| Version 6 |
Version 5 |
| Let $V_1, V_2,\ldots, V_n, W$ be vector spaces over a field $K$. A |
Let $V_1, V_2,\ldots, V_n, W$ be vector spaces over a field $K$. A |
| mapping $$M: V_1\times V_2\times \ldots \times V_n \rightarrow W$$ is |
mapping $$M: V_1\times V_2\times \ldots \times V_n \rightarrow W$$ is |
| called {\em multi-linear} or $n$-linear, if $M$ is linear in each of |
called {\em multi-linear} or $n$-linear, if $M$ is linear in each of |
| its arguments. |
its arguments. |
|
|
| \paragraph{Notes.} |
\paragraph{Notes.} |
| \begin{itemize} |
\begin{itemize} |
| \item A bilinear mapping is another name for a $2$-linear mapping. |
\item A bilinear mapping is another name for a $2$-linear mapping. |
| \item This definition generalizes in an obvious way to rings and |
\item This definition generalizes in an obvious way to rings and |
| modules. |
modules. |
| \item An excellent example of a multi-linear map is the determinant operation. |
\item An excellent example of a multi-linear map is the determinant operation. |
| \end{itemize} |
\end{itemize} |
