1 Groups and permutations 1.1 Introduction 1.2 Groups 1.3 Permutations of a finite set 1.4 The sign of a permutation 1.5 Permutations of an arbitrary set
2 The real numbers 2.1 The integers 2.2 The real numbers 2.3 Fields 2.4 Modular arithmetic
3 The complex plane 3.1 Complex numbers 3.2 Polar coordinates 3.3 Lines and circles 3.4 Isometries of the plane 3.5 Roots of unity 3.6 Cubic and quartic equations 3.7 The Fundamental Theorem of Algebra
4 Vectors in three-dimensional space 4.1 Vectors 4.2 The scalar product 4.3 The vector product 4.4 The scalar triple product 4.5 The vector triple product 4.6 Orientation and determinants 4.7 Applications to geometry 4.8 Vector equations
5 Spherical geometry 5.1 Spherical distance 5.2 Spherical trigonometry 5.3 Area on the sphere 5.4 Euler’s formula 5.5 Regular polyhedra 5.6 General polyhedra
6 Quaternions and isometries 6.1 Isometries of Euclidean space 6.2 Quaternions 6.3 Reflections and rotations
7 Vector spaces 7.1 Vector spaces 7.2 Dimension 7.3 Subspaces 7.4 The direct sum of two subspaces 7.5 Linear difference equations 7.6 The vector space of polynomials 7.7 Linear transformations 7.8 The kernel of a linear transformation 7.9 Isomorphisms 7.10 The space of linear maps
8 Linear equations 8.1 Hyperplanes 8.2 Homogeneous linear equations 8.3 Row rank and column rank 8.4 Inhomogeneous linear equations 8.5 Determinants and linear equations 8.6 Determinants
9 Matrices 9.1 The vector space of matrices 9.2 A matrix as a linear transformation 9.3 The matrix of a linear transformation 9.4 Inverse maps and matrices 9.5 Change of bases 9.6 The resultant of two polynomials 9.7 The number of surjections
10 Eigenvectors 10.1 Eigenvalues and eigenvectors 10.2 Eigenvalues and matrices 10.3 Diagonalizable matrices 10.4 The Cayley–Hamilton theorem 10.5 Invariant planes
11 Linear maps of Euclidean space 11.1 Distance in Euclidean space 11.2 Orthogonal maps 11.3 Isometries of Euclidean n-space 11.4 Symmetric matrices 11.5 The field axioms 11.6 Vector products in higher dimensions
12 Groups 12.1 Groups 12.2 Subgroups and cosets 12.3 Lagrange’s theorem 12.4 Isomorphisms 12.5 Cyclic groups 12.6 Applications to arithmetic 12.7 Product group 12.8 Dihedral groups 12.9 Groups of small order 12.10 Conjugation 12.11 Homomorphisms 12.12 Quotient groups
13 Möobius transformations 13.1 Möbius transformations 13.2 Fixed points and uniqueness 13.3 Circles and lines 13.4 Cross-ratios 13.5 Möbius maps and permutations 13.6 Complex lines 13.7 Fixed points and eigenvectors 13.8 A geometric view of infinity 13.9 Rotations of the sphere
14 Group actions 14.1 Groups of permutations 14.2 Symmetries of a regular polyhedron 14.3 Finite rotation groups in space 14.4 Groups of isometries of the plane 14.5 Group actions
15 Hyperbolic geometry 15.1 The hyperbolic plane 15.2 The hyperbolic distance 15.3 Hyperbolic circles 15.4 Hyperbolic trigonometry 15.5 Hyperbolic three-dimensional space 15.6 Finite Möbius groups
Algebra and Geometry, Alan F.Beardon - Cambridge Press
Jul 5 2007, 04:19 PM
