We discuss linear algebra of infinite-dimensional vector spaces in terms algebraic (Hamel) bases. As an application we prove the surjectivity a large class partial differential operators with smooth ($\mathcal C^\infty$-coefficients) coefficients, called article \emph{regular}, acting on dual $\mathcal D^*(\Omega)$ space test-functions D(\Omega)$. The guarantees solvability corresponding equati...