\documentclass[%
version=last,%
a5paper,%
12pt,%
%headings=small,%
bibliography=totoc,% index=totoc,%
twoside,%
%open=any,%
%parskip=half,%  this option takes 2.5% more space than parskip
% draft=true,%
titlepage=false,%
DIV=12,%
headinclude=false,%
pagesize]%
%{scrbook}
{scrartcl}

%\makeindex
%\usepackage{makeidx}
%\usepackage{showidx}
%  Run texindy -L turkish <filename>.idx on the file

\usepackage[notcite,notref]{showkeys}

%\usepackage{cclicenses}

%\usepackage{scrpage2}
%\pagestyle{scrheadings}
%\clearscrheadings
%\ofoot{\pagemark}
%\refoot{\leftmark}
%\lofoot{\rightmark}

\usepackage{moredefs,lips}

\usepackage[turkish]{babel}
%\usepackage[polutonikogreek,turkish]{babel}
%\newcommand{\eng}[1]{(\emph{#1})}
%\usepackage{gfsporson}
%\newcommand{\gr}[1]{\foreignlanguage{polutonikogreek}{\textporson{#1}}}

%\newcommand{\enquote}[1]{``#1''}
%\usepackage[utf8]{inputenc}

\usepackage{verbatim}
 \usepackage{hfoldsty}
\usepackage[perpage,symbol*]{footmisc}

% For symbol table

\usepackage{longtable}

%\makeglossary % This command must be commented when the glossary is to
	      % be printed
\newcommand{\glossaryentry}[2]{#1& #2\\ } % This is used in *.glo
% In the tex file, the \glossary command, with one argument, puts that
% argument as the first argument of \glossaryentry (the second being a
% page number) in the *.glo file.  I am using this *.glo for an index
% of symbols.  Symbols in the Introduction are not included in this
% index, but only because the relevant \glossary{---} commands are commented.

\usepackage{ifthen,calc}
\newcounter{rfp}\newcounter{ones}\newcounter{tens}
\usepackage{refcount}

\newcommand{\sayfanumaraya}[1]{%
\setcounterpageref{rfp}{#1}%
\setcounter{ones}{\value{rfp}-(\value{rfp}/10)*10}%
\setcounter{tens}{\value{rfp}/10-(\value{rfp}/100)*10}%
\arabic{rfp}'%
\ifthenelse%
   {\value{ones}=6}%
   {ya}%
   {\ifthenelse%
       {\value{ones}=9}%
       {a}%
       {\ifthenelse%
           {\value{ones}=2\or\value{ones}=7}%
           {ye}%
           {\ifthenelse%
               {\value{ones}=0}%
               {\ifthenelse%
                   {\value{tens}=2\or\value{tens}=5}
                   {ye}
                   {e}}
               {e}}}}}
\newcommand{\sayfaya}[1]{sayfa \sayfanumaraya{#1}}

\newcommand{\sayfanumarada}[1]{%
%\setcounter{rfp}{\number\numexpr\getpagerefnumber{#1}}%
\setcounterpageref{rfp}{#1}%
\setcounter{ones}{\value{rfp}-(\value{rfp}/10)*10}%
\setcounter{tens}{\value{rfp}/10-(\value{rfp}/100)*10}%
\arabic{rfp}'%
\ifthenelse%
   {\value{ones}=6\or\value{ones}=9}%
   {da}%
   {\ifthenelse%
       {\value{ones}=3\or\value{ones}=4\or\value{ones}=5}%
       {te}%
       {\ifthenelse%
           {\value{ones}=0}%
           {\ifthenelse%
               {\value{tens}=7}%
               {te}
               {\ifthenelse%
                   {\value{tens}=4\or\value{tens}=6}%
                   {ta}
                   {\ifthenelse%
                       {\value{tens}=1\or\value{tens}=3\or\value{tens}=9}%
                       {da}%
                       {de}}}}%
           {de}}}}
\newcommand{\sayfada}[1]{sayfa \sayfanumarada{#1}}
\newcommand{\Sayfada}[1]{Sayfa \sayfanumarada{#1}}

