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Linear Algebra: Concepts and Methods

Book Description

Any student of linear algebra will welcome this textbook, which provides a thorough treatment of this key topic. Blending practice and theory, the book enables the reader to learn and comprehend the standard methods, with an emphasis on understanding how they actually work. At every stage, the authors are careful to ensure that the discussion is no more complicated or abstract than it needs to be, and focuses on the fundamental topics. The book is ideal as a course text or for self-study. Instructors can draw on the many examples and exercises to supplement their own assignments. End-of-chapter sections summarise the material to help students consolidate their learning as they progress through the book.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Dedication
  6. Contents
  7. Preface
  8. Preliminaries: before we begin
    1. Sets and set notation
    2. Numbers
    3. Mathematical terminology
    4. Basic algebra
    5. Trigonometry
    6. A little bit of logic
  9. 1. Matrices and vectors
    1. 1.1 What is a matrix?
    2. 1.2 Matrix addition and scalar multiplication
    3. 1.3 Matrix multiplication
    4. 1.4 Matrix algebra
    5. 1.5 Matrix inverses
    6. 1.6 Powers of a matrix
    7. 1.7 The transpose and symmetric matrices
    8. 1.8 Vectors in R[sup(n)]
    9. 1.9 Developing geometric insight
    10. 1.10 Lines
    11. 1.11 Planes in R[sup(3)]
    12. 1.12 Lines and hyperplanes in R[sup(n)]
    13. 1.13 Learning outcomes
    14. 1.14 Comments on activities
    15. 1.15 Exercises
    16. 1.16 Problems
  10. 2. Systems of linear equations
    1. 2.1 Systems of linear equations
    2. 2.2 Row operations
    3. 2.3 Gaussian elimination
    4. 2.4 Homogeneous systems and null space
    5. 2.5 Learning outcomes
    6. 2.6 Comments on activities
    7. 2.7 Exercises
    8. 2.8 Problems
  11. 3. Matrix inversion and determinants
    1. 3.1 Matrix inverse using row operations
    2. 3.2 Determinants
    3. 3.3 Results on determinants
    4. 3.4 Matrix inverse using cofactors
    5. 3.5 Leontief input–output analysis
    6. 3.6 Learning outcomes
    7. 3.7 Comments on activities
    8. 3.8 Exercises
    9. 3.9 Problems
  12. 4. Rank, range and linear equations
    1. 4.1 The rank of a matrix
    2. 4.2 Rank and systems of linear equations
    3. 4.3 Range
    4. 4.4 Learning outcomes
    5. 4.5 Comments on activities
    6. 4.6 Exercises
    7. 4.7 Problems
  13. 5. Vector spaces
    1. 5.1 Vector spaces
    2. 5.2 Subspaces
    3. 5.3 Linear span
    4. 5.4 Learning outcomes
    5. 5.5 Comments on activities
    6. 5.6 Exercises
    7. 5.7 Problems
  14. 6. Linear independence, bases and dimension
    1. 6.1 Linear independence
    2. 6.2 Bases
    3. 6.3 Coordinates
    4. 6.4 Dimension
    5. 6.5 Basis and dimension in R[sup(n)]
    6. 6.6 Learning outcomes
    7. 6.7 Comments on activities
    8. 6.8 Exercises
    9. 6.9 Problems
  15. 7. Linear transformations and change of basis
    1. 7.1 Linear transformations
    2. 7.2 Range and null space
    3. 7.3 Coordinate change
    4. 7.4 Change of basis and similarity
    5. 7.5 Learning outcomes
    6. 7.6 Comments on activities
    7. 7.7 Exercises
    8. 7.8 Problems
  16. 8. Diagonalisation
    1. 8.1 Eigenvalues and eigenvectors
    2. 8.2 Diagonalisation of a square matrix
    3. 8.3 When is diagonalisation possible?
    4. 8.4 Learning outcomes
    5. 8.5 Comments on activities
    6. 8.6 Exercises
    7. 8.7 Problems
  17. 9. Applications of diagonalisation
    1. 9.1 Powers of matrices
    2. 9.2 Systems of difference equations
    3. 9.3 Linear systems of differential equations
    4. 9.4 Learning outcomes
    5. 9.5 Comments on activities
    6. 9.6 Exercises
    7. 9.7 Problems
  18. 10. Inner products and orthogonality
    1. 10.1 Inner products
    2. 10.2 Orthogonality
    3. 10.3 Orthogonal matrices
    4. 10.4 Gram–Schmidt orthonormalisation process
    5. 10.5 Learning outcomes
    6. 10.6 Comments on activities
    7. 10.7 Exercises
    8. 10.8 Problems
  19. 11. Orthogonal diagonalisation and its applications
    1. 11.1 Orthogonal diagonalisation of symmetric matrices
    2. 11.2 Quadratic forms
    3. 11.3 Learning outcomes
    4. 11.4 Comments on activities
    5. 11.5 Exercises
    6. 11.6 Problems
  20. 12. Direct sums and projections
    1. 12.1 The direct sum of two subspaces
    2. 12.2 Orthogonal complements
    3. 12.3 Projections
    4. 12.4 Characterising projections and orthogonal projections
    5. 12.5 Orthogonal projection onto the range of a matrix
    6. 12.6 Minimising the distance to a subspace
    7. 12.7 Fitting functions to data: least squares approximation
    8. 12.8 Learning outcomes
    9. 12.9 Comments on activities
    10. 12.10 Exercises
    11. 12.11 Problems
  21. 13. Complex matrices and vector spaces
    1. 13.1 Complex numbers
    2. 13.2 Complex vector spaces
    3. 13.3 Complex matrices
    4. 13.4 Complex inner product spaces
    5. 13.5 Hermitian conjugates
    6. 13.6 Unitary diagonalisation and normal matrices
    7. 13.7 Spectral decomposition
    8. 13.8 Learning outcomes
    9. 13.9 Comments on activities
    10. 13.10 Exercises
    11. 13.11 Problems
  22. Comments on exercises
  23. Index