tensor product

Noun

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

Usage notes

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

Meronyms

 * (if the tensor product is between algebraic structures)

Translations

 * Estonian: tensorkorrutis
 * Finnish: tensoritulo
 * French:
 * German: Tensorprodukt
 * Japanese: テンソル積
 * Romanian: produs tensorial