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.