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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
: 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 . 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 L(V1, V2; R) and let A Mn1
n2
be such that
A(i, j) = (vi, vj), i = 1, . . . , n1, j = 1, . . . , n2.

  

Source: Allard, William K. - Department of Mathematics, Duke University

 

Collections: Mathematics