Chapter 14

# Multilinear Algebra

J. A. Dias da Silva

## 14.1 Multilinear Maps

Unless otherwise stated, within this section V, U, and W as well as these letters with sub-scripts, superscripts, or accents, are finite dimensional vector spaces over a field F of characteristic zero.

Definitions:

A map φ from V1 ×···× Vm into U is a multilinear map (m-linear map) if it is linear on each coordinate, i.e., for every v′i, vi ∈, Vi,i = 1,... ,m and for every aF the following conditions hold:

1. (a) φ(v1,...,vi + ${\text{v}}_{i}^{\prime }$,...,vm)= φ(v1,...,vi,...,vm)= φ(v1,...,${\text{v}}_{i}^{\prime }$,...,vm);
2. (b) φ(v1,..., avi,..., vm) = aφ(v1,..., vi,..., vm).

The 2-linear maps and 3-linear maps are also called bilinear and trilinear

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