  We discuss linear algebra of infinite-dimensional vector spaces in terms of
algebraic (Hamel) bases. As an application we prove the surjectivity of a large
class of linear partial differential operators with smooth ($\mathcal
C^\infty$-coefficients) coefficients, called in the article \emph{regular},
acting on the algebraic dual $\mathcal D^*(\Omega)$ of the space of
test-functions $\mathcal D(\Omega)$. The surjectivity of the partial
differential operators guarantees solvability of the corresponding partial
differential equations within $\mathcal D^*(\Omega)$. We discuss our result in
contrast to and comparison with similar results about the restrictions of the
regular operators on the space of Schwartz distribution $\mathcal
D^\prime(\Omega)$, where these operators are often non-surjective.
