\documentclass[12pt,a4paper]{article}
\pagestyle{empty}

%%\usepackage[cp1250]{inputenc} 
\usepackage[cp1250]{inputenc} 
\usepackage[left=16mm,right=17mm,top=17mm,bottom=10mm]
{geometry}
\usepackage{makeidx, graphicx, amsmath, amssymb, amsthm, ae}
\usepackage[croatian, english]{babel}
\usepackage{esint, hvfloat}




\def\R{{\mbox{$\mathbb R$}}}
\def\Z{{\mbox{$\mathbb Z$}}}
\def\Q{{\mbox{$\mathbb Q$}}}
\def\F{{\mbox{$\mathbb F$}}}
\def\C{{\mbox{$\mathbb C$}}}
\def\E{{\mbox{$\mathcal{E}$}}}
\def\M{{\mbox{$\mathcal{M}$}}}
\def\S{{\mbox{$\mathcal{S}$}}}
\def\L{{\mbox{$\mathcal{L}$}}}
\def\P{{\mbox{$\mathcal{P}$}}}
\def\B{{\mbox{$\mathcal{B}$}}}
\def\X{{\mbox{$\mathcal{X}$}}}
\def\V{{\mbox{$\mathcal{V}$}}}
\def\Y{{\mbox{$\mathcal{Y}$}}}
\def\W{{\mbox{$\mathcal{W}$}}}
\def\dist{{\mbox{\rm dist}}}
\def\span{{\mbox{\rm span}}}
\def\ecc{{\mbox{\rm ecc}}}
\def\max{{\mbox{\rm max}}}
\def\spec{{\mbox{\rm spec}}}
\def\rank{{\mbox{\rm rank}}}
\def\mod{{\mbox{\rm mod}}}
\def\div{{\mbox{\rm div}}}
\def\trag{{\mbox{\rm trag}}}
\def\tr{{\mbox{\rm tr}}}
\def\dim{{\mbox{\rm dim}}}
\def\diag{{\mbox{\rm diag}}}
\def\Mat{{\mbox{\rm Mat}}}
\def\ker{{\mbox{\rm ker}}}
\def\gcd{{\mbox{\rm gcd}}}
\def\A{{\mbox{$\pmb{A}$}}}
\def\J{{\mbox{$\pmb{J}$}}}
\def\I{{\mbox{$\pmb{I}$}}}
\def\j{{\mbox{$\pmb{j}$}}}
\def\0{{\mbox{\boldmath{$0$}}}}
\def\algmult{{\mbox{\rm alg\,mult}}}
\def\geomult{{\mbox{\rm geo\,mult}}}
\def\char{{\mbox{\rm char}}}
\def\du{{\mbox{\rm \,d$u$}}}
\def\dx{{\mbox{\rm \,d$x$}}}
\def\dt{{\mbox{\rm \,d$t$}}}
\def\dy{{\mbox{\rm \,d$y$}}}
\def\dz{{\mbox{\rm \,d$z$}}}
\def\tg{{\mbox{\rm tg\,}}}
\def\arctg{{\mbox{\rm arc\,tg\,}}}
\def\arcsin{{\mbox{\rm arc\,sin\,}}}
\def\ctg{{\mbox{\rm ctg\,}}}



\newtheoremstyle{zaFiguru} %name
  {3mm}    %Space above, empty = `usual value'
  {3mm}    %Space below
  {\rm}   %Body font
  {}     %Indent amount (empty = no indent, \parindent = para indent)
  {\bf}   %Thm head font
  {}      %Punctuation after thm head
  {2mm}   %Space after thm head: \newline = linebreak
  {\thmname{#1} \thmnumber{\Large #2.}\thmnote{\,(#3)}}       %Thm head spec  
%%\swapnumbers
\theoremstyle{zaFiguru}
\newtheorem{zad}{}


