude] (top)  at ($1*(u)$) {};


% Skravering
\fill[magenta!40] ($(right)+(bot)$) -- ($(right)+(top)$) -- 
                ($(left)+(top)$) -- ($(left)+(bot)$) -- ($(right)+(bot)$) -- cycle;

% Vektorer
\draw[fedVektor] (O) to (u);
\draw[fedVektor] (O) to (v);
\draw[vektor] (O)   to node[very near end,below right] {$\underline x \in U$} ($-1/2*(u)+1/2*(v)$);


\draw[stiplet] (O) to ($-5*(x)$);
\draw[stipletVektor] (O) to ($5*(y)$);
\draw[stipletVektor] (O) to ($5*(z)$);


% linjer
\draw[prikket] (u) to (u2);
\draw[prikket] (v) to (v1);
\draw[prikket] ($-1*(v)$) to ($-1*(v1)$);

%\draw[prikket] ($-1*(v)$) to (O);
%\draw[prikket] ($-1*(u)$) to (O);


\draw[prikket] ($(right)+(bot)$) to ($(right)+(top)$);
\draw[prikket] ($(right)+(bot)$) to ($(left)+(bot)$);
\draw[prikket] ($(left)+(top)$) to ($(right)+(top)$);
\draw[prikket] ($(left)+(bot)$) to ($(left)+(top)$);



% Tekst
\node[right] at (v) {$\underline a_1$};
\node[below] at (u) {$\underline a_2$};
\node[left] at ($(left)+(bot)$) 
               {$U = \spn \{\underline a_1,\underline a_2\}$};

\end{tikzpicture}
\end{center}
\caption[Span (\nvp{4.4.5})]{Sætning \nvp{4.4.5}. To vektorer
  $\underline a_1, \underline a_2 \in \R^3$ og underrummet $U = \spn
  \{\underline a_1,\underline a_2\}$, der består 
af alle linearkombinationer af $\underline a_1$ og $\underline a_2$.}    
\label{fig_span}
\end{figure}



%######### Figur: Lineær uafhængighed

\begin{figure}[htb]
\begin{center}
\begin{tikzpicture}
  [bend angle=45,
   knude/.style = {circle, inner sep = 0pt, fill=black!80},
   punkt/.style = {circle, draw, minimum size = 1pt, inner sep =
      0pt, fill=black!80},
   akse/.style = {->, thick},
   vektor/.style = {->, >=stealth', semithick},
   prikket/.style = {semithick, dotted},
   stiplet/.style = {semithick, dashed},
   stipletVektor/.style = {->, >=stealth', semithick, dashed},
   prikketVektor/.style = {->, >=stealth', semithick, dotted},
   fedVektor/.style = {->, >=stealth', very thick},
]

% Koordinatsystemer
\node[knude] (left)  at (-4,0) {};
\node[knude] (right) at (1,0)  {};
\node[knude] (top)   at (0,3)  {};
\node[knude] (bot)   at (0,-1) {};
\node[knude] (O) at (0,0) {};

\draw[help lines] (-4,-1) grid +(5,4);

\draw[akse] (left) to node[auto,swap] {} (right);
\draw[akse] (bot)  to node[auto]      {} (top);

\node (trans) at (5,0) {};

\node[knude] (left')  at ($(trans)+(-3,0)$) {};
\node[knude] (right') at ($(trans)+(3,0)$)  {};
\node[knude] (top')   at ($(trans)+(0,3)$)  {};
\node[knude] (bot')   at ($(trans)+(0,-1)$) {};
\node[knude] (O')     at ($(trans)+(O)$) {};

\draw[help lines] ($(trans)+(-3,-1)$) grid +(6,4);

\draw[akse] (left') to node[auto,swap] {} (right');
\draw[akse] (bot')  to node[auto]      {} (top');

% Knuder
\node[knude] (A1)  at (-1,3) {};
\node[knude] (A2)  at (-4,2) {};

\node[knude] (B1)  at ($(3,2)+(trans)$) {};
\node[knude] (B2)  at ($(-2,1/2)+(trans)$) {};
\node[knude] (B3)  at ($(-1,3)+(trans)$) {};


% Vektorer
\draw[fedVektor] (O) to node[very near end, left] {$\underline a_1$} (A1);
\draw[fedVektor] (O) to node[auto] {$\underline a_2$} (A2);

