ABSTRACT
Multilinear algebra. proper, begins with the study of tensor spaces. In the most general setting, this involves dealing with m vector spaces, each having its own basis and its own inner product. Keeping track of it all requires what Elie Cartan called ''une debauche d'indices", an intimidating proliferation of superscripts and subscripts. The peak of this mountain of notation occurs when matrix representations of linear transformations on the various vector spaces are assembled to produce the matrix representation of a linear transformation on the tensor space. Following the introduction of the induced inner product, the going will be enormously simplified by setting all m vector spaces equal. (The reader may find it useful to introduce this simplification earlier and rewrite difficult passages setting V1 = V2 = • · • = Vm = V.)
Let v. ' v2' ... ' V m be finite dimensional complex vector spaces. Their Cartesian product is the set
defined by m
by m
If T e L(V, W), then T is completely and uniquely determined by its action on a basis of V . Indeed, it is common to define a linear transformation by describing its action on a basis and saying the magic words, "linear extension". As we now see, multilinear functions behave analogously.