\newcommand{\numaraya}[1]{%
\setcounterref{rfp}{#1}%
\setcounter{ones}{\value{rfp}-(\value{rfp}/10)*10}%
\setcounter{tens}{\value{rfp}/10-(\value{rfp}/100)*10}%
\arabic{rfp}'%
\ifthenelse%
   {\value{ones}=6}%
   {ya}%
   {\ifthenelse%
       {\value{ones}=9}%
       {a}%
       {\ifthenelse%
           {\value{ones}=2\or\value{ones}=7}%
           {ye}%
           {\ifthenelse%
               {\value{ones}=0}%
               {\ifthenelse%
                   {\value{tens}=2\or\value{tens}=5}
                   {ye}
                   {e}}
               {e}}}}}


\newcommand{\numarada}[1]{%
%\setcounter{rfp}{\number\numexpr\getpagerefnumber{#1}}%
\setcounterref{rfp}{#1}%
\setcounter{ones}{\value{rfp}-(\value{rfp}/10)*10}%
\setcounter{tens}{\value{rfp}/10-(\value{rfp}/100)*10}%
\arabic{rfp}'%
\ifthenelse%
   {\value{ones}=6\or\value{ones}=9}%
   {da}%
   {\ifthenelse%
       {\value{ones}=3\or\value{ones}=4\or\value{ones}=5}%
       {te}%
       {\ifthenelse%
           {\value{ones}=0}%
           {\ifthenelse%
               {\value{tens}=7}%
               {te}
               {\ifthenelse%
                   {\value{tens}=4\or\value{tens}=6}%
                   {ta}
                   {\ifthenelse%
                       {\value{tens}=1\or\value{tens}=3\or\value{tens}=9}%
                       {da}%
                       {de}}}}%
           {de}}}}

\newcommand{\numarayi}[1]{%
\setcounterref{rfp}{#1}%
\setcounter{ones}{\value{rfp}-(\value{rfp}/10)*10}%
\setcounter{tens}{\value{rfp}/10-(\value{rfp}/100)*10}%
\arabic{rfp}'%
\ifthenelse%
  {\value{ones}=1\or\value{ones}=5\or\value{ones}=8}%
  {i}%
  {\ifthenelse%
     {\value{ones}=2\or\value{ones}=7}%
     {yi}%
     {\ifthenelse%
        {\value{ones}=3\or\value{ones}=4}%
        {\"u}%
        {\ifthenelse%
           {\value{ones}=6}%
           {y\i}%
           {\ifthenelse%
              {\value{ones}=9}%
              {u}%
              {\ifthenelse%
                 {\value{tens}=7\or\value{tens}=8}%
                 {i}%
                 {\ifthenelse%
                    {\value{tens}=2\or\value{tens}=5}%
                    {yi}%
                    {\ifthenelse%
                       {\value{tens}=1\or\value{tens}=3}%
                       {u}%
                       {\ifthenelse%
                          {\value{tens}=4\or\value{tens}=6\or\value{tens}=9}%
                          {\i}
                          {\"u}}}}}}}}}}

\newcommand{\numaranin}[1]{%
\setcounterref{rfp}{#1}%
\setcounter{ones}{\value{rfp}-(\value{rfp}/10)*10}%
\setcounter{tens}{\value{rfp}/10-(\value{rfp}/100)*10}%
\arabic{rfp}'%
\ifthenelse%
  {\value{ones}=1\or\value{ones}=5\or\value{ones}=8}%
  {in}%
  {\ifthenelse%
     {\value{ones}=2\or\value{ones}=7}%
     {nin}%
     {\ifthenelse%
        {\value{ones}=3\or\value{ones}=4}%
        {\"un}%
        {\ifthenelse%
           {\value{ones}=6}%
           {n\i n}%
           {\ifthenelse%
              {\value{ones}=9}%
              {un}%
              {\ifthenelse%
                 {\value{tens}=7\or\value{tens}=8}%
                 {in}%
                 {\ifthenelse%
                    {\value{tens}=2\or\value{tens}=5}%
                    {nin}%
                    {\ifthenelse%
                       {\value{tens}=1\or\value{tens}=3}%
                       {un}%
                       {\ifthenelse%
                          {\value{tens}=4\or\value{tens}=6\or\value{tens}=9}%
                          {\i n}
                          {\"u}}}}}}}}}}

