The work is devoted to the problem of solving large systems of linear algebraic equations with irregular structure matrices. To solve them the variant of the projection method in the Petrov-Galerkin form is proposed. Most of the known projection methods is based on the use of bases of Krylov subspaces. The main difference of the proposed method is the choice of the basis from coefficients of wavelet packet decomposition of the residuals. In general, the wavelet transform can be adaptive due to the entropic criteria for the evaluation of elements of the wavelet tree. This distinguishes the proposed method from the known FOM method, the GMRES algorithm and other projection solvers. Conducted a series of computational experiments comparing the proposed algorithm with the main existing projection methods. The experiments showed that the proposed algorithm is competitive with the major existing projection type methods, and in some cases can exceed them.