exo7 1109

Dans chacun des cas ci-dessous, dire si l’application $\phi$ de $\R^{3}\times\R^{3}\times\R^{3}$ dans $\R$, est multilinéaire.

$$\begin{align*} \phi\left( \left(\begin{smallmatrix} x_{1}\\x_{2}\\x_{3} \end{smallmatrix}\right), \left(\begin{smallmatrix} y_{1}\\y_{2}\\y_{3} \end{smallmatrix}\right), \left(\begin{smallmatrix} z_{1}\\z_{2}\\z_{3} \end{smallmatrix}\right) \right) &= x_{1}+y_{2}+z_{3} \\ \phi\left( \left(\begin{smallmatrix} x_{1}\\x_{2}\\x_{3} \end{smallmatrix}\right), \left(\begin{smallmatrix} y_{1}\\y_{2}\\y_{3} \end{smallmatrix}\right), \left(\begin{smallmatrix} z_{1}\\z_{2}\\z_{3} \end{smallmatrix}\right) \right) &= x_{1}y_{3}+y_{2}z_{1}+z_{3}x_{2} \\ \phi\left( \left(\begin{smallmatrix} x_{1}\\x_{2}\\x_{3} \end{smallmatrix}\right), \left(\begin{smallmatrix} y_{1}\\y_{2}\\y_{3} \end{smallmatrix}\right), \left(\begin{smallmatrix} z_{1}\\z_{2}\\z_{3} \end{smallmatrix}\right) \right) &= x_{1}y_{2}z_{3} +x_{2}y_{3}z_{1} +x_{3}y_{1}z_{2} \\ \phi\left( \left(\begin{smallmatrix} x_{1}\\x_{2}\\x_{3} \end{smallmatrix}\right), \left(\begin{smallmatrix} y_{1}\\y_{2}\\y_{3} \end{smallmatrix}\right), \left(\begin{smallmatrix} z_{1}\\z_{2}\\z_{3} \end{smallmatrix}\right) \right) &= x_{1}x_{2}x_{3} +y_{1}y_{2}y_{3} +z_{1}z_{2}z_{3} \\ \phi\left( \left(\begin{smallmatrix} x_{1}\\x_{2}\\x_{3} \end{smallmatrix}\right), \left(\begin{smallmatrix} y_{1}\\y_{2}\\y_{3} \end{smallmatrix}\right), \left(\begin{smallmatrix} z_{1}\\z_{2}\\z_{3} \end{smallmatrix}\right) \right) &= x_{1}y_{1}z_{1} +x_{2}y_{2}z_{2} +x_{3}y_{3}z_{3} \\ \phi\left( \left(\begin{smallmatrix} x_{1}\\x_{2}\\x_{3} \end{smallmatrix}\right), \left(\begin{smallmatrix} y_{1}\\y_{2}\\y_{3} \end{smallmatrix}\right), \left(\begin{smallmatrix} z_{1}\\z_{2}\\z_{3} \end{smallmatrix}\right) \right) &= (x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3})(z_{1}+z_{3}) \\ \phi\left( \left(\begin{smallmatrix} x_{1}\\x_{2}\\x_{3} \end{smallmatrix}\right), \left(\begin{smallmatrix} y_{1}\\y_{2}\\y_{3} \end{smallmatrix}\right), \left(\begin{smallmatrix} z_{1}\\z_{2}\\z_{3} \end{smallmatrix}\right) \right) &= (x_{1}+2x_{2})(z_{1}+z_{3}) \end{align*}$$