Multilinear maps. Extensions.

1. Jan 11, 2008

WWGD

Hi, everyone:

There are standard ways of representing linear and bilinear maps.
between (fin. dim) vector spaces, after choosing a basis .Linear maps
are represented by columns T(vi) , for a basis {v1,...,vn} (assume B
defined on VxV ), bilinear maps B(x,y) with the matrix Bij=(B(ei,ej))
Is there a way of representing 3-linear, 4-linear, etc. maps with
matrices?. I have played around with matrices T(ei,ej,ek), but
I cannot see how to get a real number as a product of 3 matrices.
Any ideas?.

P.S: I don't know how to setup the spacing.In this forum I was asked
to not leave spacing. In other forums, people complain when I don't
leave spacing, because the lack of spaces force them to strain their
eyes ( where they also complain about how kids today don't understand
music, and about how Frank Sinatra was the last good singer. They also
talk about Selzer water Melba toast, and that hot new comedian Red Skelton.
. Maybe this last explains it )

2. Jan 16, 2008

WWGD

Here is the answer, given to me by someone else (Prof. R. Israel), in case anyone
else is interested:

Suppose T is an n-linear map from V^n to W, and B = {b_i: i=1..m} is a basis
of V. Then the m^n vectors T(b_i) = T(b_{i_1},...,b_{i_n}) for n-tuples
i = (i_1,...,i_n) in {1..m}^n determine T, since if each
x_j = sum_{k = 1}^m c_{j,k} b_k,
T(x_1,...,x_n) = sum_{i in {1..m}^n} product_{j=1}^n c_{j, i_j} T(b_i).

A bilinear map from V^2 to the reals R, for example, can be represented by an
m x m matrix of real numbers: each entry has a pair of indices. A trilinear
map from V^3 to the reals would be represented by a triply-indexed array
of real numbers, rather than a matrix.

3. Jan 16, 2008

mathboy

A p-linear map can be represented by a matrix (tensor) with p indices. Most of the basic theorems with linear maps and bilinear maps are generalized to the p-linear case. The proofs remain the same too, but reading and writing the proofs is really messy because of all the indices.