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Linear Algebra

Book Description

Linear Algebra is designed as a text for postgraduate and undergraduate students of Mathematics. This book explains the basics comprehensively and with clarity. The flowing narrative of the book provides a refreshing approach to the subject. Drawing on decades of experience from teaching and based on extensive discussions with teachers and students, the book simplifies proofs while doing away with needless burdensome textual details.

Table of Contents

  1. Cover
  2. Title page
  3. Contents
  4. A list of Symbols
  5. Preface
  6. A Note to Students
  7. Chapter 1. Matrices
    1. 1.1 Introduction
    2. 1.2 Basic Concepts
    3. 1.3 Matrix Operations and Their Properties
    4. 1.4 Invertible Matrices
    5. 1.5 Transpose of a Matrix
    6. 1.6 Partition of Matrices; Block Multiplication
  8. Chapter 2. Systems of Linear Equations
    1. 2.1 Introduction
    2. 2.2 Gaussian Elimination
    3. 2.3 Elementary Row Operations
    4. 2.4 Row Reduction
    5. 2.5 Invertible Matrices Again
    6. 2.6 Determinant
  9. Chapter 3. Vector Spaces
    1. 3.1 Introduction
    2. 3.2 Basic Concepts
    3. 3.3 Linear Independence
    4. 3.4 Basis and Dimension
    5. 3.5 Subspaces Again
    6. 3.6 Rank of a Matrix
    7. 3.7 Bases of Subspaces
    8. 3.8 Quotient Space
  10. Chapter 4. Linear Maps and Matrices
    1. 4.1 Introduction
    2. 4.2 Basic Concepts
    3. 4.3 Isomorphism
    4. 4.4 Algebra of Linear Maps
    5. 4.5 Matrices of Linear Maps
  11. Chapter 5. Linear Operators
    1. 5.1 Introduction
    2. 5.2 Polynomials Over Fields
    3. 5.3 Characteristic Polynomial and Eigenvalues
    4. 5.4 Minimal Polynomial
    5. 5.5 Invariant Subspaces
    6. 5.6 Some Basic Results
  12. Chapter 6. Canonical Forms
    1. 6.1 Introduction
    2. 6.2 Primary Decomposition Theorem
    3. 6.3 Jordan Forms
    4. 6.4 Rational Canonical Form
  13. Chapter 7. Bilinear Forms
    1. 7.1 Introduction
    2. 7.2 Basic Concepts
    3. 7.3 Bilinear Forms and Linear Functionals
    4. 7.4 Symmetric Bilinear Forms
    5. 7.5 Quadratic Forms
  14. Chapter 8. Inner Product Spaces
    1. 8.1 Introduction
    2. 8.2 Hermitian Forms
    3. 8.3 Inner Product Space
    4. 8.4 Gram-Schmidt Orthogonalization Process
    5. 8.5 Adjoints
    6. 8.6 Unitary and Orthogonal Operators
    7. 8.7 Normal Operators
  15. Chapter 9. Selected Topics
    1. 9.1 Introduction
    2. 9.2 Isometries
    3. 9.3 Real Quadratic Forms
    4. 9.4 Singular Value Decomposition
  16. References
  17. Copyright