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*This book is designed for use as a textbook for a formal course in linear algebra or as a supplement to all current standard texts. It aims to present an introduction to linear algebra which will be found helpful to all readers regardless of their fields of specification. More material has been included than can be covered in most first courses. This has been done to make the book more flexible, to provide a useful book of reference, and to stimulate further interest in the subject.*

*Linear algebra has in recent years become an essential part of the mathematical background required by mathematicians and mathematics teachers, engineers, computer scientists, physicists, economists, and statisticians, among others. This requirement reflects the importance and wide applications of the subject matter.*

*Overview of Linear Algebra Schaum Series PDF*

*Overview of Linear Algebra Schaum Series PDF*

Table of Contents

*Each chapter begins with clear statements of pertinent definitions, principles, and theorems together with illustrative and other descriptive material. This is followed by graded sets of solved and supplementary problems.**The solved problems serve to illustrate and amplify the theory, and to provide the repetition of basic principles so vital to effective learning. Numerous proofs, especially those of all essential theorems, are included among the solved problems.**The supplementary problems serve as a complete review of the material of each chapter.**The first three chapters treat vectors in Euclidean space, matrix algebra, and systems of linear equations.**These chapters provide the motivation and basic computational tools for the abstract investigations of vector spaces and linear mappings which follow.**After chapters on inner product spaces and orthogonality and on determinants, there is a detailed discussion of eigenvalues and eigenvectors giving conditions for representing a linear operator by a diagonal matrix. This naturally leads to the study of various canonical forms, specifically, the triangular, Jordan, and rational canonical forms. Later chapters cover linear functions and the dual space V*, and bilinear, quadratic, and Hermitian forms.**The last chapter treats linear operators on inner product spaces.*

**Improvement in the Latest Edition of Linear Algebra Schaum Series PDF**

**Improvement in the Latest Edition of Linear Algebra Schaum Series PDF**

*The main changes in this edition have been in the appendices. *

*First of all, we have expanded Appendix A on the tensor and exterior products of vector spaces where we have now included proofs of the existence and uniqueness of such products.**Also added appendices covering algebraic structures, including modules, and polynomials over a field.**Appendix D, ‘‘Odds and Ends,’’ includes the Moore–Penrose generalized inverse which appears in various applications, such as statistics.**There are also many additional solved and supplementary problems.*

*Linear Algebra Schaum Series PDF: Table of Contents *

*Linear Algebra Schaum Series PDF: Table of Contents*

CHAPTER 1: Vectors in R_{n} and C_{n}, Spatial Vectors |

1.1 Introduction 1.2 Vectors in R_{n} 1.3 Vector Addition and Scalar Multiplication 1.4 Dot (Inner) Product 1.5 Located Vectors, Hyperplanes, Lines, Curves in R_{n} 1.6 Vectors in R (Spatial Vectors), ijk Notation 1.7 Complex Numbers 1.8 Vectors in C_{n} |

CHAPTER 2: Algebra of Matrices |

2.1 Introduction 2.2 Matrices 2.3 Matrix Addition and Scalar Multiplication 2.4 Summation Symbol 2.5 Matrix Multiplication 2.6 Transpose of a Matrix 2.7 Square Matrices 2.8 Powers of Matrices, Polynomials in Matrices 2.9 Invertible (Nonsingular) Matrices 2.10 Special Types of Square Matrices 2.11 Complex Matrices 2.12 Block Matrices |

CHAPTER 3: Systems of Linear Equations |

3.1 Introduction 3.2 Basic Definitions, Solutions 3.3 Equivalent Systems, Elementary Operations 3.4 Smal Square Systems of Linear Equations 3.5 Systems in Triangular and Echelon Forms 3.6 Gaussian Elimination 3.7 Echelon Matrices, Row Canonical Form, Row Equivalence 3.8 Gaussian Elimination, Matrix Formulation 3.9 Matrix Equation of a System of Linear Equations 3.10 Systems of Linear Equations and Linear Combinations of Vectors 3.11 Homogeneous Systems of Linear Equations 3.12 Elementary Matrices 3.13 LU Decomposition |

