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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{$\mathcal{P}$}}}
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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, 09.07.2015.\\


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












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









\begin{zad}
Kanonska baza za $\R^n$ je skup $\{e_1,e_2,...,e_n\}$ vektora iz tog prostora takvih da $e_i$ na $i$-toj poziciji ima broj 1, a na svim ostalim nule. Za linearni operator $T:\R^6\rightarrow\R^4$ zadan matricom u paru kanonskih baza
$$
T=\left[\begin{matrix} 1&1&2&1&2&2\\1&2&2&1&2&2\\2&2&4&2&4&4\\0&1&1&0&1&1
\end{matrix}\right]
$$
odredite jezgru i sliku. Da li je vektor $(0,0,0,-1)^\top$ u slici od $T$?
\end{zad}









\begin{zad}
Dat je vektorski potprostor $\L$ vektorskog prostora 
$\Mat_{2\times 2}(\R)$ definisan sa
$$
\L=\{A\in\Mat_{2\times 2}(\R)\,|\,AX-XA=\0,
X=\left[
\begin{matrix} 
0 & 1\\
2 & 0\\
\end{matrix}
\right]\}.
$$
Razmatrajući standardni unutrašnji proizvod za matrice 
$
\langle A,B\rangle=\trag(A^\top B)
$
odrediti ortonormiranu bazu za $\L.$
\end{zad}




\begin{zad}
U unitarnom prostoru $\P_2=\{p(x)=ax^2+bx+c: a,b,c\in\R\}$ polinoma stepena manjeg ili jednakog 2 sa skalarnim (unutrašnjim) proizvodom $\langle p,q\rangle=\int_{-1}^1 p(x) q(x)\,\dx$ dat je potprostor
$$
\M=\span\{x^2-1,x+1\}.
$$
Odredite jednu bazu za $\M^\bot$, te nađite prikaz polinoma $p(x)=2x^2+x+5$ u obliku sume $p=p_1+p_2$, pri čemu je $p_1\in\M$, $p_2\in\M^\bot$.
\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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