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{scrartcl}
              
\usepackage{hfoldsty,relsize,paralist}
\usepackage[turkish]{babel}
\usepackage{amsmath,url,amsthm,amssymb,bm}
\newcommand{\R}{\mathbb R}
\newcommand{\lspan}[1]{\langle#1\rangle}
\newcommand{\inv}{^{-1}}
\DeclareMathOperator{\Adj}{Ek}
\newcommand{\Det}[1]{\det\left(#1\right)}

\usepackage{nicefrac}
\newcommand{\matfrac}[2]{\nicefrac{#1}{#2}}

\newtheorem{problem}{Problem}
\theoremstyle{definition}
\newtheorem*{solution}{\c C\"oz\"um}
\theoremstyle{remark}
\newtheorem*{remark}{Not}

\usepackage{verbatim}
%\let\solution=\comment
%\let\endsolution=\endcomment



\pagestyle{empty}
\begin{document}
\subtitle{S\i nav 2}
\title{Lineer Cebir (MAT \newstylenums{114})}
\author{David Pierce, MSGS\"U}
\date{12 May\i s 2017}
%\publishers{Matematik B\"ol\"um\"u, MSGS\"U}
%\mbox{}\\
%\url{dpierce@msgsu.edu.tr}\\
%\url{http://mat.msgsu.edu.tr/~dpierce/}

\maketitle\thispagestyle{empty}

\c C\"oz\"um y\"ontemlerinizi d\"u\c s\"unerek se\c cin.
%Dikkatle \c cal\i\c s\i n
\c C\"oz\"umlerinizi net bir \c sekilde yaz\i n.
M\"umk\"unse cevaplar\i n\i z\i\ kontrol edin.
\.Iyi \c cal\i\c smalar dilerim!

\begin{problem}
$V=\{x\in\R\colon x>0\}$ olsun.
Bildi\u gimiz gibi
toplaman\i n $(x,y)\mapsto xy$, skaler \c carpman\i n $(t,x)\mapsto x^t$ oldu\u gu zaman
$V$ vekt\"or uzay\i\ olur.
$\R^2$ uzay\i ndan $V$'ye giden bir $L$ fonksiyonu
\begin{equation*}
L(x,y)=2^{x+y}\cdot 4^{x-y}
\end{equation*}
ile tan\i mlans\i n, ve
\begin{align*}
	E&=\{(1,0),(0,1)\},&B&=\{2\}
\end{align*}
ile $\R^2$ uzay\i n\i n $E$ taban\i\ ve $V$'nin $B$ taban\i\ tan\i mlans\i n.
\begin{enumerate}[(a)]
\item
$L$'nin lineer oldu\u gunu g\"osterin.
\item
$L$'nin $E$ ve $B$'ye g\"ore matrisini bulun,
yani
\begin{equation*}
\bigl[L(x,y)\bigr]_B=A\bigl[(x,y)\bigr]_E
\end{equation*}
e\c sitli\u gini sa\u glayan $A$ matrisini bulun.
\end{enumerate}
\end{problem}

\begin{solution}
\begin{enumerate}[(a)]
\item
Hesaplar\i z:
\begin{multline*}
	L\bigl((x,y)+(s,t)\bigr)
=L(x+s,y+t)
=2^{x+s+y+t}\cdot 4^{x+s-y-t}\\
=2^{x+y}\cdot 4^{x-y}\cdot 2^{s+t}\cdot 4^{s-t}
=L(x,y)\cdot L(s,t),
\end{multline*}
\begin{equation*}
L\bigl(t(x,y)\bigr)
=L(tx,ty)
=2^{tx+ty}\cdot 4^{tx-ty}
={2^{x+y}}^t\cdot{4^{x-y}}^t
=L(x,y)^t.
\end{equation*}
\item
$A
=\begin{bmatrix}\bigl[L(1,0)\bigr]_B&\bigl[L(0,1)\bigr]_B\end{bmatrix}
=\begin{bmatrix}\bigl[2^3\bigr]_B&\bigl[2^{-1}\bigr]_B\end{bmatrix}
=\begin{bmatrix}3&-1\end{bmatrix}$.
\end{enumerate}
\end{solution}

\begin{remark}
\begin{equation*}
L(x,y)=2^{x+y}\cdot2^{2x-2y}=2^{3x-y}
\end{equation*}
kural\i\ da kullan\i labilir.
Problemde $x\mapsto 2^x$, $\R$'den $V$'ye giden izomorfizimdir.
Ayr\i ca $(x,y)\mapsto 3x-y$, $\R^2$ uzay\i ndan $\R$'ye giden
lineer d\"on\"u\c s\"umd\"ur,
dolay\i s\i yla $L$ de lineer d\"on\"u\c s\"umd\"ur.
\end{remark}


