In this edition, chapters I, II, which now include the theory of determinants and systems of linear equations, are printed according to the manuscript of Anatoly Ivanovich, and chapters III-VII reproduce the corresponding text of the second edition almost unchanged. The final chapter of the second edition - on tensors - Anatoly Ivanovich was going to undergo radical processing, but did not even have time to start it; this chapter is omitted in this edition.