Mir Publishers Moscow This textbook comprises lectures read by the author to the first- year students of mathematics at Moscow State University. The book is divided into two parts containing the texts of lectures read in the first and second semesters, respectively. Part One contains 29 lectures and read in the first semester. The subject matter is presented on the basis of vector axiomatics of geometry with special emphasis on logical sequence in introduction of the basic geometrical concepts. Systematic exposition and application of bivectors and trivectors enables the author to successfully combine the above course of lectures with the lectures of the following semesters. The book is intended for university undergraduates majoring in mathematics. <*Oi 'NIRO' LECTURES o v I N GEOMET l \ \ SEMESTER I ANALYTIC GEOMETRY M. M. EIOCTHUKOS JIEKqilH no rEOMETPHH CEMECTP I AHAJIHTHHECKAH rEOMETPHH MOCKBA «HAyKA» rjiasHan penamuiH $n3HKo*MaTeMaTinecKoft jTHTepaTypu M. POSTNIKOV LECTURES IN GEOMETRY SEMESTER I ANALYTIC GEOMETRY Translated from the Russian by Vladimir Shokurov MIR PUBLISHERS Moscow First published 1982 Revised from the 1979 Russian edition Ha aufjiuucKOM xattne © rnaBHan pegaKijiiH <f>nanKO-MaTeMaTniecKOH nuTepaTypu naAaTenbCTBa «HayKa», 1979 © English translation, Mir Publishers, 1982 CONTENTS Preface to the Russian edition Preface to the English edition Lecture 1 The subject-matter of analytic geometry. Vectors. Vector addition. Multiplication of a vector by a number. Vector spaces. Examples. Vector spaces over an arbitrary field Lecture 2 The simplest consequences of the vector space axioms. Independence of the sum of any number of vectors on brackets arrangement. The concept of a family Lecture 3 Linear dependence and linear independence. Linearly independent sets. The simplest properties of linear depen­ dence. Linear-dependence theorem Lecture 4 Collinear vectors. Coplanar vectors. The geometrical mean­ ing of collinearity and coplanarity. Complete families of vectors, bases, dimensionality. Dimensionality axiom. Basis criterion. Coordinates of a vector. Coordinates of the sum of vectors and those of the product of a vector by a number Lecture 5 Isomorphisms of vector spaces. Coordinate isomorphisms. The isomorphism of vector spaces of the same dimension. The method of coordinates. Affine spaces. The isomorphism of affine spaces of the same dimension. Affine coordinates. Straight lines in affine space. Segments 6 Contents Lecture 6 54 Parametric equations of a straight line. The equation of a straight line in a plane. The canonical equation of a straight line in a plane. The general equation of a straight line in a plane. Parallel lines. Relative position of two straight lines in a plane. Uniqueness theorem. Position of a straight line relative to coordinate axes. The half­ planes into which a straight line divides a plane Lecture 7 63 An intuitive notion of a bivector. A formal definition of the bivector. The coincidence of the two definitions. A zero bivector. Conditions for the equality of bivectors. Parallelism of the vector and the bivector. The role of the three-dimensionality condition. Addition of bivectors Lecture 8 71 The correctness of the definition of a bivector sum. The product of a bivector by a number. Algebraic properties of external product. The vector space of bivectors. Bivectors in a plane and the theory of areas. Bivectors in space Lecture 9 82 Planes in space. Parametric equations of a plane. The general equation of a plane. A plane passing through three noncollinear points Lecture 10 87 The half-spaces into which a plane divides space. Relative positions of two planes in space. Straight lines in space. A plane containing a given straight line and passing through a given point. Relative positions of a straight line and a plane in space. Relative positions of two straight lines in space. Change from one basis for a vector space to another Lecture 11 99 Formulas for the transformation of vector coordinates. Formulas for the transformation of the affine coordinates of points. Orientation. Induced orientation of a straight line. Orientation of a straight line given by an equation. Orientation of a plane in space Contents 7 Lecture 12 112 Deformation of bases. Sameness of the sign bases. Equiva­ lent bases and matrices. The coincidence of deformabil- ity with the sameness of sign. Equivalence of linearly independent systems of vectors. Trivectors. The product of a trivector by a number. The external product of three vectors Lecture 13 123 Trivectors in three-dimensional vector space. Addition of trivectors. The formula for the volume of a parallelepiped. Scalar product. Axioms of scalar multiplication. Euclidean spaces. The length of a vector and the angle between vectors. The Cauchy-Buniakowski inequality. The triangle inequal­ ity. Theorem on the diagonals of a parallelogram. Or­ thogonal vectors and the Pythagorean theorem Lecture 14 133 Metric form and metric coefficients. The condition of posi­ tive definiteness. Formulas for the transformation of metric coefficients when changing a basis. Orthonormal families of vectors and Fourier coefficients. Orthonormal bases and rectangular coordinates. Decomposition of posi­ tive definite matrices. The Gram-Schmidt orthogonalization process. Isomorphism of Euclidean spaces. Orthogonal matrices. Second-order orthogonal matrices. Formulas for the transformation of rectangular coordinates Lecture 15 148 Trivectors in oriented Euclidean space. Triple product of three vectors. The area of a bivector in Euclidean space. A vector complementary to a bivector in oriented Euclidean space. Vector multiplication. Isomorphism of spaces of vectors and bivectors. Expressing a vector product in terms of coordinates. The normal equation of a straight line in the Euclidean plane and the distance between a point and a straight line. Angles between two straight lines in the Euclidean plane Lecture 16 160 The plane in Euclidean space. The distance from a point to a plane. The angle between two planes, between a straight line and a plane, Detween two straight lines. The distance from a point to a straight line in space. The distance between two straight lines in space. The equations of the common perpendicular of two skew lines in space