multilinear map

A multilinear map takes several vectors as input and is linear in each slot separately, while you hold the other slots fixed. Think of a machine with k input ports: jiggle one port (scale a vector, or add two vectors there) and the output responds linearly; the other ports just sit still as constants. The determinant is the most famous example, eating n column vectors at once.

Precisely, a map f from V_1 x ... x V_k to W is multilinear (or k-linear) if for each index i, fixing all arguments except the i-th gives an ordinary linear map in that argument: f(..., a*u + b*v, ...) = a*f(..., u, ...) + b*f(..., v, ...). Linearity must hold in every slot, but NOT jointly — f(2v_1, 2v_2) is generally 4*f(v_1, v_2), not 2*f(v_1, v_2), so a multilinear map is almost never itself a linear map on the product space.

This is the right genus for the determinant. Once you see det as one species of multilinear map — specifically an alternating one — its strange-looking rules (sign flips, expansion along a row, the product formula) stop being tricks and become consequences. Multilinearity also opens the door to tensors, where a k-linear map IS literally a tensor of rank k.

Caveat: do not confuse multilinear with linear. A bilinear form like the dot product is linear in u and linear in v, but the map (u, v) -> <u, v> is not linear on the pair, since scaling both inputs scales the output by the square. Keeping 'linear in each slot, fixing the rest' as your mantra prevents this slip.