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\usepackage[turkish]{babel}
\usepackage{amsmath,url,amsthm,amssymb,bm}
\newcommand{\inv}{^{-1}}
\DeclareMathOperator{\Adj}{Ek}
\newcommand{\Det}[1]{\det\left(#1\right)}

\newcommand{\stnd}[1]{\mathbb{#1}}
\newcommand{\R}{\stnd R}
\newcommand{\C}{\stnd C}
\newcommand{\Z}{\stnd Z}
\newcommand{\N}{\stnd N}
\newcommand{\Zmod}[1]{\Z/#1\Z}

\usepackage{nicefrac}
\newcommand{\matfrac}[2]{\nicefrac{#1}{#2}}
\newcommand{\included}{\subseteq}
\newcommand{\lspan}[1]{\langle#1\rangle}

\renewcommand{\emptyset}{\varnothing}
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\swapnumbers

\newtheorem{theorem}{Teorem}
\newtheorem{corollary}[theorem]{Sonu\c c}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Tan\i m}
\newtheorem{example}[theorem]{\"Ornek}
\newtheorem*{problem}{Problem}

\theoremstyle{remark}
\newtheorem*{solution}{\c C\"oz\"um}

\begin{document}
\title{Uzaylar\i n Toplamlar\i\ ve B\"ol\"umleri}
\subtitle{Lineer Cebir}
\author{David Pierce}
\date{26 Nisan 2017}
\publishers{Matematik B\"ol\"um\"u, MSGS\"U\\
%\mbox{}\\
%\url{dpierce@msgsu.edu.tr}\\
\url{http://mat.msgsu.edu.tr/~dpierce/}}

\maketitle

Bu notlarda $F$ bir cisimdir
(\"orne\u gin $\R$ ve $\C$ veya $\Z_p$ olabilir).
$U$, $V$, ve $W$,
$F$ \"uzerinde vekt\"or uzay\i d\i r.
Tahtaya yaz\i lan $\vec u$ ve $\vec v$ vekt\"orleri,
burada $\bm u$ ve $\bm v$ bi\c ciminde yaz\i l\i yor.

\section{Toplamlar}

\begin{definition}
E\u ger $V$ ve $W$,
$U$'nun altuzay\i\ ise,
onlar\i n \textbf{toplam\i,}
\begin{equation*}
  \{\bm v+\bm w\colon\bm v\in V\And \bm w\in W\}
\end{equation*}
k\"umesidir.  Bu k\"ume,
\begin{equation*}
V+W
\end{equation*}
olarak yaz\i l\i r.
\end{definition}

\begin{example}
  $F^4$'te
	\begin{equation}
	\left.\qquad
  \begin{aligned}\label{eqn:VW}
    V&=\lspan{(1,1,1,0),(1,-1,1,0)},\\
    W&=\lspan{(1,0,1,1),(1,0,1,-1)}
  \end{aligned}\qquad	
	\right\}
	\end{equation}
  ise
  \begin{equation*}
    V+W=\lspan{(1,1,1,0),(1,-1,1,0),(1,0,1,1),(1,0,1,-1)}.
  \end{equation*}
Bu uzay,
  \begin{equation*}
    \begin{pmatrix}
      1&1&1&1\\1&-1&0&0\\1&1&1&1\\0&0&1&-1
    \end{pmatrix}
  \end{equation*}
  matrisinin s\"utun uzay\i d\i r.
  Ayr\i ca
  \begin{multline*}
    \begin{pmatrix}
      1&1&1&1\\1&-1&0&0\\1&1&1&1\\0&0&1&-1
    \end{pmatrix}
    \xrightarrow[-R_1+R_3]{-R_1+R_2}
    \begin{pmatrix}
      1&1&1&1\\0&-2&-1&-1\\0&0&0&0\\0&0&1&-1
    \end{pmatrix}\\
    \xrightarrow{R_3\leftrightarrow R_4}
    \begin{pmatrix}
      1&1&1&1\\0&-2&-1&-1\\0&0&1&-1\\0&0&0&0
    \end{pmatrix}    
  \end{multline*}
  oldu\u gundan
	\begin{compactenum}[(a)]
	\item
	$F$'de $2\neq0$ ise
  \begin{equation*}
    \{(1,1,1,0),(1,-1,1,0),
    (1,0,1,1)\}
  \end{equation*}
  k\"umesi, $V+W$'nin bir taban\i d\i r;
	\item
	$F$'de $2=0$ ise
  \begin{equation*}
    \{(1,1,1,0),(1,0,1,1)\}
  \end{equation*}
  k\"umesi, $V+W$'nin bir taban\i d\i r.
\end{compactenum}	
\end{example}