\draw[fedVektor] (O') to node[auto,swap] {$\underline b_1$} (B1);
\draw[fedVektor] (O') to node[auto,swap] {$\underline b_2$} (B2);
\draw[fedVektor] (O') to node[auto,swap] {$\underline b_3$} (B3);


% Linjer

% Tekst
\node[above] at (-4,3) {\textbf{(A)}};
\node[above] at ($(-3,3)+(trans)$) {\textbf{(B)}};


\end{tikzpicture}
\end{center}
\caption[Lineær uafhængighed (\nvp{4.5.1})]{Definition
  \nvp{4.5.1}.  \textbf{(A)} Vektorsættet $\underline a_1, \underline
  a_2$ er lineært \emph{uafhængigt}.  \textbf{(B)} Vektorsættet
  $\underline b_1, \underline b_2, \underline b_3 $ er lineært
  \emph{afhængigt} idet $\underline b_1 + 2\underline b_2 - \underline b_3 = 0 $} 
\label{fig_lin_uafh}
\end{figure}




%######### KAPITEL 5

%########  Figur : Basisskift

\begin{figure}[htb]
\begin{center}
\begin{tikzpicture}
  [bend angle=45,
   knude/.style = {circle, inner sep = 0pt, fill=black!80},
   punkt/.style = {circle, draw, minimum size = 1pt, inner sep =
      0pt, fill=black!80},
   akse/.style = {->, thick},
   vektor/.style = {->, >=stealth', semithick},
   prikket/.style = {semithick, dotted},
   stiplet/.style = {semithick, dashed},
   stipletVektor/.style = {->, >=stealth', semithick, dashed},
   prikketVektor/.style = {->, >=stealth', semithick, dotted},
   fedVektor/.style = {->, >=stealth', very thick},
]

% Koordinatsystemer
\node[knude] (left)  at (-4,0) {};
\node[knude] (right) at (1,0)  {};
\node[knude] (top)   at (0,3)  {};
\node[knude] (bot)   at (0,-1) {};
\node[knude] (O) at (0,0) {};

\draw[akse] (left) to node[auto,swap] {} (right);
\draw[akse] (bot)  to node[auto]      {} (top);


% Knuder
\node[knude] (A1)  at (-1,3) {};
\node[knude] (A2)  at (-4,2) {};
\node[knude] (A1') at (1,1) {};
\node[knude] (A2') at (-1,1) {};

% Vektorer
\draw[vektor] (O) to (A1);
\draw[vektor] (O) to (A2);
\draw[fedVektor] (O) to (A1');
\draw[fedVektor] (O) to (A2');

% Linjer
\draw[prikket] (A1) to (A1'); 
\draw[prikket] (A1) to ($2*(A2')$); 
\draw[prikket] (A2) to ($-1*(A1')$); 
\draw[prikket] (A2) to ($3*(A2')$); 
\draw[prikketVektor] (O) to ($2*(A2')$);
\draw[prikketVektor] (O) to ($-1*(A1')$);
\draw[prikketVektor] (O) to ($3*(A2')$);
 

% Tekst
\node[above] at (A1) {$\underline a_1$};
\node[left] at (A2) {$\underline a_2$};
\node[above] at (A1') {\quad $\underline{ \tilde a}_1$};
\node[above] at (A2') {\quad $\underline{ \tilde a}_2$};


\end{tikzpicture}
\end{center}
\caption[Basisskift (\nvp{5.1.2})]{Eksempel \nvp{5.1.2}. Vektorerne
  $\underline a_1$ og 
  $\underline a_2$ udgør en basis for $\R^2$. Det samme gør vektorerne
  $\underline{\tilde a}_1$ og $\underline{\tilde
    a}_2$. Bemærk, at $\underline a_1 = \underline{\tilde a}_1 +2
  \underline{\tilde a}_2$ og $\underline a_2 = -\underline{\tilde a}_1 +3
  \underline{\tilde a}_2$. 
%  Det kan bruges til at opskrive
%  koordinattransformationsmatricen for overgang fra 
%  $\underline a_1, \underline a_2$ til $\underline{\tilde a}_1,
%  \underline{\tilde a}_2$.
} 
\label{fig_basisskift}
\end{figure}


