Función multilineal

También llamada aplicación multilineal.

Definición

Sean ${\displaystyle V_{1},\ldots ,V_{n},W\quad \mathbb {K} }$-espacios vectoriales, con ${\displaystyle \mathbb {K} }$ un cuerpo ${\displaystyle {\big (}\mathbb {R} }$ o ${\displaystyle \mathbb {C} {\big )}}$.

${\displaystyle \mathbf {f} :V_{1}\times \ldots \times V_{n}\longrightarrow W}$ será una función multilineal ${\displaystyle \Leftrightarrow \forall i{\big |}1\leqslant i\leqslant n,\mathbf {v} _{i},\mathbf {w} _{i}\in V_{i},\forall \lambda ,\mu \in \mathbb {K} }$ ${\displaystyle \mathbf {f} {\big (}\mathbf {v} _{1},\ldots ,\lambda \mathbf {v} _{i}+\mu \mathbf {w} _{i},\ldots ,\mathbf {v} _{n}{\big )}=\lambda \mathbf {f} {\big (}\mathbf {v} _{1},\ldots ,\mathbf {v} _{i},\ldots ,\mathbf {v} _{n}{\big )}+\mu \mathbf {f} {\big (}\mathbf {v} _{1},\ldots ,\mathbf {v} _{i-1},\mathbf {w} _{i},\mathbf {v} _{i+1},\ldots ,\mathbf {v} _{n}{\big )}}$

Propiedad

${\displaystyle \mathbf {f} {\big (}\mathbf {v} _{1},\ldots ,\mathbf {0} ,\ldots ,\mathbf {v} _{n}{\big )}=\mathbf {0} }$

Demostración

Para ${\displaystyle \lambda =\mu =1}$ ${\displaystyle {\begin{array}{l}\mathbf {f} {\big (}\mathbf {v} _{1},\ldots ,\mathbf {0} ,\ldots ,\mathbf {v} _{n}{\big )}=\mathbf {f} {\big (}\mathbf {v} _{1},\ldots ,\mathbf {0} +\mathbf {0} ,\ldots ,\mathbf {v} _{n}{\big )}=\mathbf {f} {\big (}\mathbf {v} _{1},\ldots ,\mathbf {0} ,\ldots ,\mathbf {v} _{n}{\big )}+\\\mathbf {f} {\big (}\mathbf {v} _{1},\ldots ,\mathbf {0} ,\ldots ,\mathbf {v} _{n}{\big )}=2\mathbf {f} {\big (}\mathbf {v} _{1},\ldots ,\mathbf {0} ,\ldots ,\mathbf {v} _{n}{\big )}\Rightarrow \\\mathbf {f} {\big (}\mathbf {v} _{1},\ldots ,\mathbf {0} ,\ldots ,\mathbf {v} _{n}{\big )}=\mathbf {0} \end{array}}}$ como queríamos demostrar.

Para n=1 ${\displaystyle \mathbf {f} }$ será una aplicación lineal.

• Datos: Q16569887