Algebra and geometry / Alan F. Beardon

Includes index

Preface -- 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 determinant s- -- 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 groups -- 12.8. Dihedral groups -- 12.9. Groups of small order -- 12.10. Conjugation -- 12.11. Homomorphisms -- 12.12. Quotient groups -- 13. Möbius 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 -- Index

Describing two cornerstones of mathematics, this basic textbook presents a unified approach to algebra and geometry