Tensor product

The tensor product of two vectors is defined from their decomposition on the bases. More precisely, if

A construction of the tensor product that is basis independent can be obtained in the following way.

A consequence of this approach is that every property of the tensor product can be deduced from the universal property, and that, in practice, one may forgive the method that has been used to prove its existence.

This allows omitting parentheses in the tensor product of more than two vector spaces or vectors.

A dyadic product is the special case of the tensor product between two vectors of the same dimension.

Some vector spaces can be decomposed into direct sums of subspaces. In such cases, the tensor product of two spaces can be decomposed into sums of products of the subspaces (in analogy to the way that multiplication distributes over addition).

The most general setting for the tensor product is the monoidal category. It captures the algebraic essence of tensoring, without making any specific reference to what is being tensored. Thus, all tensor products can be expressed as an application of the monoidal category to some particular setting, acting on some particular objects.

The symmetric algebra is constructed in a similar manner, from the symmetric product