Tensor product

English

Noun

1. Template:Lb The most general bilinear operation in various contexts (as with vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, modules, and so on), denoted by ⊗.
   • Template:Quote-book

     Linear Transformations on Tensor Products of Vector Spaces

     [...]

     Proposition 16. Let V and W be finite dimensional vector spaces over the field F with bases $v_1,...,v_n$ and $w_1,...,w_m$ respectively. Then $V{\otimes }_{F}W$ is a vector space over F of dimension nm with basis $v_i\otimes w_j$, $1\le i\le n$ and $1\le j\le m$.

   • Template:Quote-av

     [The tensor product] $U\otimes V$ is the span of $\{u\otimes v:u\in U,v\in V\}$
         modulo the relations
         $\{\begin{array}{c}\cdot u\otimes (v+v^{\prime })=u\otimes v+u\otimes v^{\prime }\\ \cdot (u+u^{\prime })\otimes v=u\otimes v+u^{\prime }\otimes v\\ \cdot (\lambda u)\otimes v=\lambda (u\otimes v)=u\otimes (\lambda v)\end{array}$

Usage notes

• The ⊗ symbol can be read out as “tensor”.

Hyponyms

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• Template:L, Template:L

Meronyms

• Template:L (if the tensor product is between algebraic structures)

Holonyms

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Translations

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