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\begin{document}

\centerline{\bf MAT 2116 - \'Algebra Linear para Qu\'imica} 

\centerline{\bf 3$^a$ Prova - 29 de junho de 2017}



\noindent
    {\bf Quest\ao\ 1)} (3 pts)
    \\Seja $W$ o subespa\c{c}o de $\R^5$ gerado por
    $\mathbf{v}_{1}=(0,-1,0,-1,0)$, $\mathbf{v}_{2}=(0,1,1,1,1)$ e $\mathbf{v}_{3}=(0,1,3,1,3)$. 
    \\(a) Calcule $2\mathbf{v}_1+3\mathbf{v}_2$.
    \\(b) O conjunto $\{\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3}\}$ \'e linearmente dependente ou linearmente independente? 
    \\(c) Possui $\{\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3}\}$ um subconjunto de dois elementos que gere $W$?
    \\(d) Qual \'e a dimens\ao\ de $W$?
    \\(e) \'E verdadeira ou falsa a afirma\cao\ de que todo elemento $(x_1,x_2,x_3,x_4,x_5)$ de $W$ satisfaz $x_1=0$? 
    \\(f) \'E verdadeira ou falsa a afirma\cao\ de que, se $x_1=0$, ent\ao\ $(x_1,x_2,x_3,x_4,x_5)$ pertence a $W$? 

%\newpage
\noindent
    {\bf Quest\ao\ 2)} (2 pts)\\Encontre uma base do subespa\c{c}o vetorial de $\R^3$ que consiste dos $(x,y,z)$ tais que
   \[% ${\displaystyle
      \left[
      \begin{array}{ccc}
3&1&-2\\8&5&-8\\6&3&-5
      \end{array}
      \right]\,
       \left[
      \begin{array}{c}
x\\y\\z
      \end{array}
      \right]
       \,=\,
     \left[
      \begin{array}{c}
x\\y\\z
      \end{array}
      \right] .
   \]% }$
    

%\newpage

\noindent
    {\bf Quest\ao\ 3)} (2,5 pts) \'E diagonaliz\'avel a matriz
    ${\displaystyle  \left[\begin{array}{ccc}1&0&1\\0&1&0\\1&0&1\end{array}\right] }$? 


%\newpage

\noindent
    {\bf Quest\ao\ 4)} (2,5 pts) Considere ${\displaystyle A=
      \left[
      \begin{array}{cc}
3&6\\-2&-4
      \end{array}
      \right]}$,
${\displaystyle \mathbf{u}=    
 \left[
      \begin{array}{c}
-3\\2
      \end{array}
      \right]}$ e ${\displaystyle \mathbf{v}=    
 \left[
      \begin{array}{c}
2\\-1
      \end{array}
      \right]}$.
    \\
    (a) Calcule $A\mathbf{u}$ e $A\mathbf{v}$.
    \\
    (b) \'E $A$ diagonaliz\'avel? 
    \\
    (c) Determine uma matriz invers\'ivel $P$ e uma matriz diagonal $D$ tais que $A^{17}=PDP^{-1}$.
\end{document}
