alternating multilinear form

An alternating form is a multilinear map that collapses to zero the moment two of its inputs are equal. Geometrically this is the statement 'a flat thing has no volume': if two edge-vectors of a parallelepiped coincide, the figure is degenerate and its signed volume is 0. This single rule, layered on top of multilinearity, is the entire DNA of the determinant.

Let f be k-linear and scalar-valued. It is alternating if f(v_1, ..., v_k) = 0 whenever v_i = v_j for some i != j. A short computation forces an equivalent and more vivid property: swapping any two arguments flips the sign, f(..., v_i, ..., v_j, ...) = -f(..., v_j, ..., v_i, ...). (Expand f(..., x+y, ..., x+y, ...) = 0 to see the swap rule appear.) Over a field where 1 + 1 != 0, 'vanishes on repeats' and 'sign-flips on swaps' are interchangeable definitions.

Why it matters: alternating multilinearity is exactly the wishlist 'volume' must satisfy — scale an edge, scale the volume (multilinear); flatten the figure, kill the volume (alternating). Imposing one more normalization (value 1 on the standard cube) pins down a single function. That function is the determinant, and the alternating axiom is where its signs are born.

Caveat: 'alternating' is strictly stronger than 'antisymmetric' (sign-flipping) only in characteristic 2, where 1 + 1 = 0 and the swap rule no longer forces vanishing-on-repeats. In ordinary real or complex linear algebra you can use the two words as synonyms, but a careful algebraist keeps the distinction.