\begin{theorem}
  E\u ger $V\included U$ ve $W\included U$ ise,
  o zaman $V+W$ k\"umesi,
	\begin{enumerate}[1)]
	\item
	$U$'nun bir altuzay\i d\i r,
	\item
	$U$'nun $V$'yi ve $W$'yi kapsayan en k\"u\c c\"uk altuzay\i d\i r.
	\end{enumerate}
\end{theorem}

\begin{proof}
\begin{asparaenum}
\item
Kan\i tlanacak \"u\c c ko\c sul vard\i r.
  \begin{compactenum}[(a)]
  \item
    $V+W$ bo\c s de\u gildir, \c c\"unk\"u
    \begin{align*}
      \bm0&\in V\cap W,&
      \bm0+\bm0&=\bm0
    \end{align*}
    oldu\u gundan
    $\bm0\in V+W$.
  \item
    $\bm u_1\in V+W$ ve $\bm u_2\in V+W$ ise
    $V$'nin baz\i\ $\bm v_i$ elemanlar\i\
    ve $W$'nin baz\i\ $\bm w_i$ elemanlar\i\ i\c cin
    \begin{equation*}
      \bm u_i=\bm v_i+\bm w_i,
    \end{equation*}
    dolay\i s\i yla
    (vekt\"orler toplamas\i\ birle\c smeli ve de\u gi\c smeli oldu\u gundan)
    \begin{equation*}
      \bm u_1+\bm u_2
      =(\bm v_1+\bm w_1)+(\bm v_2+\bm w_2)
      =(\bm v_1+\bm v_2)+(\bm w_1+\bm w_2).
    \end{equation*}
    $U$ ve $V$'nin her biri toplama alt\i nda kapal\i\ oldu\u gundan
    \begin{align*}
      \bm v_1+\bm v_2&\in V,&
      \bm w_1+\bm w_2&\in W,
    \end{align*}
    dolay\i s\i yla $\bm u_1+\bm u_2\in V+W$.
  \item
    Benzer \c sekilde $\bm u\in V+W$ ve $t\in F$ ise
    $V$'nin bir $\bm v$ eleman\i\
    ve $W$'nin bir $\bm w$ eleman\i\ i\c cin
    \begin{equation*}
      \bm u=\bm v+\bm w,
    \end{equation*}
    dolay\i s\i yla
    (skalerle \c carpma da\u g\i lmal\i\ oldu\u gundan)
    \begin{equation*}
t\bm u=t(\bm v+\bm w)=t\bm v+t\bm w.
    \end{equation*}
    $U$ ve $V$'nin her biri \c carpma alt\i nda kapal\i\ oldu\u gundan
    \begin{align*}
      t\bm v&\in V,&
      t\bm w&\in W,
    \end{align*}
    dolay\i s\i yla $t\bm u\in V+W$.
  \end{compactenum}
	\item
$U'$, $U$'nun $V$'yi ve $W$'yi kapsayan bir altuzay\i\ olsun.
E\u ger $\bm u\in V+W$ ise, o zaman
    $V$'nin bir $\bm v$ eleman\i\
    ve $W$'nin bir $\bm w$ eleman\i\ i\c cin
		\begin{align*}
      \bm u&=\bm v+\bm w,&\bm v&\in U',&\bm w&\in U',&\bm v+\bm w&\in U',
		\end{align*}
    dolay\i s\i yla $\bm u\in U'$.\qedhere
	\end{asparaenum}
\end{proof}

\begin{corollary}
$C\included U$ ve $D\included U$ ise
\begin{equation*}
\lspan C+\lspan D=\lspan{C\cup D}.
\end{equation*}
\end{corollary}

\begin{proof}
  Al\i\c st\i rma.
\end{proof}

\begin{theorem}
  $V\times W$ kartezyan \c carp\i m\i,
  \begin{align*}
    (\bm v_1,\bm w_1)+(\bm v_2,\bm w_2)
    &=(\bm v_1+\bm v_2,\bm w_1+\bm w_2),\\
t(\bm v,\bm w)&=(t\bm v,t\bm w)
  \end{align*}
  kurallar\i yla vekt\"or uzay\i\ oluyor.
\end{theorem}

