: Mathematics: tensor product

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tensor product

In mathematics, the tensor product ${\\displaystyle V\\otimes W}$ of two vector spaces $V$ and $W$ (over the same field) is a vector space to which is associated a bilinear map ${\\displaystyle V\ imes W\ ightarrow V\\otimes W}$ that maps a pair ${\\displaystyle (v,w),\\ v\\in V,w\\in W}$ to an element of ${\\displaystyle V\\otimes W}$ denoted ${\\displaystyle v\\otimes w.}$ An element of the form ${\\displaystyle v\\otimes w}$ is called the tensor product of $v$ and $w$ . An element of ${\\displaystyle V\\otimes W}$ is a tensor, and the tensor product of two vectors is sometimes called an elementary tensor or a decomposable tensor. The elementary tensors span ${\\displaystyle V\\otimes W}$ in the sense that every element of ${\\displaystyle V\\otimes W}$ is a sum of elementary tensors. If bases are given for $V$ and $W$ , a basis of ${\\displaystyle V\\otimes W}$ is formed by all tensor products of a basis element of $V$ and a basis element of $W$ .
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Tensor_product)