\newcommand{\Teorem}[1]{Teorem \ref{#1}}
\newcommand{\Teoreme}[1]{Teorem \numaraya{#1}}
\newcommand{\Teoremde}[1]{Teorem \numarada{#1}}
\newcommand{\Teoremi}[1]{Teorem \numarayi{#1}}
\newcommand{\Teoremin}[1]{Teorem \numaranin{#1}}

%\usepackage{chngcntr}
%\counterwithout{figure}{chapter}
\newcommand{\Sekil}[1]{\c Sekil \ref{#1}}
\newcommand{\Sekle}[1]{\c Sekil \numaraya{#1}}
\newcommand{\Sekilde}[1]{\c Sekil \numarada{#1}}

 \renewcommand{\thefootnote}{\fnsymbol{footnote}}

\usepackage[neverdecrease]{paralist}

\usepackage{pstricks,pst-node,pst-tree}

\newcommand{\alis}{\textbf{al\i\c st\i rma}}

\usepackage{amsmath,amssymb,amsthm,url,upgreek}
\allowdisplaybreaks

\DeclareMathOperator{\isomorphisms}{Isom}
\newcommand{\Isom}[1][V]{\isomorphisms(#1)}
\DeclareMathOperator{\generallinear}{GL}
\newcommand{\GL}[1][V]{\generallinear(#1)}
\newcommand{\Orth}[1][V]{\mathrm O(#1)}
\newcommand{\SO}[1][V]{\mathrm{SO}(#1)}
\DeclareMathOperator{\matrices}{Mat}
\newcommand{\Mat}[1][n\times n]{\matrices_{#1}(\R)}
\newcommand{\trf}[1]{\mathrm T_{#1}}
\newcommand{\transp}{^\mathrm t}
\newcommand{\Id}[1][n]{\mathrm I_{#1}}
\newcommand{\trl}[1]{\mathrm t_{#1}}
\newcommand{\Trl}[1][V]{\mathrm T(#1)}
\newcommand{\gid}{\mathrm e}
\newcommand{\nsubgp}{\triangleleft}
\newcommand{\sdirect}{\rtimes}
\newcommand{\inv}{{}^{-1}}
\newcommand{\Zmod}[1]{\Z_{#1}}
\newcommand{\me}{\mathrm e}
\newcommand{\mi}{\,\mathrm i\,}
\newcommand{\conj}[1]{\overline{#1}}
\newcommand{\rfl}[1]{\mathrm r_{#1}}
\newcommand{\rot}[1]{\uprho_{#1}}
\newcommand{\abs}[1]{\lvert#1\rvert}

%\usepackage{cmbright}
%\renewcommand{\familydefault}{\rmdefault}
%\renewcommand{\rmdefault}{cmr}
%\DeclareMathAlphabet{\mathnormal}{OT1}{cmss}{m}{n}
%\usepackage{unicode-math}
\DeclareMathVersion{false}
\SetSymbolFont{operators}{false}{OT1}{cmbr}{m}{n}
\SetSymbolFont{letters}{false}{OML}{cmbrm}{m}{it}
\SetSymbolFont{symbols}{false}{OMS}{cmbrs}{m}{n}
%\SetSymbolFont{largesymbols}{false}{OMX}{cmex}{m}{n}
\usepackage{relsize}
\newenvironment{falseproof}{%
\begin{quotation}\mathversion{false}\relscale{0.9}}%
{\end{quotation}}

\newtheorem{theorem}{Teorem}
%\newtheorem{proposition}{\"Onerme}
%\newtheorem*{corollary}{Sonu\c c}
\newtheorem*{lemma}{Lemma}
%\newtheorem*{theorem*}{Teorem}
%\newtheorem{axiom}{Aks\.iyom}
%\newtheorem{axiom}{AKS\.IYOM}
%\theoremstyle{definition}
\theoremstyle{remark}

