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\def\NN{{\mbox{$\mathbb N$}}}
\def\Z{{\mbox{$\mathbb Z$}}}
\def\Q{{\mbox{$\mathbb Q$}}}
\def\F{{\mbox{$\mathbb F$}}}
\def\C{{\mbox{$\mathbb C$}}}
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\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{$\mathcal{W}$}}}
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\def\gcd{{\mbox{\rm gcd}}}
\def\A{{\mbox{$\pmb{A}$}}}
\def\J{{\mbox{$\pmb{J}$}}}
\def\I{{\mbox{$\pmb{I}$}}}
\def\j{{\mbox{$\pmb{j}$}}}
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\def\dz{{\mbox{\rm \,d$z$}}}
\def\tg{{\mbox{\rm tg\,}}}
\def\arctg{{\mbox{\rm arc\,tg\,}}}
\def\arcsin{{\mbox{\rm arc\,sin\,}}}
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\newtheorem{zad}{}


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\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, 07.07.2014.\\


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










\begin{zad}
(25\%)($a$) Objasniti koje osobine skup $\V$ mora zadovoljavati da bi bio vektorski prostor nad poljem $\R$. Da li $\V$ može biti konačan skup? Odgovor obrazložiti.


\hspace*{5mm}
(75\%)($b$)
Neka je 
$$
\L=\{
(x_1,x_2,x_3,x_4)^\top\in\R^4\,|\,-x_1+x_2+x_3+x_4=0,\,x_1-x_2+x_3+x_4=0,
$$
$$
\,x_1+x_2-x_3+x_4=0,\,x_1+x_2+x_3-x_4=0
\}.
$$
Dokazati da je $\L$ potprostor vektorskog prostora $\R^n$, odrediti mu bazu, dimenziju i neki direktni kompliment.
\end{zad}





\begin{zad}
Zadan je operator $T:\P_2\rightarrow\P_2$ sa
$$
T(a+bt+ct^2)=a+b+c+(a+3b)t+(a-b+2c)t^2
$$
Odrediti sve jednodimenzionalne potprostore koji su invarijantni u odnosu na operator $T$.
\end{zad}









\begin{zad}
U prostoru svih realnih nizova $\R^{\mathbb N}$ ($\R^{\mathbb N}=\{(a_1,a_2,a_3,...,a_n,a_{n+1},...)\,|\,a_n\in\R,n\in\NN\}$) zadan je skup
$$
\L=\{(a_n)_{n\in{\mathbb N}}\in\R^{\mathbb N}\,|\,a_{n+2}-2a_n=0,n\in\NN\}.
$$
Dokazati da je preslikavanje $T:\L\rightarrow\L$ koje nizu $(a_n)_{n\in \NN}$ pridružuje niz $(a_{n+2})_{n\in{\mathbb N}}$ (tj. niz sa općim članom $a_{n+2}$) linearni operator. Odrediti matricu operatora $T$ (matricu koordinata) u bazi $\B=\{(1,0,2,0,4,0,8,...),(0,1,0,2,0,4,0,8,...)\}$.
\end{zad}














\begin{zad}
U unitarnom prostoru $\R^4$ sa standardnim skalarnim proizvodom ortonormirati skup
$$
\left\{
\left(
\begin{matrix} 1\\ 2\\ 0\\ 1\end{matrix}
\right),\,
\left(
\begin{matrix} 0\\ 2\\ 0\\ 2\end{matrix}
\right),\,
\left(
\begin{matrix} 1\\ -1\\ 1\\ 1\end{matrix}
\right),\,
\left(
\begin{matrix} 1\\ 0\\ 0\\ 0\end{matrix}
\right)
\right\}.
$$
\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.}




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




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