| 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 dependence. Linear-dependence theorem |

| Lecture 4 |
Collinear vectors. Coplanar vectors. The geometrical meaning 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 |

| Lecture 6 |
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 |
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 |
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 |
Planes in space. Parametric equations of a plane. The general equation of a plane. A plane passing through three noncollinear points |

| Lecture 10 |
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 |
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 |

| Lecture 12 |
Deformation of bases. Sameness of the sign bases. Equivalent 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 |
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 inequality. Theorem on the diagonals of a parallelogram. Orthogonal vectors and the Pythagorean theorem |

| Lecture 14 |
Metric form and metric coefficients. The condition of positive 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 positive definite matrices. The Gram-Schmidt orthogonalization process. Isomorphism of Euclidean spaces. Orthogonal matrices. Second-order orthogonal matrices. Formulas for he transformation of rectangular coordinates |

| Lecture 15 |
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 |
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, between 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 hn space |

| Lecture 17 |
The parabola. The ellipse. The focal and directorial properties of the ellipse. The hyperbola. The focal and directorial properties of the hyperbola |

| Lecture 18 |
The equations of ellipses, parabolas and hyperbolas leferred to a vertex. Polar coordinates. The equations of ellipses, parabolas and hyperbolas in polar coordinates. Affine ellipses, parabolas, hyperbolas. Algebraic curves. Second-degree curves and associated difficulties. Complex affine geometry and its insufficiency |

| Lecture 19 |
Real-complex vector spaces. Their dimensionality. Isomorphism of real-complex vector spaces. Complexification. Real-complex affine spaces. The complexification of affine spaces. Real-complex Euclidean spaces. Real and imaginary curves of second degree. |

| Lecture 20 |
Introductory remarks. The centre of a second-degree curve. Centres of symmetry. Central and nonceniral curves of second degree. Straight lines of non-asymptotic direction. Tangents. Straight lines of asymptotic direction |

| Lecture 21 |
Singular and nonsingular directions. Diameters. Diameters and centres. Conjugate directions and conjugate diameters. Simplification of the equation of the second-degree central curve. Necessary refinements. Simplification of the equation of the second-degree rioncentral curve |

| Lecture 22 |
Second-degree curves in the complex affine plane. Second-degree curves in the real-complex affine plane. The uniqueness of the equation of a second-degree curve. Second-degree curves in the Euclidean plane. Circles |

| Lecture 23 |
Ellipsoids. Imaginary ellipsoids. Second-degree imaginary cones. Hyperboloids of two sheets. Hyperboloids of one sheet. Rectilinear generators of a hyperboloid of one sheet.
Second-degree cones. Elliptical paraboloids. Hyporbolic paraboloids. Elliptical cylinders. Other second-degree surfaces. The statement of the classification theorem |

| Lecture 24 |
Coordinates of a straight line. Pencils of straight lines. Ordinary and ideal pencils. Extended planes. Models of projective-affine geometry |

| Lecture 25 |
Homogeneous affine coordinates. Equations of straight lines in homogeneous coordinates. Second-degree curves in the projective-affine plane. Circles in the projective-Euclidean real-complex plane. Projective planes. Homogeneous afftne coordinates in the bundle of straight lines. Formulas for the transformation of homogeneous affine coordinates. Projective coordinates. Second-degree curves in the projec-tive plane |

| Lecture 26 |
Coordinate isomorphisms of vector spaces. Coordinate isomorphisms of affine spaces. Projective-affine spaces. Projective spaces. Pencils of planes. Bundles of planes. Extending space with ideal elements. Orthogonal, affine and projective transformations |

| Lecture 27 |
Expressing an affine transformation in terms of coordinates. Examples of affine transformations. Factorization of affine transformations. Orthogonal transformations. Motions of a plane. Symmetries and glide symmetries. A motion of a plane as a composition of two symmetries. Rotations of a space |

| Lecture 28 |
The Desargues theorem. The Pappus-Pascal theorem. The Fano theorem. The duality principle. Models of the projective plane. Models of the projective straight line and of the projective space. The complex projective straight line |

| Lecture 29 |
Linear fractional transformations. Linear transformations. Inversion. Inversions and linear fractional transformations. Two properties of linear fractional transformations.
Fixed points of linear fractional transformations. Parabolic, elliptical, hyperbolic and loxodromic linear fractional transformations. The three-point theorem. The multiplier of linear fractional nonparabolic transformation. Classification of linear fractional transformations. Stereographic projection formulas. Rotations of a sphere as linear fractional transformations of a plane. Isometries of a cube
Subject index |