%######### KAPITEL 6

%########## Figur : Egenvektorer

\begin{figure}[htb]
\begin{center}
\begin{tikzpicture}
  [bend angle=45,
   knude/.style = {circle, inner sep = 0pt, fill=black!80},
   punkt/.style = {circle, draw, minimum size = 1pt, inner sep =
      0pt, fill=black!80},
   akse/.style = {->, thick},
   vektor/.style = {->, >=stealth', semithick},
   prikket/.style = {semithick, dotted},
   stiplet/.style = {semithick, dashed},
   stipletVektor/.style = {->, >=stealth', semithick, dashed},
   prikketVektor/.style = {->, >=stealth', semithick, dotted},
   fedVektor/.style = {->, >=stealth', very thick},
]

% Koordinatsystem
\node        (left)  at (-11,0) {};
\node        (right) at (1,0)  {};
\node        (top)   at (0,3)  {};
\node        (bot)   at (0,-1) {};
\node[knude] (O) at (0,0) {};

\draw[stipletVektor] (left) to (right);
\draw[stipletVektor] (bot) to (top);

% Knuder

\node[knude] (V-2) at ($1/2*(1,-1)$) {};
\node[knude] (V3) at  ($1/2*(-7,2)$) {};
\node (tekst) at (-4,-1) {};

% Vektorer
\draw[fedVektor] (O) to (V-2);
\draw[fedVektor] (O) to (V3);
\draw[vektor] (O) to ($-2*(V-2)$);
\draw[vektor] (O) to node[near end, above] 
     {$\quad f(\underline y) = 3\underline y$} ($3*(V3)$);

% tekst
\node[below] at (V-2) {$\underline x$};
\node[above] at (V3) {$\underline y$};
\node[above] at ($-2*(V-2)$) {$f(\underline x) = -2 \underline x \quad$}; 
\node[below] at (tekst) {$\underline x = \left( \begin{array}{r}
                                               1 \\
                                               -1   
                                               \end{array} \right) 
                                        \in V_{-2} \quad , \quad
                          \underline y = \left( \begin{array}{r}
                                               -7 \\
                                               3   
                                               \end{array} \right)
                                          \in V_{3} $};
 

\end{tikzpicture}
\end{center}
\caption[Egenvektorer (\nvp{6.1.10})]{Egenvektorer for den lineære
  afbildning $f \colon \R^2 \to \R^2$ fra eksempel \nvp{6.1.10}. }
\label{fig_eks_egenvektor}
\end{figure}



%######### KAPITEL 7

%######### Figur til definition 7.1.4 - Ortonormalbasis

\begin{figure} [htb]
\begin{center}
\begin{tikzpicture}
  [bend angle=45,
   knude/.style = {circle, inner sep = 0pt, fill=black!80},
   punkt/.style = {circle, draw, minimum size = 1pt, inner sep =
      0pt, fill=black!80},
   akse/.style = {->, thick},
   vektor/.style = {->, >=stealth', semithick},
   prikket/.style = {semithick, dotted},
   stiplet/.style = {semithick, dashed},
   stipletVektor/.style = {->, >=stealth', semithick, dashed},
   prikketVektor/.style = {->, >=stealth', semithick, dotted},
   fedVektor/.style = {->, >=stealth', very thick},
]

% Koordinatsystem
\node (x)  at \treD{1}{0}{0} {};
\node (y)  at \treD{0}{1}{0} {};
\node (z)  at \treD{0}{0}{1} {};
\node[knude] (O) at (0,0) {};

\draw[stipletVektor] ($-2.5*(x)$) to ($2.5*(x)$);
\draw[stipletVektor] ($-2.5*(y)$) to ($2.5*(y)$);
\draw[stipletVektor] ($-2.5*(z)$) to ($2.5*(z)$);

% Knuder
\node[knude] (A1) at \treD{-2/sqrt(5)}{0}{4/sqrt(5)}  {};
\node[knude] (A2) at \treD{4/sqrt(5)}{0}{2/sqrt(5)} {};
\node[knude] (A3) at \treD{0}{2}{0} {};

\node[knude] (P11) at \treD{-2/sqrt(5)}{0}{0} {};
\node[knude] (P13) at \treD{0}{0}{4/sqrt(5)} {};
 
\node[knude] (P21) at \treD{4/sqrt(5)}{0}{0} {};
\node[knude] (P23) at \treD{0}{0}{2/sqrt(5)} {};

% Vektorer
\draw[fedVektor] (O) to (A1);
\draw[fedVektor] (O) to (A2);
\draw[fedVektor] (O) to node[very near end, below right] {$\underline a_3$} (A3);

% Linjer
\draw[prikket] (A1) to (P11);
\draw[prikket] (A1) to (P13);

