Page "Rank (linear algebra)" Paragraph 35

The rank can also be characterized as the decomposition rank: the minimum k such that A can be factored as, where C is an m × k matrix and R is a k × n matrix.

Like the " dimension of image " characterization this can be generalized to a definition of the rank of a linear map: the rank of a linear map f from V → W is the minimal dimension k of an intermediate space X such that f can be written as the composition of a map V → X and a map X → W. While this definition does not suggest an efficient manner to compute the rank ( for which it is better to use one of the alternative definitions ), it does allow to easily understand many of the properties of the rank, for instance that the rank of the transpose of A is the same as that of A.