%  The template for the following is taken from
%  http://en.wikibooks.org/wiki/LaTeX/Theorems 
%  The punctuation dot is needed (default is apparently no dot).
%  ``Space after theorem head'' is needed (otherwise errors).
\newtheoremstyle{xca}% name of the style to be used
  {}% measure of space to leave above the theorem. E.g.: 3pt
  {}% measure of space to leave below the theorem. E.g.: 3pt
  {\sffamily}% name of font to use in the body of the theorem
  {}% measure of space to indent
  {\sffamily\itshape}% name of head font
  {.}% punctuation between head and body
  {0.5em}% space after theorem head; " " = normal interword space
  {}% Manually specify head

\theoremstyle{xca}
\newtheorem{xca}{Al\i\c st\i rma}

\newcommand{\ktk}[1][]{\begin{xca}Teoremi kan\i tlay\i n.#1\end{xca}}
\newcommand{\ksk}[1][]{\begin{xca}Sonucu kan\i tlay\i n.#1\end{xca}}
%\newcommand{\kok}[1][]{\begin{xca}\"Onermeyi kan\i tlay\i n.#1\end{xca}}
\newcommand{\klk}[1][]{\begin{xca}Lemmay\i\ kan\i tlay\i n.#1\end{xca}}

\usepackage{bm}

\newcommand{\lto}{\Rightarrow}
\newcommand{\liff}{\Leftrightarrow}
\newcommand{\Forall}[1]{\forall{#1}\;}
\newcommand{\Exists}[1]{\exists{#1}\;}

\newcommand{\norm}[1]{\lVert#1\rVert}
%\newcommand{\denk}{\;\text{denktir}\;}
\newcommand{\denk}{\quad\text{denktir}\quad}
\newcommand{\universe}{\mathbf V}
%\newcommand{\inv}{^{-1}}
%\newcommand{\inv}[1]{{#1}^{-1}}
\newcommand{\conv}[1]{\breve{#1}}
\newcommand{\on}{\mathbf{ON}}
\newcommand{\cn}{\mathbf{KN}}
\newcommand{\zf}{\mathrm{ZF}}
\newcommand{\ac}{\mathrm{AC}}
\newcommand{\zfc}{\mathrm{ZFC}}
\newcommand{\ch}{\mathrm{KH}}
\newcommand{\gch}{\mathrm{GKH}}
\newcommand{\ord}{\operatorname{ord}}
\DeclareMathOperator{\cardinal}{kard}
\newcommand{\card}[1]{\cardinal(#1)}
%\newcommand{\cardsum}{+^{\mathrm k}}
\newcommand{\cardsum}{\oplus}
%\newcommand{\cardprod}{\cdot^{\mathrm k}}
\newcommand{\cardprod}{\otimes}
\newcommand{\cof}[1]{\operatorname{kf}(#1)}

\renewcommand{\deg}{\operatorname{der}}
\renewcommand{\max}{\operatorname{maks}}
\renewcommand{\vec}[1]{\bm{#1}}

\newcommand{\symdiff}{\vartriangle}
\newcommand{\comp}{^{\mathrm c}}

\usepackage{mathrsfs}
\newcommand{\pow}[2][]{\mathscr P^{#1}(#2)}
\newcommand{\powf}[1]{\mathscr P_{\upomega}(#1)}
%\usepackage{MnSymbol}
%\newcommand{\pow}[1]{\powerset(#1)}

\newcommand{\included}{\subseteq}
\newcommand{\nincluded}{\nsubseteq}
\newcommand{\includes}{\supseteq}
\newcommand{\pincluded}{\subset}

\newcommand{\N}{\mathbb N}
\newcommand{\Z}{\mathbb Z}
\newcommand{\Q}{\mathbb Q}
\newcommand{\Qp}{\mathbb Q^+}
\newcommand{\R}{\mathbb R}
\newcommand{\Rp}{\mathbb R^+}
\newcommand{\C}{\mathbb C}


\newcommand{\divides}{\mid}
\newcommand{\ndivides}{\nmid}

\newcommand*{\twoheadrightarrowtail}{\mathrel{\rightarrowtail\kern-1.9ex\twoheadrightarrow}}

\renewcommand{\emptyset}{\varnothing}
\renewcommand{\phi}{\varphi}
\renewcommand{\leq}{\leqslant}
\renewcommand{\geq}{\geqslant}
\renewcommand{\setminus}{\smallsetminus}