\setlength{\textheight}{280mm}
\setlength{\textwidth}{180mm}
\begin{document}




\raggedright





\includegraphics{logo_unze}


\vspace*{-25mm}
\hspace*{30mm}
\textbf{Univerzitet u Zenici}\\
\hspace*{30mm}
\textbf{Pedago\v ski fakultet}\\
\hspace*{30mm}
Odsjek: Matematika i informatika\\
\hspace*{30mm}
Zenica, 10.06.2013.\\


\vspace*{5mm}
\renewcommand{\thesection}{}
\section{\centerline{Pismeni ispit iz predmeta Linearna algebra}}




\vspace*{5mm}
\textbf{Bitna napomena:} Svaku formulu koju mislite koristit, u sva 4 zadatka, obavezno napisati, kao i značenja simbola iz formule. Ispit pisati isključivo hemiskom olovkom plave ili crne tinte. Prije rješenja prepisati postavku (tekst) zadatka.\\




\begin{zad}
Neka je $\M=\{X\in\Mat_{2\times 2}(\R)\,|\,AX=\0\}$ gdje je $A=\left(
\begin{matrix} 
2 & 1\\
6 & 3\\
\end{matrix}
\right).$ Odrediti bazu i dimenziju vektorskog prostora $\M$. Odredite mu i neku bazu za (direktni) komplement (koji nije otrogonalni komplement).
\end{zad}

\begin{zad}
(20\%)($a$) Dat je linearni operator
$$
T:\R^2\longrightarrow\R^2
$$
$$
\,\,\,\,\,\,\,\,\,\,\,\,x\longrightarrow Ax
$$
gdje je $A=\left(
\begin{matrix} 
1 & 3\\
2 & 6\\
\end{matrix}
\right).$ Odredite jezgru i sliku od $T$ i izračunati njihove dimenzije.\\


\hspace*{5mm}(80\%)($b$)
Zadan je linearni operator $T:\R^3\longrightarrow\Mat_{2\times 2}(\R)$ sa
$$
T(a,b,c)=\left(
\begin{matrix} 
a+2b & a-b+c\\
b-2c & 5a+5c\\
\end{matrix}
\right).
$$
Prikažite operator $T$ (odredite matricu koordinata) u paru standardnih baza, te mu odredite po jednu bazu za jezgru i sliku.
\end{zad}



\begin{zad}
Neka je $T$ linearni operator na prostoru $\R^2$ koji vektoru pridružuje njegovu ortogonalnu projekciju na pravac $y=-x$. Odredite djelovanje linearnog operatora $T$ na proizvoljnom vektoru 
$\left(
\begin{matrix} 
x\\
y\\
\end{matrix}
\right)$
i odredite mu matrični prikaz u standardnoj bazi, te u bazi $\B=\left\{
\left(
\begin{matrix} 
1\\
1\\
\end{matrix}
\right),
\left(
\begin{matrix} 
2\\
1\\
\end{matrix}
\right)
\right\}$. 
Odrediti i matricu koordinata operatora $T$ u odnosu na standardnu bazu, te u odnosu na bazu $\B$.
\end{zad}




\begin{zad}
U unitarnom prostoru $\Mat_{2\times 2}(\R)$ sa standardnim skalarnim (unutrašnjim) proizvodom $\langle A,B \rangle=\trag(A^\top B)$ dat je podprostor
$$
\M=\span
\left\{
\left(
\begin{matrix} 
2 & 1\\
0 & -1\\
\end{matrix}
\right),
\left(
\begin{matrix} 
0 & 1\\
1 & 0\\
\end{matrix}
\right)
\right\}
$$
Odredite jednu bazu za $\M^\bot$, te nađite prikaz matrice $A
=\left(
\begin{matrix} 
4 & -2\\
-1 & 2\\
\end{matrix}
\right)$ u obliku $A=B+C$, gdje je $B\in\M$, $C\in\M^\bot$.
\end{zad}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\newpage
%%\vspace*{5mm}
\centerline{
Zadaci su skinuti sa stranice 
\underline{pf.unze.ba\slash nabokov}.}
\centerline{
Za uočene greške pisati na {\it infoarrt@gmail.com}
}




\end{document}