= Tensor factor
A tensor factor is one of the spaces or modules in a specified <tensor product> decomposition, such as $V$ or $W$ in $V\otimes_F W$. The decomposition is part of the data; an abstract vector space need not come with a preferred choice of tensor factors.
For finite-dimensional spaces, $\dim(V\otimes W)=\dim V\,\dim W$. Thus two two-dimensional factors produce a four-dimensional space, with basis $e_i\otimes f_j$. A vector $v$ from one factor is not itself a tensor-product vector without choosing additional data in the other factor.
A linear operator acting only on the first factor is represented on the product by $A\otimes I_W$, with $(A\otimes I_W)(v\otimes w)=Av\otimes w$. Operators on separate factors commute: $(A\otimes I)(I\otimes B)=A\otimes B=(I\otimes B)(A\otimes I)$. This notation makes independent actions precise, including local operations on composite quantum systems.
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