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\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{$\mathcal{Y}$}}}
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\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{$0$}}}}
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\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\,}}}



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\begin{document}




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\includegraphics{logo_unze}


\vspace*{-25mm}
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\textbf{Univerzitet u Zenici}\\
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\textbf{Filozofski fakultet}\\
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Odsjek: Matematika i informatika\\
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Zenica, 30.01.2015.\\


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\renewcommand{\thesection}{}
\section{\centerline{Linearna algebra, pismeni ispit}}












\begin{zad}
Zadan je vektorski prostor $$\M = \{(z_1, z_2, z_3, z_4)\in\C^4 : z_1 + 2z_2 + z_3 = 0, 2z_1+z_2+z_3 = 0, z_1 + 5z_2 + 4z_3 = 0\}.$$ Odredite mu bazu, dimenziju, te po jedan direktni komplement ako $\M$ shvatimo kao 

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(i) kompleksan vektorski potprostor (kao potprostor vektorskog prostora $\C^4$ nad poljem $\C$);

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(ii) realan vektorski potprostor (kao potprostor vektorskog prostora $\C^4$ nad poljem $\R$).
\end{zad}









\begin{zad}
Neka je $A$ linearni operator na $\R^2$ koji djeluje tako da vektor prvo zarotim za ugao $\frac{\pi}{2}$ u pozitivnom smjeru, a dobiveni vektor zatim zrcali (reflektuje) u odnosu na $x$-osu.

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a) Odredite matricu operatora $A$ u standardnoj bazi $\{\vec{i},\vec{j}\}$

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b) Odredite $\vec{v}\in\R^2$ takav da je $A\vec{v} = \vec{v}$ i $\|v\|= 2\sqrt{2}$.
\end{zad}









\begin{zad}
Neka je $\L$ potprostor unitarnog prostora $\P_3$, polinoma stepena $\le 3$, generisan (razapet) vektorima $p_1(t) = 1 - t$ i $p_2(t)=t^2-t.$ Odredite neku bazu za ortogonalni komplement od $\L.$ Nadalje, prikažite $p(t)=1-2t+5t^3$ u obliku $q(t) + r(t)$, gdje je $q\in L$, a $r\in\L^\bot$. (Standardni skalarni proizvod u $\P_3$ je $\langle p,q\rangle=\int_0^1p(t)q(t)\dt$).
\end{zad}




\begin{zad}
Neka je operator $A\in\L(\C^4)$ zadan izrazom
$$
A(x_1,x_2,x_3,x_4)=(x_1-x_2,x_2-x_3,x_3-x_4,2x_4).
$$
Odredite spektar operatora $A$, te algebarse i gometriske višestrukosti svake svojstvene vrijednosti. Može li se ovaj operator dijagonalizirati?
\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{
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\underline{ff.unze.ba\slash nabokov}.}
\centerline{
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}




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