alternating form

An alternating form is a multilinear measurement that automatically returns zero whenever two of its inputs are identical — like a volume gauge that reads zero for any flattened, degenerate box. This single 'collapse to zero on repeats' demand encodes the entire sign-and-orientation behaviour of determinants and signed volumes.

Let V be a module over R. A k-multilinear form f: V × ... × V -> R is alternating if f(v_1, ..., v_k) = 0 whenever v_i = v_j for some i ≠ j. From multilinearity this forces antisymmetry: swapping any two arguments multiplies the value by −1, and more generally permuting the arguments by σ multiplies by the sign sgn(σ). Over a field where 2 is invertible, the vanishing-on-repeats and antisymmetry conditions are equivalent; in characteristic 2 the vanishing condition is the correct, stronger one.

Alternating k-forms on V are precisely the linear functionals on Λ^k(V): the space of them is Λ^k(V)*, with dimension C(n, k) when dim V = n. The unique (up to scalar) alternating n-form on an n-dimensional space, normalized to send a fixed basis to 1, is the determinant. Alternating forms are the linear-algebra shadow of differential forms in geometry.