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\def\Z{{\mbox{$\mathbb Z$}}}
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\def\F{{\mbox{$\mathbb F$}}}
\def\C{{\mbox{$\mathbb C$}}}
\def\E{{\mbox{$\mathcal{E}$}}}
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\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\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}}}
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\def\tg{{\mbox{\rm tg\,}}}
\def\arctg{{\mbox{\rm arc\,tg\,}}}
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\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, 27.01.2014.\\


\vspace*{5mm}
\renewcommand{\thesection}{}
\section{\centerline{Pismeni ispit iz Linearne algebre}}




\vspace*{5mm}
{\large \textbf{Pravila:} 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}
Zadatak prepisati sa table.
\end{zad}





\begin{zad}
Neka je $Q:\P_3\rightarrow\P_3$ ($\P_3$ označava prostor svih polinoma stepena $\le 3$) linearni operator dat sa
$$
Q(p)=\mbox{polinom stepena 2 čiji graf prolazi tačkama}(-1;p(-1)),\,(0;p(0))\mbox{ i } (1;p(1)).
$$


($a$) Odrediti matricu operatora $Q$ (matricu koordinata) u odnosu na standardnu bazu.


($b$) Odrediti (direktni) komplement prostora $\ker(Q)$ (koji nije ortogonalni komplement).
\end{zad}








\begin{zad}
Zadan je linearni operator $T:\Mat_{2\times 2}(\R)\rightarrow \P_3$
$$
T\left(
\begin{matrix} 
a & b\\
c & d\\
\end{matrix}
\right)=(3a+2c-d)x^3+(3b-c-d)x^2+(2a+b+c-d)x+(a+2b-d)
$$
Odrediti ortonormiranu bazu za $\im(T)$ (Koristiti standardni unutrašnji proizvod u $\P_3$ definisan sa $\langle a_3x^3+a_2x^2+a_1x+a_0, b_3x^3+b_2x^2+b_1x+b_0\rangle=a_3b_3+a_2b_2+a_1b_1+a_0b_0$).
\end{zad}












\begin{zad}
U unitarnom prostoru $\R^4$, sa skalarnim proizvodom 
$$
\langle x,y\rangle=x_1y_1+2x_2y_2+x_3y_3+2x_4y_4
$$
zadan je podprostor $\V$ razapet (generisan) vektorima $v_1=(1,0,1,0)^\top$ i $v_2=(1,0,1,1)^\top$. Prikažite vektor $x=(4,2,2,4)^\top$ u obliku $x=v+w$, gdje je $v\in\V$, $w\in\V^\bot$.
\end{zad}


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




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