\draw[prikket] (A2) to (P21);
\draw[prikket] (A2) to (P23);

% Tekst

\node[right] at (A2) {$\underline a_2$};
\node[left]  at (A1) {$\underline a_1$};
  
\end{tikzpicture}
\end{center}
\caption[Ortonormalbasis (\nvp{7.1.4})]{Definition
  \nvp{7.1.4}. Vektorsættet $\underline a_1, 
  \underline a_2,\underline a_3$ er en ortonormalbasis for $\R^3$. De
  naturlige ortonormalbaser for $\R^2$ og $\R^3$ er vist på figur
  \ref{fig_enhedsvektorer}.}
\label{fig_ortonormalbasis}
\end{figure}


%######### Figur til sætning 7.1.6 - ortogonalsaet 

\begin{figure} [htb]
\begin{center}
\begin{tikzpicture}
  [bend angle=45,
   knude/.style = {circle, inner sep = 0pt, fill=black!80},
   punkt/.style = {circle, draw, minimum size = 1pt, inner sep =
      0pt, fill=black!80},
   akse/.style = {->, thick},
   vektor/.style = {->, >=stealth', semithick},
   prikket/.style = {semithick, dotted},
   stiplet/.style = {semithick, dashed},
   stipletVektor/.style = {->, >=stealth', semithick, dashed},
   prikketVektor/.style = {->, >=stealth', semithick, dotted},
   fedVektor/.style = {->, >=stealth', very thick},
]

% Koordinatsystem
\node (left)  at (-5,0) {};
\node (right) at (5,0)  {};
\node (top)   at (0,5)  {};
\node (bot)   at (0,-2) {};
\node[knude] (O) at (0,0) {};

\draw[stipletVektor] (left) to node[auto,swap] {} (right);
\draw[stipletVektor] (bot)  to node[auto]      {} (top);

% Knuder
\node[knude] (A1)  at (2,4) {};
\node[knude] (A1+) at (2.5,5) {};
\node[knude] (A2)  at (-3,1.5) {};
\node[knude] (A2+) at (-5,2.5) {};
\node[knude] (A2-) at (5,-2.5) {};
\node[knude] (A)  at (3,2) {};
\node[knude] (P1) at ($(A1)!(A)!(A1+)$) {};
\node[knude] (P2) at ($(A2)!(A)!(A2+)$) {};

% Vektorer
\draw[vektor] (O) to node[very near end, right] {$\underline a_1$} (A1);
\draw[vektor] (O) to node[auto] {$\underline a_2$} (A2);
\draw[vektor] (O) to node[auto] {$\underline a$} (A);
\draw[fedVektor] (O) to node[auto] {} (P1);
\draw[fedVektor] (O) to node[auto] {} (P2);


% Linjer
%\draw[stiplet] (A1) to (A1+);
%\draw[stiplet] (A2-) to (A2+);
\draw[prikket] (A) to (P1);
\draw[prikket] (A) to (P2);

% Tekst
\node[left]  at (P1) {$\frac{\underline a \cdot \underline a_1}
                     {\underline a_1 \cdot \underline a_1} \underline a_1$};
\node[below] at (P2) {$\frac{\underline a \cdot \underline a_2}
                     {\underline a_2 \cdot \underline a_2} \underline a_2$};
  
\end{tikzpicture}
\end{center}
\caption[Ortogonalsæt (\nvp{7.1.6})]{Vektorerne $\underline a_1$ og
  $\underline a_2$ udgør et ortogonalsæt, så de er
  lineært uafhængige, og sætning \nvp{ 7.1.6 } viser hvordan en
  vektor $\underline a \in \spn \{\underline a_1 , \underline a_2 \}$
  kan skrives som en linearkombination af $\underline a_1$ og $\underline a_2$.}
\label{fig_ortogonal_linear_kombination}
\end{figure}

%######### Figur til sætning 7.1.7

\begin{figure}
\begin{center}
\begin{tikzpicture}
  [bend angle=45,
   knude/.style = {circle, inner sep = 0pt, fill=black!80},
   punkt/.style = {circle, draw, minimum size = 1pt, inner sep =
      0pt, fill=black!80},
   akse/.style = {->, thick},
   vektor/.style = {->, >=stealth', semithick},
   prikket/.style = {semithick, dotted},
   stiplet/.style = {semithick, dashed},
   stipletVektor/.style = {->, >=stealth', semithick, dashed},
   prikketVektor/.style = {->, >=stealth', semithick, dotted},
   fedVektor/.style = {->, >=stealth', very thick},
]

