From the Rank Theorem to the Vaschy-Buckingham Theorem

This pedagogical note presents the mathematical foundations of the Vaschy-Buckingham theorem, known in fluid mechanics as the -theorem. The goal is to make explicit the link between this central result of dimensional analysis and the rank theorem of a linear map, as taught in first- or second-year undergraduate linear algebra. We show that the dimensionless condition can be reformulated as the search for the kernel of a linear map --- the dimensional matrix --- and that the number of independent dimensionless quantities is precisely the dimension of this kernel. The running example is the Stokes problem: the viscous drag exerted by a Newtonian fluid on a moving sphere, for which the method recovers the canonical form F = (Re) \, v² R², bringing out the Reynolds number.