||The book is a substantially revised version of a book by the same author "Introduction to the theory of linear spaces" (Gostehizdat, 1952 and 1956). Edition corresponds to the main program of the university course of linear algebra and is designed primarily for students of mathematical, physical and other natural-science specialities. To read it is necessary, as a rule, possess only elementary mathematics; in some cases, use the information in mathematical analysis with the relevant references. In chapter 1 outlines the theory of determinants. In chapters 2-7 considered affine theory of linear spaces over arbitrary numeric field), in chapters 8-10-theory of euclidean and unitary spaces. In chapter 11 describes the algebra of linear operators in finite-dimensional spaces and in chapter 12 of the relevant category.