\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}
%%\usepackage{systeme,mathtools}




\def\R{{\mbox{$\mathbb R$}}}
\def\NN{{\mbox{$\mathbb N$}}}
\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\N{{\mbox{$\mathcal{N}$}}}
\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\im{{\mbox{\rm im}}}
\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
  {5mm}    %Space above, empty = `usual value'
  {5mm}    %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{Filozofski fakultet}\\
\hspace*{30mm}
Odsjek: Matematika i informatika\\
\hspace*{30mm}
Zenica, 22.09.2014.\\


\vspace*{5mm}
\renewcommand{\thesection}{}
\section{\centerline{Linearna algebra, pismeni ispit}}












\begin{zad}
U $\Mat_{2\times 2}(\R)$ zadani su potprostori
$$
\M=\left\{
\left(
\begin{matrix} 
a & b\\
c & d
\end{matrix}
\right):
a-2b=0,\,\,a+c+d=0
\right\}\,\,\,\,\mbox{i}
$$
$$
\N=\left\{
\left(
\begin{matrix} 
a & b\\
c & d
\end{matrix}
\right):
a+c=0,\,\,a-2b+d=0
\right\}.
$$
Odrediti po jednu bazu za $\M$, $\N$, $\M+\N$ i $\M\cap\N$.
\end{zad}









\begin{zad}
Zadana je linearna transformacija $T:\P_3\rightarrow\Mat_{2\times 2}(\R)$
$$
T(a_0+a_1t+a_2 t^2+a_3 t^3)=\left[\begin{matrix} 
a_0+a_3-a_2&a_0+2a_1-a_2\\a_3&a_0-a_2
\end{matrix}\right]
$$
Prikažite transformaciju $T$ u paru standardnih baza, te joj odredite $\ker(T)$, $\im(T)$, rang $\rho(T)$ i defekt $\delta(T)$ (rang i defekt linearnog preslikavanja $T$ označavamo redom sa $\rho(T)$ i $\delta(T)$ i definišemo na sljedeći način $\rho(T):=\dim\,\im(T)$, $\delta(T):=\dim\,\ker(T)$). ($\P_3$ je prostor svih polinoma stepena $\le 3$).
\end{zad}









\begin{zad}
Zadan je linearni operator $T:\Mat_{2\times 2}(\R)\rightarrow\Mat_{2\times 2}(\R)$ sa
$$
T(\left[\begin{matrix} 
a&b\\c&d
\end{matrix}\right])=
\left(\begin{matrix} 
a-b&4a-4b\\-a+2b+c&b+c
\end{matrix}\right)
$$
Odrediti sve jednodimenzionalne potprostore koji su invarijantni u odnosu na operator $T$.
\end{zad}




\begin{zad}
Prostor $\L$ je zadan kao skup rješenja sistema
$$
\begin{aligned}
2x_1  & \,\,+\,\, & x_2 &\,\,+\,\,&  x_3 &\,\,+\,\,&  3x_4 &= 0\\
3x_1  & \,\,+\,\, & 2x_2 &\,\,+\,\,&  2x_3 &\,\,+\,\,&  x_4 &= 0\\
x_1  & \,\,+\,\, & 2x_2 &\,\,+\,\,&  2x_3 &\,\,-\,\,&  9x_4 &= 0\,\,\, .
\end{aligned}
$$
Prikažite vektor $x=(7,-4,-1,2)^\top$ u obliku $x=y+z$, pri čemu je $y\in\L$, a $z$ iz ortogonalnog komplimenta od $\L$ u $\R^4$.
\end{zad}










\vspace*{25mm}
{\textbf{Važno:} Ovaj papir treba predati zajedno s rješenjima zadataka! 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.}




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


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



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



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




\end{document}