CHAPTER 4: Vector Spaces |

4.1 Introduction 4.2 Vector Spaces 4.3 Examples of Vector Spaces 4.4 Linear Combinations, Spanning Sets 4.5 Subspaces 4.6 Linear Spans, Row Space of a Matrix 4.7 Linear Dependence and Independence 4.8 Basis and Dimension 4.9 Application to Matrices, Rank of a Matrix 4.10 Sums and Direct Sums 4.11 Coordinates |

CHAPTER 5: Linear Mappings |

5.1 Introduction 5.2 Mappings, Functions 5.3 Linear Mappings (Linear Transformations) 5.4 Kernel and Image of a Linear Mapping 5.5 Singular and Nonsingular Linear Mappings, Isomorphisms 5.6 Operations with Linear Mappings 5.7 Algebra A(V) of Linear Operators |

CHAPTER 6: Linear Mappings and Matrices |

6.1 Introduction 6.2 Matrix Representation of a Linear Operator 6.3 Change of Basis 6.4 Similarity 6.5 Matrices and General Linear Mappings |

CHAPTER 7: Inner Product Spaces, Orthogonality |

7.1 Introduction 7.2 Inner Product Spaces 7.3 Examples of Inner Product Spaces 7.4 Cauchy-Schwarz Inequality, Applications 7.5 Orthogonality 7.6 Orthogonal Sets and Bases 7.7 Gram – Schmidt Orthogonalization Process 7.8 Orthogonal and Positive Definite Matrices 7.9 Complex Inner Product Spaces 7.10 Normed Vector Spaces (Optional) |

CHAPTER 8: Determinants |

8.1 Introduction 8.2 Determinants of Orders 1 and 2 8.3 Determinants of Order 3 8.4 Permutations 8.5 Determinants of Arbitrary Order 8.6 Properties of Determinants 8.7 Minors and Cofactors 8.8 Evaluation of Determinants 8.9 Classical Adjoint 8.10 Applications to Linear Equations, Cramer’s Rule 8.11 Submatrices, Minors, Principal Minors 8.12 Block Matrices and Determinants 8.13 Determinants and Volume 8.14 Determinant of a Linear Operator 8.15 Multilinearity and Determinants |

CHAPTER 9 : Diagonalization: Eigenvalues and Eigenvectors |

9.1 Introduction 9.2 Polynomials of Matrices 9.3 Characteristic Polynomial, Cayley-Hamilton Theorem 9.4 Diagonalization, Eigenvalues, and Eigenvectors 9.5 Computing Eigenvalues and Eigenvectors, Diagonalizing Matrices 9.6 Diagonalizing Real Symmetric Matrices and Quadratic Forms 9.7 Minimal Polynomial 9.8 Characteristic and Minimal Polynomials of Block Matrices |

CHAPTER 10: Canonical Forms |

10.1 Introduction 10.2 Triangular Form 10.3 Invariance 10.4 Invariant Direct-Sum Decompositions 10.5 Primary Decomposition 10.6 Nilpotent Operators 10.7 Jordan Canonical Form 10.8 Cyclic Subspaces 10.9 Rational Canonical Form 10.10 Quotient Spaces |

CHAPTER 11: Linear Functionals and the Dual Space |

11.1 Introduction 11.2 Linear Functionals and the Dual Space 11.3 Dual Basis 11.4 Second Dual Space 11.5 Annihilators 11.6 Transpose of a Linear Mapping |

CHAPTER 12: Bilinear, Quadratic, and Hermitian Forms |

12.1 Introduction 12.2 Bilinear Forms 12.3 Bilinear Forms and Matrices 12.4 Alternating Bilinear Forms 12.5 Symmetric Bilinear Forms, Quadratic Forms 12.6 Real Symmetric Bilinear Forms, Law of Inertia 12.7 Hermitian Forms |

CHAPTER 13: Linear Operators on Inner Product Spaces |

13.1 Introduction 13.2 Adjoint Operators 13.3 Analogy Between A(V) and C, Special Linear Operators 13.4 Self-Adjoint Operators 13.5 Orthogonal and Unitary Operators 13.6 Orthogonal and Unitary Matrices 13.7 Change of Orthonormal Basis 13.8 Positive Definite and Positive Operators 13.9 Diagonalization and Canonical Forms in Inner Product Spaces 13.10 Spectral Theorem |

APPENDIX A Multilinear Products |

APPENDIX B Algebraic Structures |

APPENDIX C Polynomials over a Field |

APPENDIX D Odds and Ends |

List of Symbols |

Index |

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