The tensor product of two vector spaces 
and 
, denoted 
and also called the tensor
direct product, is a way of creating a new vector
space analogous to multiplication of integers. For instance,
 |
(1)
|
In particular,
 |
(2)
|
Also, the tensor product obeys a distributive law with the direct sum operation:
 |
(3)
|
The analogy with an algebra is the motivation behind K-theory. The tensor product of two tensors 
and 
can be implemented in the Wolfram
Language as:
TensorProduct[a_List, b_List] := Outer[List, a, b]
Algebraically, the vector space 
is spanned by
elements of the form 
, and the following rules are satisfied, for any scalar
.
The definition is the same no matter which scalar field
is used.
 |
(4)
|
 |
(5)
|
 |
(6)
|
One basic consequence of these formulas is that
 |
(7)
|
A vector basis 
of 
and 
of 
gives a basis for 
, namely 
, for all pairs 
. An arbitrary element of 
can be written uniquely as 
, where 
are scalars. If 
is 
dimensional and 
is 
dimensional, then 
has dimension 
.
Using tensor products, one can define symmetric tensors, antisymmetric tensors, as well as the exterior algebra. Moreover, the tensor product is generalized to the vector bundle tensor product. In particular, tensor products of the tangent bundle and its dual bundle are studied in Riemannian geometry and physics. Sections of these bundles are often called tensors. In addition, it is possible to take the representation tensor product to get another representation.
All of these versions of tensor product can be understood as module tensor products. The trick is to find the right way to think of these spaces as modules.
