Linear algebra is one of the central disciplines in mathematics. A student of pure mathematics must know linear algebra if he is to continue with modern algebra or functional analysis. Much of the mathematics now taught to engineers and physicists requires it.
This well-known and highly regarded text makes the subject accessible to undergraduates with little mathematical experience. Written mainly for students in physics, engineering, economics, and other fields outside mathematics, the book gives the theory of matrices and applications to systems of linear equations, as well as many related topics such as determinants, eigenvalues, and differential equations.
Table of Contents:
l. The Algebra of Matrices
2. Linear Equations
3. Vector Spaces
4. Determinants
5. Linear Transformations
6. Eigenvalues and Eigenvectors
7. Inner Product Spaces
8. Applications to Differential Equations
For the second edition, the authors added several exercises in each chapter and a brand new section in Chapter 7. The exercises, which are both true-false and multiple-choice, will enable the student to test his grasp of the definitions and theorems in the chapter. The new section in Chapter 7 illustrates the geometric content of Sylvester's Theorem by means of conic sections and quadric surfaces. 6 line drawings. lndex. Two prefaces. Answer section.
Preface to the Second Edition; Preface to the First Edition 1. The Algebra of Matrices 1. Matrices: Definitions 2. Addition and Scalar Multiplication of Matrices 3. Matrix Multiplication 4. Square Matrices, Inverses, and Zero Divisors 5. Transposes, Partitioning of Matrices, and Direct Sums 2. Linear Equations 1. Equivalent Systems of Equations 2. Row Operations on Matrices 3. Row Echelon Form 4. Homogeneous Systems of Equations 5. The Unrestricted Case: A Consistency Condition 6. The Unrestricted Case: A General Solution 7. Inverses of Nonsingular Matrices 3. Vector Spaces 1. Vectors and Vector Spaces 2. Subspaces and Linear Combinations 3. Linear Dependence and Linear Independence 4. Bases 5. Bases and Representations 6. Row Spaces of Matrices 7. Column Equivalence 8. Row-Column Equivalence 9. Equivalence Relations and Canonical Forms of Matrices 4. Determinants 1. Introduction as a Volume Function 2. Permutations and Permutation Matrices 3. Uniqueness and Existence of the Determinant Function 4. Practical Evaluation and Transposes of Determinants 5. Cofactors, Minors, and Adjoints 6. Determinants and Ranks 5. Linear Transformations 1. Definitions 2. Representation of Linear Transformations 3. Representations Under Change of Bases 6. Eigenvalues and Eigenvectors 1. Introduction 2. Relation Between Eigenvalues and Minors 3. Similarity 4. Algebraic and Geometric Multiplicites 5. Jordan Canonical Form 6. Functions of Matrices 7. Application: Markov Chains 7. Inner Produce Spaces 1. Inner Products 2. Representation of Inner Products 3. Orthogonal Bases 4. Unitary Equivalence and Hermitian Matrices 5. Congruence and Conjunctive Equivalence 6. Central Conics and Quadrics 7. The Natural Inverse 8. Normal Matrices 8. Applications to Differential Equations 1. Introduction 2. Homogeneous Differential Equations 3. Linear Differrential Equations: The Unrestricted Case 4. Linear Operators: The Global View Answers; Symbols; Index
