Alternating multilinear map

Definition

Let $F$ be a field, let $n \ge 1$, and let $V$ and $W$ be vector spaces over $F$. A multilinear map $f \colon V^n \to W$ 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 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.