Exercice 5

1) $$ M = \begin{pmatrix} 1 & -1 & 2 \\ -1 & 1 & 1 \\ 3 & -2 & 1 \end{pmatrix} $$ 2) $$ f(x, y, z) = (x - y + 2z,\ -x + y + z,\ 3x - 2y + z) $$ 3) \[ B' = (e'_1, e'_2, e'_3), \quad e'_1 = 2e_1 + e_2,\ e'_2 = 2e_1 - 2e_2,\ e'_3 = 2e_3 \] B' est une base car elle est libre. 4) $$ P = \begin{pmatrix} 2 & 2 & 0 \\ 1 & -2 & 0 \\ 0 & 0 & 2 \end{pmatrix}, \quad P^{-1} = \begin{pmatrix} \frac{1}{3} & \frac{1}{3} & 0 \\ \frac{1}{6} & -\frac{1}{3} & 0 \\ 0 & 0 & \frac{1}{2} \end{pmatrix} $$ 5) $$ X = e_1 - e_3 = \begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix}, \quad [X]_{B'} = P^{-1} \cdot X = \begin{pmatrix} \frac{1}{3} \\ \frac{1}{6} \\ -\frac{1}{2} \end{pmatrix} $$ 6) $$ A_{B'} = P^{-1} M P = \begin{pmatrix} 0 & 0 & 2 \\ 1/2 & 2 & 0 \\ 2 & 5 & 1 \end{pmatrix} $$

Exercice 6

1) $$ \begin{cases} 5x - 8y = 1 \\ -7x + 3y = -4 \end{cases}, \quad \Rightarrow x = \frac{29}{41}, \quad y = \frac{13}{41} $$ 2) $$ \begin{cases} 3x - 2y + z = 0 \\ -2x + y - z = -1 \\ 2x - 4y + 5z = 2 \end{cases}, \quad \Rightarrow x = \frac{4}{7}, \quad y = \frac{11}{7}, \quad z = \frac{10}{7} $$ 3) $$ \begin{cases} 2i x + y = -3 + i \\ 2x + (1 + i)z = 6 \\ (1 - i)y - 6z = 3i \end{cases}, \quad i^2 = -1 $$ $$ \Rightarrow x = \frac{29}{10} + \frac{21}{20}i, \quad y = -\frac{9}{10} - \frac{24}{5}i, \quad z = -\frac{19}{20} - \frac{23}{20}i $$