Definition. Suppose V1, . . . , Vm and W are vector spaces. We say a function : V1 Vm W Summary: Definition. Suppose V1, . . . , Vm and W are vector spaces. We say a function µ : V1 × · · · × Vm W is multilinear if it is linear in each of its m arguments when the other m - 1 are held fixed. Let L(V1, . . . , Vm; W) be the set of such µ. Note that L(V1, . . . , Vm; W) is a linear subspace of the vector space of all W-valued functions on V1 × · · · × Vm and is thus a vector space with respect to pointwise addition and scalar multipli- cation. Suppose i V i , i = 1, . . . , m and w W. Define 1 . . . mw : V1 × · · · × Vm W to have the value 1(v1) · · · m(vm)w at (v1, . . . , vm) V1 × · · · × Vm and note that 1 . . . mw L(V1, . . . , Vm; W). In case W = R and w = 1 one customarily writes 1 · · · m for 1 · · · mw. Problem 1. Suppose for each i = 1, 2, Vi is a finite dimensional vector space of dimension ni and with ordered basis vi. Let µ L(V1, V2; R) and let A Mn1 n2 be such that A(i, j) = µ(vi, vj), i = 1, . . . , n1, j = 1, . . . , n2. Collections: Mathematics