Meaning of tensor product | Babel Free

Definitions

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 ⊗.

Equivalents

Examples

“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#95;1,...,v#95;n and w#95;1,...,w#95;m respectively. Then V#92;otimes#95;FW is a vector space over F of dimension nm with basis v#95;i#92;otimesw#95;j, 1#92;lei#92;len and 1#92;lej#92;lem.”

“[The tensor product] U#92;otimesV is the span of #92;#123;u#92;otimesv#58;u#92;inU,v#92;inV#92;#125; modulo the relations #92;begin#123;cases#125;#92;cdot#92;u#92;otimes(v#43;v')#61;u#92;otimesv#43;u#92;otimesv'#92;#92;#92;cdot#92;(u#43;u')#92;otimesv#61;u#92;otimesv#43;u'#92;otimesv#92;#92;#92;cdot#92;(#92;lambdau)#92;otimesv#61;#92;lambda(u#92;otimesv)#61;u#92;otimes(#92;lambdav)#92;end#123;cases#125;”

CEFR level