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Vectors, Pure and Applied

Book Description

Many books in linear algebra focus purely on getting students through exams, but this text explains both the how and the why of linear algebra and enables students to begin thinking like mathematicians. The author demonstrates how different topics (geometry, abstract algebra, numerical analysis, physics) make use of vectors in different ways and how these ways are connected, preparing students for further work in these areas. The book is packed with hundreds of exercises ranging from the routine to the challenging. Sketch solutions of the easier exercises are available online.

Table of Contents

  1. Cover Page
  2. Title Page
  3. Copyright
  4. Contents
  5. Introduction
  6. PART I FAMILIAR VECTOR SPACES
    1. 1 Gaussian elimination
      1. 1.1 Two hundred years of algebra
      2. 1.2 Computational matters
      3. 1.3 Detached coefficients
      4. 1.4 Another fifty years
      5. 1.5 Further exercises
    2. 2 A little geometry
      1. 2.1 Geometric vectors
      2. 2.2 Higher dimensions
      3. 2.3 Euclidean distance
      4. 2.4 Geometry, plane and solid
      5. 2.5 Further exercises
    3. 3 The algebra of square matrices
      1. 3.1 The summation convention
      2. 3.2 Multiplying matrices
      3. 3.3 More algebra for square matrices
      4. 3.4 Decomposition into elementary matrices
      5. 3.5 Calculating the inverse
      6. 3.6 Further exercises
    4. 4 The secret life of determinants
      1. 4.1 The area of a parallelogram
      2. 4.2 Rescaling
      3. 4.3 3 x 3 determinants
      4. 4.4 Determinants of n x n matrices
      5. 4.5 Calculating determinants
      6. 4.6 Further exercises
    5. 5 Abstract vector spaces
      1. 5.1 The space Cn
      2. 5.2 Abstract vector spaces
      3. 5.3 Linear maps
      4. 5.4 Dimension
      5. 5.5 Image and kernel
      6. 5.6 Secret sharing
      7. 5.7 Further exercises
    6. 6 Linear maps from Fn to itself
      1. 6.1 Linear maps, bases and matrices
      2. 6.2 Eigenvectors and eigenvalues
      3. 6.3 Diagonalisation and eigenvectors
      4. 6.4 Linear maps from C2 to itself
      5. 6.5 Diagonalising square matrices
      6. 6.6 Iteration's artful aid
      7. 6.7 LU factorisation
      8. 6.8 Further exercises
    7. 7 Distance preserving linear maps
      1. 7.1 Orthonormal bases
      2. 7.2 Orthogonal maps and matrices
      3. 7.3 Rotations and reflections in R2 and R3
      4. 7.4 Reflections in Rn
      5. 7.5 QR factorisation
      6. 7.6 Further exercises
    8. 8 Diagonalisation for orthonormal bases
      1. 8.1 Symmetric maps
      2. 8.2 Eigenvectors for symmetric linear maps
      3. 8.3 Stationary points
      4. 8.4 Complex inner product
      5. 8.5 Further exercises
    9. 9 Cartesian tensors
      1. 9.1 Physical vectors
      2. 9.2 General Cartesian tensors
      3. 9.3 More examples
      4. 9.4 The vector product
      5. 9.5 Further exercises
    10. 10 More on tensors
      1. 10.1 Some tensorial theorems
      2. 10.2 A (very) little mechanics
      3. 10.3 Left-hand, right-hand
      4. 10.4 General tensors
      5. 10.5 Further exercises
  7. PART II GENERAL VECTOR SPACES
    1. 11 Spaces of linear maps
      1. 11.1 A look at L(U, V)
      2. 11.2 A look at L(U, U)
      3. 11.3 Duals (almost) without using bases
      4. 11.4 Duals using bases
      5. 11.5 Further exercises
    2. 12 Polynomials in L(U,U)
      1. 12.1 Direct sums
      2. 12.2 The Cayley-Hamilton theorem
      3. 12.3 Minimal polynomials
      4. 12.4 The Jordan normal form
      5. 12.5 Applications
      6. 12.6 Further exercises
    3. 13 Vector spaces without distances
      1. 13.1 A little philosophy
      2. 13.2 Vector spaces over fields
      3. 13.3 Error correcting codes
      4. 13.4 Further exercises
    4. 14 Vector spaces with distances
      1. 14.1 Orthogonal polynomials
      2. 14.2 Inner products and dual spaces
      3. 14.3 Complex inner product spaces
      4. 14.4 Further exercises
    5. 15 More distances
      1. 15.1 Distance on L(U, U)
      2. 15.2 Inner products and triangularisation
      3. 15.3 The spectral radius
      4. 15.4 Normal maps
      5. 15.5 Further exercises
    6. 16 Quadratic forms and their relatives
      1. 16.1 Bilinear forms
      2. 16.2 Rank and signature
      3. 16.3 Positive definiteness
      4. 16.4 Antisymmetric bilinear forms
      5. 16.5 Further exercises
  8. References
  9. Index