Tensor product
is formed by all tensor products of a basis element of V and a basis element of W. The tensor product of two vector spaces captures the properties of all bilinear maps in the sense that a bilinear map from
to Z. Tensor products are used in many application areas, including physics and engineering.
Most consist of defining explicitly a vector space that is called a tensor product, and, generally, the equivalence proof results almost immediately from the basic properties of the vector spaces that are so defined.
When this definition is used, the other definitions may be viewed as constructions of objects satisfying the universal property and as proofs that there are objects satisfying the universal property, that is that tensor products exist.
making these maps similar to a Schauder basis for the vector space
In either construction, the tensor product of two vectors is defined from their decomposition on the bases.
A construction of the tensor product that is basis independent can be obtained in the following way.
Let R be the linear subspace of L that is spanned by the relations that the tensor product must satisfy.
It is straightforward to prove that the result of this construction satisfies the universal property considered below.
In this section, the universal property satisfied by the tensor product is described.
It follows that this is a (non-constructive) way to define the tensor product of two vector spaces.
The "universal-property definition" of the tensor product of two vector spaces is the following (recall that a bilinear map is a function that is separately linear in each of its arguments): Like the universal property above, the following characterization may also be used to determine whether or not a given vector space and given bilinear map form a tensor product.
denote the tensor product of n copies of the vector space V. For every permutation s of the first n positive integers, the map: induces a linear automorphism of
, their tensor product: is the unique linear map that satisfies: One has: In terms of category theory, this means that the tensor product is a bifunctor from the category of vector spaces to itself.
By choosing bases of all vector spaces involved, the linear maps f and g can be represented by matrices.
is the dual vector space (which consists of all linear maps f from V to the ground field K).
[6] The interplay of evaluation and coevaluation can be used to characterize finite-dimensional vector spaces without referring to bases.
The first two properties make φ a bilinear map of the abelian group
a module structure under some extra conditions: For vector spaces, the tensor product
Higher Tor functors measure the defect of the tensor product being not left exact.
All higher Tor functors are assembled in the derived tensor product.
A particular example is when A and B are fields containing a common subfield R. The tensor product of fields is closely related to Galois theory: if, say, A = R[x] / f(x), where f is some irreducible polynomial with coefficients in R, the tensor product can be calculated as:
In the larger field B, the polynomial may become reducible, which brings in Galois theory.
There is an analogous operation, also called the "tensor product," that makes Hilbert spaces a symmetric monoidal category.
It is essentially constructed as the metric space completion of the algebraic tensor product discussed above.
However, such a construction is no longer uniquely specified: in many cases, there are multiple natural topologies on the algebraic tensor product.
Vector spaces endowed with an additional multiplicative structure are called algebras.
Thus, all tensor products can be expressed as an application of the monoidal category to some particular setting, acting on some particular objects.
, the nth exterior power of V. The latter notion is the basis of differential n-forms.
Other array languages may require explicit treatment of indices (for example, MATLAB), and/or may not support higher-order functions such as the Jacobian derivative (for example, Fortran/APL).
