DECIMOTESTEINDEXAÇÃO pdf

2). Tarefas adicionais Q5, e)\nEncontrar uma base de um espaço-solu-\nço de um SEL homogêneo:\ne) \([-1\ 1\ 1\ 0\ 0\ -1]\n[1\ 2\ -1\ 0\ 1\ 0]\n[2\ 0\ 1, 2\ -1, 1, 1]\) \({}^{\!(1)}_2\) \to \(\left[ \begin{array}{ccccc} 1 & 1 & 1 & 0 & -1 \\ 0 & 3 & 0 & 1 & -1 \\ 0 & 2 & 3 & 2 & 1 -1 \end{array} \right]^{\!(1)}_2\)\n\n\(\left[ \begin{array}{ccccc} 1 & 1 & 1 & 0 & -1 \\ 0 & 1 & -3 & 2 & 0 \\ 0 & 0 & 9 & 6 & 5 & -1 \end{array} \right]^{\!(\!-2)})\)\n\n\(x_1, x_2, x_3\text{-pivôs.}\ x_4, x_5, x_6\text{-var. livres.}\)\n\n\(9x_3 + 6x_4 - 5x_5 + x_6 = 0;\\\n9x_3 = -6x_4 + 5x_5 + x_6;\)\n\(x_3 = \frac{-6x_4 + 5x_5 + x_6}{9}.\)\n\n\(x_2 - 3x_3 - 2x_4 + 2x_5 = 0;\)\n\(x_2 = 3x_3 + 2x_4 - 2x_5 = 3 \left( -\frac{2x_4}{3} + \frac{5x_5}{9}x_6 \right) + 2x_4 - \)\n\(-\frac{18}{9} x_5 = -2x_4 + \frac{15}{9}x_5 + \frac{1}{3} x_6 + 2x_4 \right. \left. - \frac{18}{9} x_5 = -\frac{3}{9} x_5 + x_6 \)\n\(x_2 = -\frac{1}{3} x_5 + \frac{1}{3} x_6.\)\n\n\(-x, + x_2 + x_3 - x_6 = 0;\)\n\(x_1 = x_2 + x_3 - x_6 = \left( -\frac{1}{3} x_5 + \frac{1}{3} x_6 \right) + \left( \frac{-6}{9} x_4 \underline{+}\frac{5}{9} x_5 + \frac{1}{9} x_6 \right).\)\n\(-x_6 = -\frac{2}{3} x_4 + \frac{2}{9} x_5 - \frac{5}{9} x_6;\)\n\(x_1 = -\frac{2}{3} x_4 \underline{+}\frac{2}{9}x_5-\frac{5}{9} x_6\) \(x_1 = -\frac{2}{3} x_4^{\!(100)}_{09} - \frac{2}{9} x_5^{\!(10)}_{0,1} \; x_6,\)\n\(x_2 = -\frac{1}{3} x_5 \underline{+}\frac{1}{3} x_6;\)\n\(x_3 = -\frac{6}{9} x_4 \underline{+}\frac{5}{9} x_5 \underline{+}\frac{1}{9} x_6;\ x_4, x_5, x_6 = \text{livres.}\)\n\[\begin{array}{c|c|c|c|c|c} x_1 & x_2 & x_3 & x_4 & x_5 & x_6 \\ \hline -\frac{2}{3} & 0 & -\frac{6}{9} & 1 & 0 & 0 \\ \hline -\frac{2}{9} & -\frac{1}{3} & \frac{5}{9} & 0 & 1 & 0 \\ \hline -\frac{5}{9} & \frac{1}{3} & \frac{1}{9} & 0 & 0 & 1 \end{array}\] = \vec{\alpha}_1 \\ = \vec{\alpha}_2 \\ = \vec{\alpha}_3.\)\n\n\text{Resposta:} \text{Uma base do espaço-solução:}\\\n\{ (-\frac{2}{3}, 0, -\frac{6}{9}, 1, 0, 0), (-\frac{2}{9}, -\frac{1}{3}, \frac{5}{9}, 0, 1, 0), \\(-\frac{5}{9}, \frac{1}{3}, \frac{1}{9}, 0, 0, 1) \}.\