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Algebra and Geometry

Book Description

Describing two cornerstones of mathematics, this basic textbook presents a unified approach to algebra and geometry. It covers the ideas of complex numbers, scalar and vector products, determinants, linear algebra, group theory, permutation groups, symmetry groups and aspects of geometry including groups of isometries, rotations, and spherical geometry. The book emphasises the interactions between topics, and each topic is constantly illustrated by using it to describe and discuss the others. Many ideas are developed gradually, with each aspect presented at a time when its importance becomes clearer. To aid in this, the text is divided into short chapters, each with exercises at the end. The related website features an HTML version of the book, extra text at higher and lower levels, and more exercises and examples. It also links to an electronic maths thesaurus, giving definitions, examples and links both to the book and to external sources.

Note:The ebook version does not provide access to the companion files.

Table of Contents

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