%\renewcommand{\theequation}{\fnsymbol{equation}}
\renewcommand{\epsilon}{\varepsilon}
\usepackage[all]{xy}

\begin{document}
%\frontmatter
\title{D\"uzlemin \.Izometrileri\\
TASLAK}
\author{David Pierce}
\date{\today}
%\date{8 \c Subat 2016}
\publishers{Matematik B\"ol\"um\"u\\
Mimar Sinan G\"uzel Sanatlar \"Universitesi\\
\.Istanbul\\
\url{dpierce@msgsu.edu.tr}\\
\url{http://mat.msgsu.edu.tr/~dpierce/}}

\maketitle

%\tableofcontents

%\listoffigures

$V$, $\R^n$ ger\c cel i\c c\c carp\i m uzay\i\ olsun.
$V$'nin bir \textbf{izometri,}
uzunluklar\i\ koruyan bir e\c sle\c smedir.
\begin{align*}
  \Isom&=\{\text{$V$'nin izometrileri}\},\\
\GL&=\{\text{$V$'nin lineer e\c sle\c slemeleri}\},\\
\Orth&=\Isom\cap\GL
\end{align*}
olsun.
Bunlar, bile\c ske alma alt\i nda \emph{grupturlar.}

\begin{theorem}
  $\Orth=\{g\in\Isom\colon g(\vec 0)=\vec0\}$.
\end{theorem}

\begin{proof}
$\Isom$ grubunun her $g$ eleman\i,
k\"o\c seleri $\vec0$, $\vec a$, $\vec b$, ve $\vec a+\vec b$ olan her paralelkenar\i n\i n kenarlar\i n\i n ve k\"o\c segenlerinin uzunluklar\i n\i\ korur.
\"Ozel olarak $g(\vec0)=\vec0$ ise
\begin{equation*}
  g(\vec a+\vec b)=g(\vec a)+g(\vec b).\qedhere
\end{equation*}
\end{proof}

$A\in\Mat$ ise
\begin{align*}
\trf A&\colon V\to V,&
  \trf A(\vec x)&=A\vec x
\end{align*}
olsun.  Burada
\begin{equation*}
  \vec x=
  \begin{pmatrix}
    x_1\\\vdots\\x_n
  \end{pmatrix}
=
\begin{pmatrix}
  x_1&\cdots&x_n
\end{pmatrix}\transp.
\end{equation*}

\begin{theorem}
  $\begin{gathered}[t]
   \GL=\{\trf X\colon\det(X)\neq0\},\\
\Orth=\{\trf X\colon X\transp X=\Id\}.
  \end{gathered}$
\end{theorem}

E\u ger $\vec a\in V$ ise
\begin{align*}
  \trl{\vec a}&\colon V\to V,&
\trl{\vec a}(\vec x)=\vec a+\vec x
\end{align*}
olsun, ve
\begin{equation*}
  \Trl=\{\trl{\vec x}\colon\vec x\in V\}
\end{equation*}
olsun.
Burada $\trl{\vec a}$, $\vec a$ ile \textbf{\"otelemedir.}
O zaman
\begin{gather*}
  \Trl<\Isom,\\
\vec x\mapsto\trl{\vec x}\colon V\xrightarrow{\cong}\Trl.
\end{gather*}

\begin{theorem}
$\Trl\times\Orth\approx\Isom$,
yani bu iki k\"ume e\c sleniktir.
Asl\i nda
\begin{equation*}
  (\trl{\vec x},\xi)\mapsto\trl{\vec x}\circ\xi,
%\colon\Trl\times\Orth\xrightarrow{\approx}\Isom.
\end{equation*}
bir e\c slemedir.
\end{theorem}

\begin{proof}
Verilen g\"onderme, birebirdir (\alis).
G\"ondermenin tersi,
\begin{equation*}
  \xi\mapsto(\trl{\xi(\vec0)},\trl{-\xi(\vec0)}\circ\xi)
\end{equation*}
\c c\"unk\"u $\alpha\in\Isom$ ve $\beta=\trl{-\alpha(\vec0)}\circ\alpha$ ise
$\beta(\vec0)=\vec0$,
dolay\i s\i yla $\beta\in\Orth$;
ve $\alpha=\trl{\alpha(\vec0)}\circ\beta$.
\end{proof}