\begin{proof}
  Al\i\c st\i rma.
\end{proof}

\begin{definition}
Vekt\"or uzay\i\ olarak anla\c s\i l\i nca $V\times W$ \c carp\i m\i
\begin{equation*}
  V\oplus W
\end{equation*}
olarak yaz\i l\i r
ve ona
$V$ ve $W$'nin \textbf{direkt toplam\i} denir.
\end{definition}

\begin{example}
$F^2=F\oplus F$.
\end{example}

\begin{example}
$F^{k+m}=F^k\oplus F^m$.
\end{example}

\begin{example}
  $F^n=\underbrace{F\oplus\dots\oplus F}_n$.
\end{example}

\begin{example}
  $\Z$ cisim de\u gildir,
  \c c\"unk\"u \c carpmaya g\"ore sadece $1$ ve $-1$'in tersleri vard\i r;
  ama $\underbrace{\Z\oplus\dots\oplus\Z}_n$ toplamlar\i\ olu\c sturulabilir.
\end{example}

\begin{definition}
$U$'dan $V$'ye giden,
\begin{align*}
  h(\bm u_1+\bm u_2)&=h(\bm u_1)+h(\bm u_2),\\
  h(t\bm u)&=t\cdot h(\bm u)
\end{align*}
kurallar\i n\i\ sa\u glayan bir $h$ g\"ondermesine
\textbf{do\u grusal d\"on\"u\c s\"um} denir.
E\u ger $h$ birebir ve $V$'yi \"orten ise,
o zaman $h$ alt\i nda $U$ ve $V$ \textbf{izomorftur} ve
\begin{equation*}
U\cong V
\end{equation*}
yaz\i l\i r.
Bazen izomorfluk, e\c sitlik olarak anla\c s\i l\i r.
\end{definition}

\begin{theorem}
E\u ger $V\included U$ ve $W\included U$ ise,
o zaman
$V\oplus W$'den $V+W$'ye giden
\begin{equation*}
  (\bm v,\bm w)\mapsto\bm v+\bm w
\end{equation*}
g\"ondermesi,
$V+W$'yi \"orten bir $h$ do\u grusal d\"on\"u\c s\"um\"ud\"ur,
ve bu d\"on\"u\c s\"um i\c cin
a\c sa\u g\i daki ko\c sullar birbirine denktir:
\begin{enumerate}
\item
  $h$ birebirdir.
\item
  $V\cap W=\{\bm0\}$.
\item
  $h(\bm v,\bm w)=\bm 0$ denkleminin 
	sadece a\c sik\^ar $(\bm0,\bm0)$ \c c\"oz\"um\"u vard\i r.
\item
  E\u ger $B$, $V$'nin bir taban\i\ ise ve $C$, $W$'nin bir taban\i\ ise,
  o zaman $B\cup C$, $V+W$'nin bir taban\i d\i r.
\item
  E\u ger $B\included V$, $C\included W$, ve her biri do\u grusal ba\u g\i ms\i z ise,
  o zaman $B\cup C$ de do\u grusal ba\u g\i ms\i zd\i r.
\end{enumerate}
\end{theorem}

\begin{proof}
  Al\i\c st\i rma.
\end{proof}

\begin{definition}
Teoremde $h$ birebir ise $V+W$ toplam\i, 
$V$ ve $W$'nin \textbf{i\c c direkt toplam\i d\i r}
ve $V\oplus W$ olarak yaz\i labilir.
\end{definition}

\begin{example}
\eqref{eqn:VW} tan\i mlar\i\ alt\i nda
\begin{equation*}
V+W=V\oplus\lspan{(1,0,1,1)};
\end{equation*}
$F$'de $2=0$ ise $V+W=V\oplus W$.
\end{example}

