tensor product

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tensor product

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tensor product (plural tensor products)

1. (mathematics) 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 ⊗.
   • 2004, David S. Dummit with Richard M. Foote, chapter 11, in Abstract Algebra‎, →ISBN, →OCLC, page 421:

     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 ${\displaystyle v_{1},...,v_{n}}$ and ${\displaystyle w_{1},...,w_{m}}$ respectively. Then ${\displaystyle V\otimes _{F}W}$ is a vector space over F of dimension nm with basis ${\displaystyle v_{i}\otimes w_{j}}$, ${\displaystyle 1\leq i\leq n}$ and ${\displaystyle 1\leq j\leq m}$.

   • 2012, 27:30 from the start, in Lecture 1 . Hopf Algebras and Combinatorics (Federico Ardila)‎, spoken by Federico Ardila (Federico Ardila), Federico Ardila, via YouTube:

     ${\displaystyle U\otimes V}$ is the span of ${\displaystyle \{u\otimes v:u\in U,v\in V\}}$
         modulo the relations
         ${\displaystyle {\begin{cases}\cdot \ u\otimes (v+v')=u\otimes v+u\otimes v'\\\cdot \ (u+u')\otimes v=u\otimes v+u'\otimes v\\\cdot \ (\lambda u)\otimes v=\lambda (u\otimes v)=u\otimes (\lambda v)\end{cases}}}$

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

• Kronecker product
• exterior product, wedge product

• tensor (if the tensor product is between algebraic structures)

• monoidal category