Product bases

After one more word of preliminary explanation we shall be ready to discuss the formal definition of tensor products. It turns out to be technically preferable to get at indirectly, by defining it as the dual of another space; we shall make tacit use of reflexivity to obtain itself. Since we have proved reflexivity for finite-dimensional spaces only, we shall restrict the definition to such spaces.

Definition 1. The tensor product of two finite-dimensional vector spaces and (over the same field) is the dual of the vector space of all bilinear forms on . For each pair of vectors and , with in and in , the tensor product is the element of defined by for every bilinear form .

This definition is one of the quickest rigorous approaches to the theory, but it does lead to some unpleasant technical complications later. Whatever its disadvantages, however, we observe that it obviously has the two desired properties: it is clear, namely, that dimension is multiplicative (see Section: Bilinear forms , Theorem 2, and Section: Dual bases , Theorem 2), and it is clear that depends linearly on each of its factors.

Another possible (and deservedly popular) definition of tensor product is by formal products. According to that definition is obtained by considering all symbols of the form , and, within the set of such symbols, making the identifications demanded by the linearity of the vector operations and the bilinearity of tensor multiplication. (For the purist: in this definition stands merely for the ordered pair of and ; the multiplication sign is just a reminder of what to expect.) Neither definition is simple; we adopted the one we gave because it seemed more in keeping with the spirit of the rest of the book. The main disadvantage of our definition is that it does not readily extend to the most useful generalizations of finite-dimensional vector spaces, that is, to modules and to infinite-dimensional spaces.

For the present we prove only one theorem about tensor products. The theorem is a further justification of the product terminology, and, incidentally, it is a sharpening of the assertion that dimension is multiplicative.

Theorem 1. If and are bases in and respectively, then the set of vectors ( ; ) is a basis in .

Proof. Let be the bilinear form on such that ( ; ); the existence of such bilinear forms, and the fact that they constitute a basis for all bilinear forms, follow from Section: Bilinear forms , Theorem 2. Let be the dual basis in , so that . If is an arbitrary bilinear form on , then The conclusion follows from the fact that the vectors do constitute a basis of . ◻

EXERCISES

Exercise 1. If and are vectors in and respectively, find the coordinates of in with respect to the product basis , where and .

Exercise 2. Let be the space of all polynomials with complex coefficients, in two variables and , such that either or else the degree of is for each fixed and for each fixed . Prove that there exists an isomorphism between and such that the element of that corresponds to ( in , in ) is given by .

Exercise 3. To what extent is the formation of tensor products commutative and associative? What about the distributive law ?

Exercise 4. If is a finite-dimensional vector space, and if and are in , is it true that ?

Exercise 5.

Suppose that is a finite-dimensional real vector space, and let be the set of all complex numbers regarded as a (two-dimensional) real vector space. Form the tensor product . Prove that there is a way of defining products of complex numbers with elements of so that whenever and are in and is in .
Prove that with respect to vector addition, and with respect to complex scalar multiplication as defined in (a), the space is a complex vector space.
Find the dimension of the complex vector space in terms of the dimension of the real vector space .
Prove that the vector space is isomorphic to a subspace in (when the latter is regarded as a real vector space).

The moral of this exercise is that not only can every complex vector space be regarded as a real vector space, but, in a certain sense, the converse is true. The complex vector space is called the complexification of .

Exercise 6. If and are finite-dimensional vector spaces, what is the dual space of ?

EXERCISES

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