Linear Algebra

Linear Algebra is intended to be used as a text for a one-semester course in linear algebra at the undergraduate level. The treatment of the subject will be both useful to students of mathematics and those interested primarily in applications of the theory. The major prerequisite for mastering the material is the readiness of the student to reason abstractly. Specifically, this calls for an understanding of the fact that axioms are assumptions and that theorems are logical consequences of one or more axioms. Familiarity with calculus and linear differential equations is required for understanding some of the examples and exercises. This book sets itself apart from other similar textbooks through its dedication to the principle that, whenever possible, definitions and theorems should be stated in a form which is independent of the notion of the dimension of a vector space. A second feature of this book which is worthy of mention is the early introduction of inner product spaces and the associated metric concepts. Students soon feel at ease with this class of spaces because they share so many properties with physical space when equipped with a rectangular coordinate system. Finally, the book includes a chapter concerned with several applications to other fields of the theory that have been developed.

Preface

Symbols

Chapter 1 Vector Spaces

1 Vectors

2 Definitions of a Vector Space

3 Subspaces and Their Algebra

4 Vector Spaces over Arbitrary Fields

Chapter 2 Further Properties of Vector Spaces

1 Bases and Dimension

2 Isomorphism

3 Calculation Methods

4 Change of Basis

5 Geometric Aspects of Vector Spaces

Chapter 3 Inner-Product Spaces

1 Euclidean Spaces

2 Orthonormal Bases

3 Distances and Norms

4 Orthogonal Complements and Orthogonal Projections

5 Unitary Spaces

Chapter 4 Linear Transformations

1 Definition of a Linear Transformation

2 Range, Null Space, Rank, and Nullity

3 The Vector Spaces L(V,W) and L(V,V)

4 Linear Functionals and Dual Spaces

5 Annihilators

6 Adjoints

7 Unitary and Orthogonal Transformations

Chapter 5 Matrices

1 Rank

2 Similar Linear Transformations and Matrices

3 Elementary Matrices

4 Triangular Matrices

5 Determinants

Chapter 6 Algebraic Properties of Linear Transformations

1 Polynomial Rings

2 Minimal Polynomials

3 Characteristic Values and Vectors

4 Diagonalization of Self-Adjoint Transformations

5 Characteristic Polynomials

6 Triangulable Linear Transformations

Chapter 7 Bilinear Forms and Quadratic Forms

1 Bilinear Forms

2 Quadratic Forms

3 External Properties of Characteristic Values of a Symmetric Matrix

Chapter 8 Decomposition Theorems for Normal Transformations

1 Direct Sums and Projections

2 A Decomposition Theorem

3 Normal Transformations

4 The Jordan Normal Form

Chapter 9 Several Applications of Linear Algebra

1 Linear Differential Equations

2 Economics: Interactions among Industries and Consumers

3 Chemistry: Analysis of Multicomponent Mixtures

4 Physics: Coupled Oscillations and Normal Modes

5 Chemical Physics : The Harmonic Oscillator

Appendix: Notions of Set Theory

Index