%####### A

Koordinatsystem
\node[knude] (x)  at \treD{1}{0}{0} {};
\node[knude] (y)  at \treD{0}{1}{0} {};
\node[knude] (z)  at \treD{0}{0}{1} {};
\node[knude] (O)  at \treD{0}{0}{0} {};

\draw[stiplet] (O) to ($5*(x)$);
\draw[stiplet] (O) to ($4*(y)$);
\draw[stiplet] (O) to ($4*(z)$);

\draw[stiplet] ($-2*(x)$) to (O);
\draw[stiplet] ($-2*(y)$) to (O);
\draw[stiplet] ($-1*(z)$) to (O);

% Knuder
\node[knude] (A1)  at \treD{3}{0}{0} {};
\node[knude] (A2)  at \treD{-1}{3}{0} {};
\node[knude] (A3)  at \treD{4}{2}{3} {};

\node[knude] (A2proj)  at \treD{-1}{0}{0} {};
\node[knude] (A2orto)  at \treD{0}{3}{0} {};

\node (A3orto) at  \treD{0}{0}{3} {};
\node (A3proj) at  \treD{4}{2}{0} {};
\node (A3proj1) at  \treD{4}{0}{0} {};
\node (A3proj2) at  \treD{0}{2}{0} {};


% Vektorer
\draw[fedVektor] (O) to (A1); 
\draw[fedVektor] (O) to (A2); 
\draw[fedVektor] (O) to node[very near end, left] {$\underline a_3$} (A3); 
\draw[vektor] (O) to (A2orto); 
\draw[vektor] (O) to node[very near end, above] 
          {$\frac{\underline a_2 \cdot \underline b_1}
                 {\underline b_1 \cdot \underline b_1}
                 \underline b_1$}(A2proj); 

% Linjer
\draw[prikket] (A2) to (A2proj);
\draw[prikket] (A2) to (A2orto);

% Tekst
\node at (-5,4) {\textbf{(A)}};
\node[above] at (A1) {$\underline b_1 = \underline a_1$};
\node[left] at  (A2) {$ \underline a_2$};
\node[below, right] at  (A2orto) {$ \quad \underline b_2 = 
                            \underline a_2 - 
                            \frac{\underline a_2 \cdot \underline b_1}
                                 {\underline b_1 \cdot \underline b_1}
                            \underline b_1$};


%########## B


% Koordinatsystem
\node[knude] (trans) at \treD{0}{0}{-8} {};
\node[knude] (O')  at ($(O)+(trans)$) {};

\node (x+')  at \treD{5}{0}{-8} {};
\node (y+')  at \treD{0}{4}{-8} {};
\node (z+')  at \treD{0}{0}{-4} {};
\node (x-')  at \treD{-2}{0}{-8} {};
\node (y-')  at \treD{0}{-2}{-8} {};
\node (z-')  at \treD{0}{0}{-9} {};

\draw[stiplet] (x-') to (x+'); 
\draw[stiplet] (y-') to (y+'); 
\draw[stiplet] (z-') to (z+'); 

% Knuder

\node[knude] (A1')  at  ($(A1)+(trans)$) {};
\node[knude] (A2')  at  ($(A2)+(trans)$) {};
\node[knude] (A3')  at  ($(A3)+(trans)$) {};

\node[knude] (B2')  at  ($(A2orto)+(trans)$) {};

\node[knude] (A3orto') at ($(A3orto) + (trans)$) {};
\node[knude] (A3proj') at ($(A3proj) + (trans)$) {};
\node[knude] (A3proj1') at ($(A3proj1) + (trans)$) {};
\node[knude] (A3proj2') at ($(A3proj2) + (trans)$) {};

% vektorer
\draw[fedVektor] (O') to (A1'); 
\draw[fedVektor] (O') to (A2'); 
\draw[fedVektor] (O') to node[very near end, left] {$\underline a_3$} (A3'); 
\draw[vektor] (O') to (B2'); 
\draw[vektor] (O') to (A3proj'); 
\draw[vektor] (O') to (A3orto'); 


% linjer
\draw[prikket] (A3orto') to (A3');
\draw[prikket] (A3proj') to (A3');
\draw[prikket] (A3proj') to (A3proj1');
\draw[prikket] (A3proj') to (A3proj2');