\newpage

\begin{problem}
$A=\begin{bmatrix}
a_{1\,1}&\cdots&a_{1\,6}\\
\vdots  &      &\vdots\\
a_{4\,1}&\cdots&a_{4\,6}
\end{bmatrix}$ ve
$\begin{bmatrix}
0&1&17&0&3&-1\\
0&0& 0&1&5& 4\\
0&0& 0&0&0& 0\\
0&0& 0&0&0& 0
\end{bmatrix}$
ile sat\i rca denk olsun.
$A$'n\i n s\"utunlar\i\ ve sat\i rlar\i\ i\c cin
\begin{align*}
	\bm b_j&=(a_{1\,j},\dots,a_{4\,j}),&
\bm c_i&=(a_{i\,1},\dots,a_{i\,6})
\end{align*}
k\i saltmalar\i\ kullan\i labilir.
S\i ras\i yla $A$'n\i n
\begin{enumerate}[(a)]
\item
sat\i r uzay\i,
\item
s\"utun uzay\i, ve
\item
$\{\bm x\in\R^6\colon A\bm x=\bm0\}$ s\i f\i r uzay\i
\end{enumerate}
i\c cin bir taban bulun.
\end{problem}

\begin{solution}
\begin{enumerate}[(a)]
\item
$\{(0,1,17,0,3,-1),(0,0,0,1,5,4)\}$
\item
$\{\bm b_2,\bm b_4\}$
\item\label{item:null}
$\{(1,0,0,0,0,0),(0,-17,1,0,0,0),(0,-3,0,-5,1,0),(0,1,0,-4,0,1)\}$
\end{enumerate}
\end{solution}

\begin{remark}
\eqref{item:null} \c s\i kk\i nda
\begin{equation*}
\begin{array}{*{13}r}
0x_1&+&1x_2&+&17x_3&+&0x_4&+&3x_5&+&-1x_6&=&0\\
0x_1&+&0x_2&+& 0x_3&+&1x_4&+&5x_5&+& 4x_6&=&0
\end{array}
\end{equation*}
homojen sisteminin \c c\"oz\"um k\"umesinin bir taban\i\ aran\i yor.
Her serbest de\u gi\c sken i\c cin taban\i n bir eleman\i\ vard\i r.
Serbest de\u gi\c skenler $x_1$, $x_3$, $x_5$, ve $x_6$.
Cevap kontrol edilebilir.
\end{remark}

\newpage

\begin{problem}
$\R^4$ uzay\i nda
\begin{align*}
&	\begin{aligned}
		\bm b_1&=(2,2,2,-4),\\
		\bm b_2&=(-1,0,0,0),\\
		\bm b_3&=(3,2,2,-4),
	\end{aligned}&
&\begin{aligned}
	\bm c_1&=(1,5,5,-10),\\
	\bm c_2&=(0,2,0,0),\\
	\bm c_3&=(5,6,9,-18),
\end{aligned}&
&\begin{aligned}
B&=\{\bm b_1,\bm b_2,\bm b_3\},\\
C&=\{\bm c_1,\bm c_2,\bm c_3\}		
\end{aligned}
\end{align*}
olsun.
A\c sa\u g\i daki vekt\"or uzaylar\i n\i n her biri i\c cin bir taban bulun.
\begin{enumerate}[(a)]
\item
$\lspan B\oplus\lspan C$
\item
$\lspan B+\lspan C$
\item
$\lspan{B\cup C}/\lspan B$
\item
$\R^n/\lspan{B\cup C}$
\end{enumerate}
\end{problem}
		
