Completion and decomposition of a clutter into representable matroids

This paper deals with the question of completing a monotone increasing family of subsets Γ of a finite set Ω to obtain the linearly dependent subsets of a family of vectors of a vector space. Specifically, we prove that such vectorial completions of the family of subsets Γ exist and, in addition, we show that the minimal vectorial completions of the family Γ provide a decomposition of the clutter Λ of the inclusion-minimal elements of Γ. The computation of such vectorial decomposition of clutters is also discussed in some cases.