Definition: Multilinear Map

Let \(F\) be a field and let \(V_{1},\ldots ,V_{n}\) and \(W\) be vector spaces over \(F\). A map. \[\Phi \colon V_{1}\times \cdots \times V_{n}\longrightarrow W\,\]

is called a multilinear map, if for each \(i\in \{1,\ldots ,n\}\) and each \((n-1)\)-tuple \((v_{1},\ldots ,v_{i-1},v_{i+1},\ldots ,v_{n})\) with \(v_{j}\in V_{j}\), the induced map

\[V_{i}\longrightarrow W,\,v_{i}\longmapsto \Phi (v_{1},\ldots ,v_{i-1},v_{i},v_{i+1},\ldots ,v_{n})\,,\]

is a linear map over \(F\).

  1. Definition: Alternating Multilinear Map

Definitions: 1

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Adapted from CC BY-SA 3.0 Sources:

  1. Brenner, Prof. Dr. rer. nat., Holger: Various courses at the University of Osnabrück