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\def\N{{\mbox{$\mathcal{N}$}}}
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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{$\mathcal{Y}$}}}
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\def\gcd{{\mbox{\rm gcd}}}
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\def\J{{\mbox{$\pmb{J}$}}}
\def\I{{\mbox{$\pmb{I}$}}}
\def\j{{\mbox{$\pmb{j}$}}}
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\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, 01.11.2013.\\


\vspace*{5mm}
\renewcommand{\thesection}{}
\section{\centerline{Pismeni ispit iz predmeta Linearna algebra}}




\vspace*{5mm}
{\large \textbf{Uputa:} 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}
U $\Mat_{2\times 2}(\R)$ zadani su podprostori
$$
\M=\left\{
\left(
\begin{matrix} 
a & b\\
c & d
\end{matrix}
\right):
a-2b=0,\,\,a+c+d=0
\right\}\,\,\,\,\mbox{i}
$$
$$
\N=\left\{
\left(
\begin{matrix} 
a & b\\
c & d
\end{matrix}
\right):
a+c=0,\,\,a-2b+d=0
\right\}.
$$
Odrediti po jednu bazu za $\M$, $\N$, $\M+\N$ i $\M\cap\N$.
\end{zad}




\begin{zad}
Zadan je linearni operator $T:\R^2\longrightarrow\R^2$ svojom matricom 
$T=\left[
\begin{matrix} 
-1 & 0\\
1 & 1\\
\end{matrix}
\right]$ u kanonskoj bazi $\{\vec{i},\vec{j}\}$. Neka su $\vec{a}=\vec{i}+\vec{j}$, $\vec{b}=\vec{i}-2\vec{j}$.\\
\hspace*{5mm}($a$) Odrediti $T(\vec{a})$, $T(\vec{b})$.\\
\hspace*{5mm}($b$) Za koje $\alpha\in\R$ su vektori $T(\vec{a})$, $T(\vec{a}+\alpha\vec{b})$ kolinearni?
\end{zad}



\begin{zad}
Neka je $T$ linearan operator na prostoru $\R^2$ koji vektor najprije reflektuje (zrcali) s obzirom na pravac $y=-x$, zatim ga rotira za ugao $\frac{\pi}{4}$ oko koordinatnog početka (oko izvorišta) u negativnom smjeru, te zatim reflektuje (zrcali) s obzirom na pravac $y=x$. Naći matricu (matricu koordinata) operatora $T$ u bazi 
$\B=\left\{
2\left(
\begin{matrix} 
1\\
0\\
\end{matrix}
\right)-\left(
\begin{matrix} 
0\\
1\\
\end{matrix}
\right),
-\left(
\begin{matrix} 
1\\
0\\
\end{matrix}
\right)+2\left(
\begin{matrix} 
0\\
1\\
\end{matrix}
\right)
\right\}$.
\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{
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