Vectors, Matrices and Geometry

CONTENT

CONTENT Preface vu Chapter One Vectors and geometry in the plane 1.1 Vectors in the plane 1.2 The vector space R 2 1.3 Linear combinations 1.4 Plane geometry 1.5 The dot product 1.6 Equations of a straight line Chapter Two Vectors and geometry in space 2 4 10 15 22 32 2.1 Cartesian coordinates in space 39 2.2 Vectors in space 43 2.3 Linear independence 46 2.4 Geometry in space 52 2.5 The dot product 58 2.6 Planes in space 64 2.7 The cross product 71 2.8 Lines in space 79 2.9 Cylindrical coordinates and spherical coordinates 85 Chapter Three Conic sections 3.1 Focus, directrix and eccentricity 3.2 The parabola 3.3 The ellipse 3.4 The hyperbola 3.5 Plane sections of a cone 3.6 The Dandelin sphere 3.7 Parallel translation 3.8 Rotation 3.9 General quadratic equation 3.10 Tangents v 90 91 95 106 114 122 126 138 147 154 Oontent Chapter Four Quadric surfaces 4.1 Surfaces of revolution 4.2 Cylinders and cones 4.3 The ellipsoid 4.4 The hyperboloid of two sheets 4.5 The hyperboloid of one sheet 4.6 The elliptic paraboloid 4.7 The hyperbolic paraboloid 4.8 Coordinate transformations 4.9 The general quadratic equation Chapter Five Higher dimensional vector spaces 5.1 The vector space Rn 5.2 Linear combinations and subspaces 5.3 Linear independence 5.4 Elementary transformations of matrices 5.5 The rank of a matrix 5.6 Dimension and base Chapter Six Matrix and determinant 6.1 Terminology 6.2 Scalar multiple and sum 6.3 Product 6.4 Square matrix 6.5 Invertible matrix 6.6 Determinant of order 2 6.7 Determinant of order 3 6.8 Determinant of higher order Chapter Seven Linear equations 7.1 Terminology 7.2 Condition for consistency 7.3 Homogeneous linear equations 7.4 Inhomogeneous system 7.5 Cramer's rule Numerical answers to exercises Index vi 167 174 178 183 185 189 191 193 201 207 210 218 221 229 236 243 245 247 253 259 266 267 279 281 283 289 300 309 315 341 ...