 
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 Wvalued
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.
