tensor product (vector spaces)
One then represented relative to this particular choice of bases as the vector space of mappings . These mappings were called “second-order contravariant tensors” and their values were customarily denoted by superscripts, a.k.a. contravariant indices:
was defined by representing , relative to the chosen bases, as the tensor
In this system, the products
These gave a basis of .
The construction is independent of the choice of bases in the following sense. Let
be different bases of and with indexing sets and respectively. Let
be the corresponding change of basis matrices determined by
One then stipulated that tensors and represent the same element of if
for all . This relation corresponds to the fact that the products
constitute an alternate basis of , and that the change of basis relations are
Notes. Historically, the tensor product was called the outer product, and has its origins in the absolute differential calculus (the theory of manifolds). The old-time tensor calculus is difficult to understand because it is afflicted with a particularly lethal notation that makes coherent comprehension all but impossible. Instead of talking about an element of a vector space, one was obliged to contemplate a symbol , which signified a list of real numbers indexed by , and which was understood to represent relative to some specified, but unnamed basis.
What makes this notation truly lethal is the fact a symbol was taken to signify an alternate list of real numbers, also indexed by , and also representing , albeit relative to a different, but equally unspecified basis. Note that the choice of dummy variables make all the difference. Any sane system of notation would regard the expression
as representing a list of symbols
because where, after all, is the all important dummy variable to indicate choice of basis?
Thankfully, it is possible to shed some light onto this confusion (I have read that this is credited to Roger Penrose) by interpreting the symbol as a mapping from some finite index set to , whereas is interpreted as a mapping from another finite index set (of equal cardinality) to .
My own surmise is that the source of this notational difficulty stems from the reluctance of the ancients to deal with geometric objects directly. The prevalent superstition of the age held that in order to have meaning, a geometric entity had to be measured relative to some basis. Of course, it was understood that geometrically no one basis could be preferred to any other, and this leads directly to the definition of geometric entities as lists of measurements modulo the equivalence engendered by changing the basis.
It is also worth remarking on the contravariant nature of the relationship between the actual elements of and the corresponding representation by tensors relative to a basis — compare equations (1) and (2). This relationship is the source of the terminology “contravariant tensor” and “contravariant index”, and I surmise that it is this very medieval pit of darkness and confusion that spawned the present-day notion of “contravariant functor”.
Levi-Civita, “The Absolute Differential Calculus.”
|Title||tensor product (vector spaces)|
|Date of creation||2013-03-22 12:21:40|
|Last modified on||2013-03-22 12:21:40|
|Last modified by||rmilson (146)|