Vector Geometry

This brief undergraduate-level text by a prominent Cambridge-educated mathematician explores the relationship between algebra and geometry. An elementary course in plane geometry is the sole requirement for Gilbert de B. Robinson's text, which is the result of several years of teaching and learning the most effective methods from discussions with students. Topics include lines and planes, determinants and linear equations, matrices, groups and linear transformations, and vectors and vector spaces. Additional subjects range from conics and quadrics to homogeneous coordinates and projective geometry, geometry on the sphere, and reduction of real matrices to diagonal form. Exercises appear throughout the text, with complete answers at the end. Title Page Copyright Page Dedication Preface Reference Contents 1 LINES AND PLANES 1.1 Coordinate Geometry 1.2 Equations of a Line 1.3 Vector Addition 1.4 the Inner Product 1.5 Linear Dependence 1.6 Equations of a Plane 2 DETERMINANTS AND LINEAR EQUATIONS 2.1 The problem defined 2.2 Determinants 2.3 Evaluation of a determinant 2.4 Intersections of three planes 2.5 Homogeneous equations 3 MATRICES 3.1 Matrix addition and multiplication 3.2 Transpose of a matrix 3.3 Inverse of a matrix 3.4 Reduction of a matrix to canonical form 3.5 Inverse of a matrix 3.6 The approximate inverse of a matrix 3.7 Linear transformations 4 GROUPS AND LINEAR TRANSFORMATION 4.1 Definition of a group 4.2 The symmetric group 4.3 The group of a square 4.4 Rotations and reflections 4.5 The group of the cube 4.6 Euler’ formula 4.7 The regular polyhedra 4.8 Polytopes 5 VECTORS AND VECTOR SPACES 5.1 Basis vectors 5.2 Gram-Schmidt orthogonalization process 5.3 The vector product U × V 5.4 Distance between two skew lines 5.5 n-Dimensional volume 5.6 Subspaces of υn 5.7 Equations of a subspace 5.8 Orthogonal projection 6 CONICS AND QUADRICS 6.1 Circles and spheres 6.2 Conics in Cartesian coordinates 6.3 Quadrics and the lines on them 6.4 Cones, cylinders, and surfaces of revolution 6.5 Pairs of lines and planes 6.6 A quadric to contain three skew lines 6.7 The intersection of two quadrics 7 HOMOGENEOUS COORDINATES AND PROJECTIVE GEOMETRY 7.1 Euclidean geometry 7.2 Homogeneous coordinates 7.3 Axioms of projective geometry 7.4 Theorems of Desargues and Pappus 7.5 Affine and Euclidean geometry 7.6 Desargues’ theorem in the Euclidean plane 7.7 Pappus’ theorem in the Euclidean plane 7.8 Cross ratio 8 GEOMETRY ON THE SPHERE 8.1 Spherical trigonometry 8.2 The polar triangle 8.3 Area of a spherical triangle 8.4 The inversion transformation 8.5 Geometrical properties of inversion 8.6 Stereographic projection 8.7 Elliptic geometry 8.8 Hyperbolic geometry 9 REDUCTION OF REAL MATRICES TO DIAGONAL FORM 9.1 Introduction 9.2 Change of basis 9.3 Characteristic vectors 9.4 Collineations 9.5 Reduction of a symmetric matrix 9.6 Similar matrices 9.7 Orthogonal reduction of a symmetric matrix 9.8 The real classical groups 9.9 Reduction of the general conic to normal form APPENDIX ANSWERS TO EXERCISES INDEX This concise undergraduate-level text explores the relationship between algebra and geometry. Topics include determinants and linear equations, matrices, linear transformations, projective geometry, geometry on the sphere, and much more. An elementary course in plane geometry is the sole requirement, and answers to the exercises appear at the end. 1962 edition.