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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, 01.09.2014.\\


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










\begin{zad}
U vektorskom prostoru $\P_4$ realnih polinoma stepena $\le 4$ dat je skup
$$
\M=\{p\in\P_4\,|\, p'(0)=p(1),\,p''(0)=2p(-1) \}.
$$
Dokažite da je $\M$ vektorski potprostor prostora $\P_4$, odredite mu jednu bazu i dimenziju, te jedan direktni komplement.
\end{zad}





\begin{zad}
Neka je
$$
\left[\begin{matrix} 
2&1\\1&4
\end{matrix}\right]
$$
matrica linearnog operatora $T:\V^2(0)\rightarrow\V^2(0)$ u kanonskoj bazi $\left\{\vec{i}=\left(\begin{matrix} 1\\0  \end{matrix}\right),\vec{j}=\left(\begin{matrix} 0\\1  \end{matrix}\right)\right\}$. Odrediti matricu operatora $T$ u bazi $\{\vec{i}+2\vec{j},\vec{i}+3\vec{j}\}$. Da li postoji vektor $\vec{v}\in\V^2(0)$ takav da je $T(\vec{v})=3\vec{i}+5\vec{j}$?
\end{zad}









\begin{zad}
($20\%$)($a$) Objasniti šta će se desiti kada se Gram-Schmidtov proces primjeni na linearno zavisan skup vektora.

\hspace*{5mm}
($80\%$)($b$)Zadan je unitarni prostor $\Mat_{2\times 2}(\R)$ sa skalarnim (unutrašnjim) proizvodom $\langle A,B\rangle=\trag(A^\top B)$ i neka je $\L$ vektorski potprostor $\Mat_{2\times 2}(\R)$ definiran kao
$$
\L=\span\left\{
\left(
\begin{matrix} 
0 & 3\\ 
3 & -3
\end{matrix}
\right),
\left(
\begin{matrix} 
-2 & -2\\ 
6 & -2
\end{matrix}
\right),
\left(
\begin{matrix} 
-1 & 1\\ 
-1 & 1
\end{matrix}
\right)
\right\}.
$$
Nađite ortonormiranu bazu za $\L$.
\end{zad}














\begin{zad}
Data je matrica 
$$
A=\left(\begin{matrix} 1&a&1&0\\1&-1&0&1\\0&0&1&0\\1&b&0&1 \end{matrix}\right).
$$
Odrediti parametre $a$ i $b$ ako je poznato da je $A$ singularna matrica čije sve svojstvene vrijednosti imaju algebarsku višestrukost $2$.
\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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%%\vspace*{5mm}
\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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