= Tensor rank
{title2=$R_k(T)$}
For a <tensor> $T\in V_1\otimes\cdots\otimes V_d$ over a <field> $k$, its <tensor rank> is the smallest number of <pure tensors> needed in an exact expression
$$
T=\sum_{j=1}^r v_{1j}\otimes\cdots\otimes v_{dj}.
$$
The zero tensor has rank zero. The field and the specified tensor factors are part of the definition. For two factors, the rank agrees with the <matrix rank> after choosing <bases>; for three or more factors, one matrix does not capture the full decomposition problem.<Geometry and complexity theory by J. M. Landsberg>[opening definitions of tensor rank and its relation to bilinear algorithms]{ref}
For instance, $e_1\otimes f_1+e_2\otimes f_2$ has rank two. It cannot be a <pure tensor>, because its coefficient <matrix> has two independent columns. Every <pure tensor> corresponds to an outer-product matrix of rank at most one. In a <bilinear algorithm>, rank counts the scalar products of linear input expressions needed to form all outputs. <Border rank> asks the different question of how many terms suffice after allowing a limiting process.
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