tensor product

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Noun

tensor product (plural tensor products)

1. (math) 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 ⊗.

   • 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.

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

Translations

• French: produit tensoriel
• German: Tensorprodukt

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