The paper is concerned with a linear method for solving linear inverse problems with discrete data. Extending the earlier approach of G. Backus and F. Gilbert [Geophys. J. Roy. Astron. Soc. 16, 169-205 (1968; Zbl 0177.541) and Philos. Trans. Roy. Soc. London, Ser. A 266, 123-192 (1970; MR 56.7763)] the authors give the general formulation of the method defining a mapping from a Hilbert space to a finite- dimensional vector space. The singular system of this mapping is used to define natural bases in both spaces. The method is compared with techniques developed in other fields and several examples are discussed in detail. The numerical instabilities are mentioned briefly but the discussion of the methods for overcoming these difficulties is deferred to the second part of the paper.