% Tekst
\node at ($(-5,4)+(trans)$)  {\textbf{(B)}};
\node[above] at (A1') {$\underline b_1$};
\node[left]  at (A2') {$ \underline a_2$};
\node[below,right] at  (B2') {$\quad \underline b_2$};
\node[left] at  (A3orto') {$\underline b_3 = 
                            \underline a_3 - 
                            \frac{\underline a_3 \cdot \underline b_1}
                                 {\underline b_1 \cdot \underline b_1}
                            \underline b_1 -
                            \frac{\underline a_3 \cdot \underline b_2}
                                 {\underline b_2 \cdot \underline b_2}
                            \underline b_2$};
\node[right] at  (A3proj') {$ 
                            \frac{\underline a_3 \cdot \underline b_1}
                                 {\underline b_1 \cdot \underline b_1}
                            \underline b_1 +
                            \frac{\underline a_3 \cdot \underline b_2}
                                 {\underline b_2 \cdot \underline b_2}
                            \underline b_2$};


\end{tikzpicture}
\end{center}
\caption[Gram-Schmidt ortogonalisering (\nvp{7.1.7})]{Gram-Schmidt
  ortogonalisering (sætning  \nvp{7.1.7}) af vektorerne $\underline
  a_1,\underline a_2, \underline a_3$.\textbf{(A)} Konstruktion af
  $\underline b_2$ udfra $\underline b_1$ og $\underline
  a_2$. \textbf{(B)} Konstruktion af 
  $\underline b_3$ udfra $\underline b_1, \underline b_2$ og $\underline a_3$}
\label{fig_GS}
\end{figure}




%######### Figur til sætning 7.3.4

\begin{figure} 
\begin{center}
\begin{tikzpicture}
  [bend angle=45,
   knude/.style = {circle, inner sep = 0pt, fill=black!80},
   punkt/.style = {circle, draw, minimum size = 1pt, inner sep =
      0pt, fill=black!80},
   akse/.style = {->, thick},
   vektor/.style = {->, >=stealth', semithick},
   prikket/.style = {semithick, dotted},
   stiplet/.style = {semithick, dashed},
   stipletVektor/.style = {->, >=stealth', semithick, dashed},
   prikketVektor/.style = {->, >=stealth', semithick, dotted}, 
   fedVektor/.style = {->, >=stealth', very thick},
]

% Koordinatsystem
\node (A1)  at (6,0) {};
\node (A2)   at (-3,-3)  {};
\node (C)   at (0,3) {};
\node (A1-)  at (-2,0) {};
\node (A2-)   at (1.4,1.4)  {};
\node (C-)   at (0,-1.5) {};


\node[knude] (O) at (0,0) {};

\draw[vektor] (O) to node[auto] {} (A1);
\draw[vektor] (O) to node[auto]      {} (A2);
\draw[stiplet] (O) to node[auto]      {} (A1-);
\draw[stiplet] (O) to node[auto]      {} (A2-);
\draw[stiplet] (C-) to node[auto]      {} (C);

% Knuder
\node[knude] (x) at (3,-2) {};
\node[knude] (x1) at (5,0) {};
\node[knude] (x2) at (-2,-2) {};
\node[knude] (A) at (3,2) {};

% Vektorer
\draw[vektor] (O) to node[auto] {$\underline x$} (x);
\draw[fedVektor] (O) to node[near end, above] {$\underline a \;$} (A);
\draw[fedVektor] (O) to (x1);
\draw[fedVektor] (O) to (x2);

% Linjer
\draw[prikket] (x) to node[auto] {} (x1);
\draw[prikket] (x) to node[auto] {} (x2);
\draw[prikket] (x) to node[auto] {} (A);

% Tekst
\node[below] at (A1) {$\underline a_1$};
\node[left]  at (A2) {$\underline a_2$};
\node[above] at (x1) {$(\underline a \cdot \underline a_1) \underline a_1$};
\node[left]  at (x2) {$(\underline a \cdot \underline a_2) \underline
  a_2 \;$};


\end{tikzpicture}
\end{center}
\caption[Ortogonalprojektion (\nvp{7.3.4})]{Ortogonalprojektion $\underline x$ af $\underline a$ på $U = \spn
  \{ \underline a_1, \underline a_2 \}$ (sætning \nvp{7.3.4}).
   Bemærk, at $x \in U$ og $\underline a -
  \underline x \in U^\perp$.}
\label{fig_ortogonalprojektion}
\end{figure}



\end{document}