This book was conceived as a text combining the course of linear algebra and analytic geometry. It originated as a course of lectures delivered by N. V. Efimov at Moscow State University(mechanics and mathematics department) in 1964-1966. However, the material of these lectures has been completely reworked and substantially expanded. We have tried to bear in mind the requi­rements of other mathematical disciplines and also of mechanics and physics. We hope that all parts of the text will be useful. The only preparation required for this text can be given an a first- semester course of analytic geometry and algebra at the most ele­mentary level. All that is needed is a firm grasp of the elements of these subjects. For Chapter XII the student should be ac­quainted with projective transformations and the projective pro­perties of figures in the plane. Also, in Chapter X the reader may simplify his task by skipping Subsections 13 to 23 (Section 3) and Subsection 10 of Section 7. What is left of Chapter X can serve as a minimal algebraic basis for the theory of multidimensional in­tegration.It may be noted in conclusion that the first five chapters already contain material with broad applications in mathematics, mecha­nics, and physics. These chapters, supplemented with some of the material of subsequent chapters, can be utilized in higher tech­nical schools with a more advanced mathematics curriculum.