Bir $G$ grubunun $H$ ve $K$ altgruplar\i\ i\c cin
\begin{align*}
  G&=HK,&
H\cap K&=\{\gid\},
\end{align*}
olsun.  Bu \c sekilde
$(x,y)\mapsto xy\colon H\times K\xrightarrow{\approx} G$.
E\u ger
\begin{equation*}
 H\nsubgp G  
\end{equation*}
ise, o zaman
\begin{equation*}
  G=H\sdirect K
\end{equation*}
yazar\i z,
ve bu durumda $h_i\in H$, $k_i\in K$ ise
\begin{align*}
  h_1k_1\cdot h_2k_2&=h_1(k_1h_2k_1{}\inv)\cdot k_1k_2,&
k_1h_2k_1{}\inv&\in H.
\end{align*}
$G=HK$ durumunda $H\nsubgp G$ g\"ostermek i\c cin,
$k\in K$ oldu\u gunda $kHk\inv=H$ g\"ostermek yeter.

\begin{theorem}
$\Isom=\Trl\sdirect\Orth$. 
\end{theorem}

\begin{proof}
  $\Trl\nsubgp\Isom$ g\"ostermek yeter.
E\u ger $\vec a\in V$ ve $\alpha\in\Orth$ ise,
o zaman
\begin{equation*}
  \alpha\circ\trl{\vec a}\circ\alpha\inv=\trl{\alpha(\vec a)},
\end{equation*}
\c c\"unk\"u $\alpha$ lineer oldu\u gundan
\begin{equation*}
  \alpha\circ\trl{\vec a}\circ\alpha\inv(\vec x)
=\alpha(\vec a+\alpha\inv(\vec x))
=\alpha(\vec a)+\vec x
=\trl{\alpha(\vec a)}(\vec x).\qedhere
\end{equation*}
\begin{comment}

E\u ger $\vec b\in V$ ise
\begin{align*}
  (\trl{\vec b}\circ\alpha)\circ\trl{\vec a}\circ(\trl{\vec b}\circ\alpha)\inv
&= \trl{\vec b}\circ(\alpha\circ\trl{\vec a}\circ\alpha\inv)\circ\trl{-\vec b}\\
&= \trl{\vec b}\circ\trl{\alpha(\vec a)}\circ\trl{-\vec b}\\
&=\trl{\alpha(\vec a)}.\qedhere
\end{align*}

\end{comment}
\end{proof}

\c Sonuc olarak $\Isom\cong V\sdirect\Orth$.
\"Ozel olarak
\begin{equation*}
  \Isom[\R]\cong\R\sdirect{\Zmod2}
\end{equation*}
\c c\"unk\"u $\Orth[\R]\cong\{x\in\R\colon x^2=1\}=\{\pm1\}$.

Genelde
\begin{gather*}
  \SO=\{\trf X\colon\det X=1\},\\
\Orth=\{\trf X\colon\det X=\pm1\}.
\end{gather*}

\c Simdi $n=2$ olsun.  Bu durumda $V=\R^2$ ve
\begin{gather*}
  \SO=\{\text{$\vec0$ etraf\i nda d\"ond\"urmeler}\},\\
\Orth\setminus\SO
=\{\text{$\vec0$'dan ge\c cen do\u grular etraf\i ndan yans\i malar}\}.
\end{gather*}
Asl\i nda
\begin{gather*}
  \Orth=\left\{\trf X\colon
X=    \begin{pmatrix}
      \cos\theta&\mp\sin\theta\\\sin\theta&\pm\cos\theta
    \end{pmatrix}\land0\leq\theta<2\uppi\right\},\\
  \SO=\left\{\trf X\colon
X=    \begin{pmatrix}
      \cos\theta&-\sin\theta\\\sin\theta&\cos\theta
    \end{pmatrix}\land0\leq\theta<2\uppi\right\}.
\end{gather*}
K\i saltma olarak
\begin{align*}
  \rot{\theta}&=\trf{
    \begin{pmatrix}
      \cos\theta&-\sin\theta\\\sin\theta&\cos\theta
    \end{pmatrix}},&
\rfl0&=\trf{
    \begin{pmatrix}
      1&0\\0&-1
    \end{pmatrix}}
\end{align*}
olsun.  $V=\C$ d\"u\c s\"un\"ulebilir, ve bu \c sekilde
\begin{align*}
	\rot{\theta}(z)&=\me^{\mi\theta}\cdot z,&
	\rfl0(z)&=\conj z.
\end{align*}

