---
tag: 0MSWYWHYF5
kind: definition
name: Alternating multilinear map
msc: [15A69, 15A75]
defines: 0K6CH0MT0F
uses: [0SDMPMCN43]
sources:
  - url: https://en.wikipedia.org/wiki/Alternating_multilinear_map
    locator: Definition
---

## Definition

Let $F$ be a field, let $n \ge 1$, and let $V$ and $W$ be vector spaces over $F$.
A [[multilinear map|0443XNARDV]] $f \colon V^n \to W$ is *alternating* when $f(v_1, \dots, v_n) = 0$ for each family with $v_i = v_j$ for some $i \ne j$.

## Remarks

An alternating map changes sign when two arguments are exchanged.
If the characteristic of $F$ is not $2$, then this sign rule implies that the map is alternating.
An alternating map is zero on each linearly dependent family.
The [[determinant|00KDZRW4C0]] is an alternating multilinear map of the rows of a square matrix.

## Explanations

### L1 human

An alternating map gives zero as soon as two inputs are the same.
It follows that swapping two inputs changes the sign of the value.
The signed area of the parallelogram of two plane vectors is an alternating bilinear map.