Vector Space Tensor Product

The tensor product of two vector spaces V and W, denoted V tensor W and also called the tensor direct product, is a way of creating a new vector space analogous to multiplication of integers. For instance,

 R^n tensor R^k=R^(nk).

In particular,

 r tensor R^n=R^n.

Also, the tensor product obeys a distributive law with the direct sum operation:

 U tensor (V direct sum W)=(U tensor V) direct sum (U tensor W).

The analogy with an algebra is the motivation behind K-theory. The tensor product of two tensors a and b can be implemented in the Wolfram Language as:

  TensorProduct[a_List, b_List] := Outer[List, a, b]

Algebraically, the vector space V tensor W is spanned by elements of the form v tensor w, and the following rules are satisfied, for any scalar alpha. The definition is the same no matter which scalar field is used.

 (v_1+v_2) tensor w=v_1 tensor w+v_2 tensor w
 v tensor (w_1+w_2)=v tensor w_1+v tensor w_2
 alpha(v tensor w)=(alphav) tensor w=v tensor (alphaw)

One basic consequence of these formulas is that

 0 tensor w=v tensor 0=0.

A vector basis v_i of V and w_j of W gives a basis for V tensor W, namely v_i tensor w_j, for all pairs (i,j). An arbitrary element of V tensor W can be written uniquely as suma_(i,j)v_i tensor w_j, where a_(i,j) are scalars. If V is n dimensional and W is k dimensional, then V tensor W has dimension nk.

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.

See also

Antisymmetric Tensor, Exterior Algebra, Field, K-Theory, Module, Module Tensor Product, Representation Tensor Product, Symmetric Tensor, Tensor, Tensor Direct Product, Vector Space

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Vector Space Tensor Product." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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