\begin{theorem}
  $\Orth=\SO\sdirect\langle \rfl0\rangle$.
\end{theorem}

\begin{proof}
  $\rfl0{}\inv=\rfl0$ ve $\rfl0\circ\rot{\theta}\circ \rfl0=\rot{-\theta}$.
	(Bu e\c sitliker, ya matrisler ya da karma\c s\i k say\i lar ile do\u grulanabilir.)
\end{proof}

\c Simdi
\begin{equation*}
\Isom=\Trl\sdirect\bigl(\SO\sdirect\langle \rfl0\rangle\bigr).  
\end{equation*}
\"Ozel olarak $\Isom$ grubunun her $\alpha$ eleman\i\ i\c cin,
$V$'nin bir ve tek bir $\vec a$ eleman\i\ i\c cin,
bir ve tek bir $\theta$ a\c c\i s\i\ i\c cin
\begin{equation*}
  \alpha\in
\bigl\{\trl{\vec a}\circ\rot{\theta},\ \trl{\vec a}\circ\rot{\theta}\circ \rfl0\bigr\}.
\end{equation*}
Ama ba\c ska bir bi\c cimde $\alpha$ daha iyi anla\c s\i labilir.

Tekrar $\vec a\in V$ ve $\rot{\theta}\in\SO$ olsun.  Burada
\begin{align*}
	\trl{\vec a}{}\inv&=\trl{-\vec a},&
	\rot{\theta}\inv(\vec x)&=\rot{-\theta}
	(\vec x).
\end{align*}
O zaman
\begin{itemize}
\item
\fbox{$\trl{\vec a}$,} $\vec a$ ile \textbf{\"otelemedir;}
\item
\fbox{$\rot{\theta}$,}
$\vec0$ noktas\i\ etraf\i nda $\theta$ a\c c\i s\i ndan \textbf{d\"ond\"urmedir;}
\item
\fbox{$\rfl0$,}
$\{t(1,0)\colon t\in\R\}$ do\u grusu etraf\i nda \textbf{yans\i mad\i r;}
\item
\fbox{$\trl{\vec a}\circ\rot{\theta}\circ\trl{-\vec a}$,}
$\vec a$ noktas\i\ etraf\i nda $\theta$ a\c c\i s\i ndan \textbf{d\"ond\"urmedir;}
\item
\fbox{$\rot{\theta}\circ \rfl0\circ\rot{-\theta}$,}
$\{t(\cos\theta,\sin\theta)\colon t\in\R\}$ do\u grusu etraf\i nda \textbf{yans\i mad\i r;}
\item
\fbox{$\trl{\vec a}\circ\rot{\theta}\circ \rfl0\circ\rot{-\theta}\circ\trl{-\vec a}$,}
$\{\vec a+t(\cos\theta,\sin\theta)\colon t\in\R\}$ do\u grusu etraf\i nda \textbf{yans\i mad\i r;}
\item
$b\in\R$ ise
\fbox{$\trl{b(\cos\theta,\sin\theta)}\circ\trl{\vec a}\circ\rot{\theta}\circ \rfl0\circ\rot{-\theta}\circ\trl{-\vec a}$,}
$b(\cos\theta,\sin\theta)$ ile
$\{\vec a+t(\cos\theta,\sin\theta)\colon t\in\R\}$ do\u grusu etraf\i nda \textbf{kayd\i rma yans\i mas\i d\i r.}
\end{itemize}

