Dual finite frames for vector spaces over an arbitrary field with applications

In the present paper, we study frames for finite-dimensional vector spaces over an arbitrary field. We develop a theory of dual in order to obtain and different representations elements space provided by frame. relate introduced with classical one Hilbert apply it three types spaces: conjugate closed subfields complex numbers (in particular, cyclotomic fields), metric spaces, ultrametric normed complete non-archimedean valued fields. Finally, consider matrix representation operators using its application solution equations Petrov-Galerkin scheme.