\section{B\"ol\"umler}

\begin{example}
  $n\in\N$ ise
  \begin{equation*}
    n\Z=\{nx\colon x\in\Z\}
  \end{equation*}
  tan\i m\i n\i\ yapar\i z.
  Bu durumda $a\in\Z$ ise
  \begin{equation*}
    [a]
		%=a+n\Z
		=\{a+x\colon x\in\Z\}
  \end{equation*}
  olsun.  O halde
  \begin{equation*}
    \Z_n=\{[x]\colon x\in\Z\},
  \end{equation*}
  ve bu k\"ume i\c cin
  \begin{equation*}
    \Zmod n
  \end{equation*}
  ifadesi kullan\i labilir.
  Bu k\"umede toplama ve \c carpma
  \begin{align*}
    [a]+[b]&=[a+b],\\
    [a]\cdot[b]&=[ab]
  \end{align*}
  e\c sitlikleriyle tan\i mlanabilir,
  \c c\"unk\"u
  \begin{equation*}
    [a]=[b]\iff a-b\in n\Z,
  \end{equation*}
  dolay\i s\i yla
  \begin{equation*}
      [a_1]=[a_2]\And[b_1]=[b_2]
  \end{equation*}
  ise
  \begin{equation*}
    a_1-a_2\in n\Z\And b_1-b_2\in n\Z,
  \end{equation*}
  ve sonu\c c olarak
  \begin{gather*}
    (a_1-a_2)+(b_1-b_2)\in n\Z,\\
    (a_1+b_1)-(a_2+b_2)\in n\Z,\\
    [a_1+b_1]=[a_2+b_2],
  \end{gather*}
ve  ayr\i ca
  \begin{gather*}
    (a_1-a_2)\cdot b_1+ a_2\cdot(b_1-b_2)\in n\Z,\\
    a_1b_1-a_2b_2\in n\Z,\\
      [a_1b_1]=[a_2b_2].
  \end{gather*}
  \c Simdi
	\begin{align*}
[x]^1&=[x],&[x]^{k+1}&=[x]^k\cdot[x]		
	\end{align*}\
	\"ozyinelemeli tan\i m\i yla
	  $(\Zmod n)\times\N$ kartezyan \c carp\i m\i ndan
$\Zmod n$ b\"ol\"um\"ne giden
\begin{equation*}
  ([x],y)\mapsto[x]^y
\end{equation*}
g\"ondermesi tan\i mlanabilir.
Ayr\i ca t\"umevar\i mla
  \begin{equation*}
    [x]^y=[x^y].
  \end{equation*}
Bununla beraber
\begin{equation*}
  [x]^{[y]}=[x^y]
\end{equation*}
kural\i yla $\Zmod n$ b\"ol\"um\"unda
$2$-konumlu
\begin{equation*}
  ([x],[y])\mapsto[x]^{[y]}
\end{equation*}
i\c slemi tan\i mlanamaz \c c\"unk\"u, \"orne\u gin
\begin{align*}
1&\equiv4,&2^1&\equiv2,&2^4&\equiv1\pmod3,
\end{align*}
dolay\i s\i yla $n=3$ durumunda
\begin{align*}
  [1]&=[4],&[2^1]&\neq[2^4].
\end{align*}
K\i saca $\Zmod n$ b\"ol\"um\"unde
$([x],[y])\mapsto[x]^{[y]}$ i\c slemi \textbf{iyitan\i ml\i}
de\u gildir:
b\"oyle bir i\c slem yoktur.
\end{example}

\begin{definition}
E\u ger $V\included U$ ve $\bm a\in U$ ise
\begin{equation*}
  \bm a+V=\{\bm a+\bm v\colon v\in V\}.
\end{equation*}
\end{definition}

\begin{definition}
E\u ger $V\included U$ ise
\begin{equation*}
  U/V=\{\bm u+V\colon\bm u\in U\}.
\end{equation*}
\end{definition}

\begin{theorem}
  $V\included U$ ise
  \begin{align*}
    (\bm u_1+V)+(\bm u_2+V)&=(\bm u_1+\bm u_2)+V,\\
    t(\bm u_1+V)&=t\bm u_1+V
  \end{align*}
  kurallar\i yla $U/V$ b\"ol\"um\"u,
  iyitan\i mlanm\i\c s bir vekt\"or uzay\i d\i r.
\end{theorem}

\begin{proof}
  Al\i\c st\i rma.
\end{proof}

\begin{example}
\eqref{eqn:VW} tan\i mlar\i\ alt\i nda
\begin{equation*}
(V+W)/V\cong\lspan{(1,0,1,1)};
\end{equation*}
$F$'de $2=0$ ise $(V+W)/V\cong W$.
\end{example}