\begin{lemma}
Her $\theta$ a\c c\i s\i\ 
ve $a$ karma\c s\i k say\i s\i\ i\c cin
\begin{gather*}
\rfl0\circ\rot{\theta}=\rot{-\theta}\circ\rfl0,\\
\rfl0\circ\trl a=\trl{\conj a}\circ\rfl0,\\
\rot{\theta}\circ\trl a=\trl{a\me^{\mi\theta}}\circ\rot{\theta}.
\end{gather*}
\end{lemma}

\begin{theorem}
$\Isom[\R^2]$ grubunun a\c sik\^ar olmayan her eleman\i,
ya \"oteleme, ya d\"ond\"urme, ya yans\i ma, ya da kayd\i rma yans\i mas\i d\i r.
\end{theorem}

\begin{proof}
G\"ord\"u\u g\"um\"uz gibi
$\Isom[\C]$ grubunun her $\alpha$ eleman\i\ i\c cin,
$\C$'nin bir $a$ eleman\i\ i\c cin,
bir $\theta$ a\c c\i s\i\ i\c cin
ya $\alpha=\trl a\circ\rot{\theta}$,
ya da $\alpha=\trl a\circ\rot{\theta}\circ \rfl0$.

Genelde $\C$'nin her $b$ eleman\i\ i\c cin
\begin{gather*}
	\trl a\circ\rot{\theta}=\trl b\circ\rot{\theta}\circ\trl{-b}
\iff\trl{a-b}\circ\rot{\theta}=\rot{\theta}\circ\trl{-b},\\
a-b=-b\me^{\mi\theta}
\iff a=b(1-\me^{\mi\theta}).
\end{gather*}
O zaman $\theta\neq0$ ise lemmaya g\"ore
\begin{equation*}
%\trl a\circ\rot{\theta}=\trl{\frac a{1-\me^{\mi\theta}}}\circ\rot{\theta}\circ\trl{\frac{-a}{1-\me^{\mi\theta}}}
\trl a\circ\rot{\theta}=\trl{a/(1-\me^{\mi\theta})}\circ\rot{\theta}\circ\trl{-a/(1-\me^{\mi\theta})};
\end{equation*}
\"ozel olarak $\trl a\circ\rot{\theta}$, bir d\"ond\"urmedir.

Lemmaya g\"ore de
\begin{equation*}
\rot{\theta}\circ\rfl0=\rot{\theta/2}\circ\rfl0\circ\rot{-\theta/2},
\end{equation*}
ve bu bir yans\i mad\i r.
Genelde
\begin{equation*}
\rfl{\theta}=\rot{\theta}\circ\rfl0\circ\rot{-\theta}
\end{equation*}
olsun.  O zaman bir kayd\i rma yans\i mas\i n\i
\begin{equation*}
\trl{r\me^{\mi\theta}}\circ\trl a\circ\rfl{\theta}\circ\trl{-a}
\end{equation*}
bi\c ciminde yazabiliriz.
Lemman\i n ikinci e\c sitli\u gini
\begin{equation*}
\rfl0\circ\trl a=\trl{\rfl0(a)}\circ\rfl0
\end{equation*}
bi\c ciminde yazabiliriz,
ve benzer \c sekilde
\begin{equation*}
\rfl{\theta}\circ\trl a=\trl{\rfl{\theta}(a)}\circ\rfl0.
\end{equation*}
\c Simdi $\phi$ a\c c\i s\i\ ve $\C$'nin $a$ eleman\i\ verilirse
\begin{align*}
	b&=\frac{a+\rfl{\phi}(a)}2,&
	c&=\frac{a-\rfl{\phi}(a)}4
\end{align*}
olsun.  O zaman $b=\abs b\cdot\me^{\mi\phi}$ ve
\begin{equation*}
\trl b\circ\trl c\circ\rfl{\phi}\circ\trl{-c}
=\trl{b+c+\rfl{\phi}(-c)}\circ\rfl{\phi}
=\trl a\circ\rot{2\phi}\circ\rfl0.
\end{equation*}
Burada $2\phi=\theta$ olabildi\u ginde
$\trl a\circ\rot{\theta}\circ\rfl0$ izometrisi,
$b=0$ durumunda yans\i mad\i r,
di\u ger durumlarda kayd\i rma yans\i mas\i d\i r.
\end{proof}

\end{document}