\begin{solution}
\"Once hesaplar\i z:
\begin{gather*}
\relscale{0.9}
\left[\begin{array}{c|c|c}\bm c_1&\bm c_2&\bm c_3\end{array}\right]=
\begin{bmatrix}1&0&5\\5&2&6\\5&0&9\\-10&0&-18\end{bmatrix}
\xrightarrow{2R_3+R_4}
\begin{bmatrix}1&0&5\\5&2&6\\5&0&9\\0&0&0\end{bmatrix}
\xrightarrow[-5R_1+R_3]{-5R_1+R_2}
\begin{bmatrix}1&0&5\\0&2&-19\\0&0&-16\\0&0&0\end{bmatrix}\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\left[\begin{array}{c|c|c|c|c|c|c}\bm b_1&\bm b_2&\bm b_3&\bm c_1&\bm c_2&\bm c_3&I\end{array}\right]
=
\begin{bmatrix}
 2&-1& 3&  1& 0&  5& 1&0&0&0\\
 2& 0& 2&  5& 2&  6& 0&1&0&0\\
 2& 0& 2&  5& 0&  9& 0&0&1&0\\
-4& 0&-4&-10& 0&-18& 0&0&0&1
\end{bmatrix}
\xrightarrow[2R_1+R_4]{\substack{-R_1+R_2\\-R_1+R_3}}\\
\begin{bmatrix}
 2&-1& 3&  1& 0&  5& 1&0&0&0\\
 0& 1&-1&  4& 2&  1&-1&1&0&0\\
 0& 1&-1&  4& 0&  4&-1&0&1&0\\
 0&-2& 2& -8& 0& -8& 2&0&0&1
\end{bmatrix}
\xrightarrow[2R_2+R_4]{-R_2+R_3}\\
\begin{bmatrix}
2&-1& 3& 1& 0& 5& 1& 0&0&0\\
0& 1&-1& 4& 2& 1&-1& 1&0&0\\
0& 0& 0& 0&-2& 3& 0&-1&1&0\\
0& 0& 0& 0& 4&-6& 0& 2&0&1
\end{bmatrix}
\xrightarrow{2R_3+R_4}
\begin{bmatrix}
2&-1& 3& 1& 0&5& 1& 0&0&0\\
0& 1&-1& 4& 2&1&-1& 1&0&0\\
0& 0& 0& 0&-2&3& 0&-1&1&0\\
0& 0& 0& 0& 0&0& 0& 0&2&1
\end{bmatrix}
\end{gather*}
\begin{enumerate}[(a)]
\item
$\{\bm b_1,\bm b_2\}$, $B$'nin gergisinin bir taban\i\ ve
$\{\bm c_1,\bm c_2,\bm c_3\}$, $C$'nin gergisinin bir taban\i\ oldu\u gundan
\begin{equation*}
\{(\bm b_1,\bm0),(\bm b_2,\bm0),(\bm0,\bm c_1),(\bm0,\bm c_2),(\bm0,\bm c_3)\},
\end{equation*}
$\lspan B\oplus\lspan C$ direkt toplam\i n\i n bir taban\i d\i r.
\item
$\lspan B+\lspan C$ toplam\i,
$\left[\begin{array}{c|c|c|c|c|c}\bm b_1&\bm b_2&\bm b_3&\bm c_1&\bm c_2&\bm c_3\end{array}\right]$
matrisinin s\"utun uzay\i\ oldu\u gundan
\begin{equation*}
\{\bm b_1,\bm b_2,\bm c_2\},
\end{equation*}
bir taban\i d\i r.
\item
$\{\bm c_2+\lspan B\}$
\item
$\{(0,0,1,0)+\lspan{B\cup C}\}$
\end{enumerate}
 		\end{solution}
		
		\newpage
		
		\begin{problem}
		A\c sa\u g\i da tan\i mlanan k\"umelerin biri,
		$\R^4$ uzay\i n\i n bir altuzay\i d\i r.
		\begin{align*}
S_1&=\{s(1,0,1,1)+(0,1,1,-1)\colon s\in\R\},\\
S_2&=\{s(1,0,1,1)+t(0,1,1,-1)\colon(s,t)\in\R^2\},\\
S_3&=\{s(1,0,1,1)+t^2(0,1,1,-1)\colon(s,t)\in\R^2\},\\
S_4&=\{\bm x\in\R^4\colon x_1{}^2+x_2{}^2+x_3{}^2+x_4{}^2=0\},\\
S_5&=\{\bm x\in\R^4\colon x_1+x_2+x_3+x_4=1\}.
		\end{align*}
		Altuzay, $V$ olsun.
		\begin{enumerate}[(a)]
		\item
		Hangi $k$ i\c cin $V=S_k$?
		\item
		$V$ hangi matrisin s\i f\i r uzay\i d\i r?
		Yani, hangi $A$ matrisi i\c cin $V=\{\bm x\in\R^4\colon A\bm x=\bm0\}$?
		\end{enumerate}
		\end{problem}
		
		\begin{solution}
		\begin{enumerate}[(a)]
		\item
		$k=2$ veya $4$.
		\item
		$V=S_2$ ise
		\begin{equation*}
		\begin{bmatrix}1&0&x_1\\0&1&x_2\\1&1&x_3\\1&-1&x_4\end{bmatrix}
		\xrightarrow[-R_1+R_4]{-R_1+R_3}
		\begin{bmatrix}1&0&x_1\\0&1&x_2\\0&1&-x_1+x_3\\0&-1&-x_1+x_4\end{bmatrix}
		\xrightarrow[R_2+R_4]{-R_2+R_3}
		\begin{bmatrix}1&0&x_1\\0&1&x_2\\0&0&-x_1-x_2+x_3\\0&0&-x_1+x_2+x_4\end{bmatrix}
		\end{equation*}
		oldu\u gundan
		\begin{align*}
			V&=\{\bm x\in\R^4\colon-x_1-x_2+x_3=0\And-x_1+x_2+x_4=0\}\\
			&=\left\{\bm x\in\R^4\colon
			\begin{bmatrix}-1&-1&1&0\\-1&1&0&1\end{bmatrix}\bm x=\bm0\right\},
		\end{align*}
		\begin{equation*}
		A=\begin{bmatrix}-1&-1&1&0\\-1&1&0&1\end{bmatrix}.
		\end{equation*}
		$V=S_4=\{\bm0\}$ ise $A=I$ olabilir.
		\end{enumerate}
		\end{solution}
\end{document}