\begin{example}
  E\u ger
  \begin{equation*}
    U=\lspan{(1,1,1,0),(1,-1,1,0),(1,0,1,1),(1,0,1,-1)}
  \end{equation*}
  ise
  \begin{multline*}
    \begin{pmatrix}
      1& 1&1& 1&1&0&0&0\\
			1&-1&0& 0&0&1&0&0\\
			1& 1&1& 1&0&0&1&0\\
			0& 0&1&-1&0&0&0&1
    \end{pmatrix}
    \xrightarrow[-R_1+R_3]{-R_1+R_2}\\
    \begin{pmatrix}
      1& 1& 1& 1& 1&0&0&0\\
			0&-2&-1&-1&-1&1&0&0\\ 
			0& 0& 0& 0&-1&0&1&0\\
			0& 0& 1&-1& 0&0&0&1
    \end{pmatrix}
    \xrightarrow{R_3\leftrightarrow R_4}\\
    \begin{pmatrix}
      1& 1& 1& 1& 1&0&0&0\\
			0&-2&-1&-1&-1&1&0&0\\ 
			0& 0& 1&-1& 0&0&0&1\\
			0& 0& 0& 0&-1&0&1&0
    \end{pmatrix}    
  \end{multline*}
  oldu\u gundan
  \begin{equation*}
    F^4/V\cong\lspan{(1,0,0,0)}.
  \end{equation*}
\end{example}

\begin{theorem}
  E\u ger $ B\cap C=\emptyset$
  ve $B\cup C$ do\u grusal ba\u g\i ms\i z ise
  \begin{equation*}
    \lspan{B\cup C}/\lspan B\cong\lspan C.
  \end{equation*}
\end{theorem}

\begin{proof}
$B=\{\bm b_1,\dots,\bm b_{\ell}\}$ ve
$C=\{\bm c_1,\dots,\bm c_m\}$
  olsun.
O zaman $\lspan{B\cup C}$ gergisinin her $\bm a$ eleman\i,
  tek bir \c sekilde,
  $u_i\in F$ ve $v_j\in F$ olmak \"uzere
  bir
  \begin{equation*}
    u_1\bm b_1+\dots+u_{\ell}\bm b_{\ell}+v_1\bm c_1+\dots+v_m\bm c_m
  \end{equation*}
  do\u grusal bile\c simi olarak yaz\i labilir.
  Bu durumda
  \begin{equation*}
    h(\bm a)=v_1\bm c_1+\dots+v_m\bm c_m
  \end{equation*}
  e\c sitli\u gi ile $\lspan{B\cup C}$ gergisinden
  $\lspan C$ gergisine giden bir $h$ do\u grusal d\"on\"u\c s\"um\"u
  tan\i mlanabilir.
  \c Simdi
  \begin{equation*}
    \bm a'=
    u_1'\bm b_1+\dots+u_{\ell}'\bm b_{\ell}+v_1'\bm c_1+\dots+v_m'\bm c_m
  \end{equation*}
  olsun.
  E\u ger
  \begin{equation}\label{eqn:a+B}
   \bm a'+\lspan B=\bm a+\lspan B 
  \end{equation}
ise, o zaman $\bm a'-\bm a\in\lspan B$,
  yani
  \begin{multline*}
    (u_1'-u_1)\bm b_1+\dots+(u_{\ell}'-u_{\ell})\bm b_{\ell}\\
    +(v_1'-v_1)\bm c_1+\dots+(v_m'-v_m)\bm c_m\in\lspan B.
  \end{multline*}
  Bu durumda $B\cup C$ do\u grusal ba\u g\i ms\i z oldu\u gundan
  \begin{equation*}
    v_1'-v_1=\dots=v_m'-v_m=0,
  \end{equation*}
  dolay\i s\i yla $v_1'=v_1$, \dots, $v_m'=v_m$, ve sonu\c c olarak
  \begin{equation}\label{eqn:h(a)}
    h(\bm a')=h(\bm a).
  \end{equation}
  Bu \c sekilde $\lspan{B\cup C}/\lspan B$ b\"ol\"um\"unden
  $\lspan C$ gergisine giden
  \begin{equation*}
    \tilde h(\bm u+\lspan B)=h(\bm u)
  \end{equation*}
  e\c sitli\u gi ile iyitan\i mlanm\i\c s do\u grusal d\"on\"u\c s\"um\"u
  vard\i r.
  Benzer \c sekilde \eqref{eqn:h(a)} e\c sitli\u gi do\u gru ise
  \eqref{eqn:a+B} e\c sitli\u gi de do\u grudur,
  dolay\i s\i yla $\tilde h$, birebirdir.
  Son olarak $\tilde h$ d\"on\"u\c s\"um\"un\"un
  $\lspan C$ gergisini \"orten oldu\u gu apa\c c\i kt\i r.
\end{proof}

\end{document}