Tensor product

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In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general bilinear operation. In some contexts, this product is also referred to as outer product. The term "tensor product" is also used in relation to monoidal categories.


Tensor product of vector spaces

The tensor product V\otimesW of two vector spaces V and W over a field K can be defined by the method of generators and relations.

To construct V ⊗ W, one begins with the set of ordered pairs in the Cartesian product V × W. For the purposes of this construction, regard this Cartesian product as a set rather than a vector space. The free vector space F on V × W is defined by taking the vector space in which the elements of V × W are a basis. In set-builder notation,

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