Multilinear algebra: determinants

Abstract

Let n be a positive integer, and let V1,...,V n ,W be vector spaces. A function $$ f:{V_1}{\text{x}} \cdots {\text{x}}{V_n} \to W $$ is called a multilinear form iff for each integer i, 1 ≤ i ≤ n, and each (n − 1)tuple (v1,..., v i +1,..., v n ) the function $$ F:{V_i} \to W $$ definced by $$ F(v) = f(v,...,{v_{i - 1}},v,{v_{i + 1}},...,{v_n}) $$ is a linear transformation. For n = 1 a multilinear form is simply a linear transformation. For n = 2 we speak of a bilinear form. Often we just say form for a multilinear form.