Matrices and Linear Algebra (Dover Books on Mathematics)

✍ Scribed by Hans Schneider, George Phillip Barker

• Publisher: Dover Publications
• Year: 1989
• Tongue: English
• Leaves: 302
• Edition: 2nd Revised ed.
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Preface to the Second Edition
Preface to the First Edition
CHAPTER 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
CHAPTER 2 LINEAR EQUATIONS
1. EQUIVALENT SYSTEMS OF EQUATIONS
2. ROW OPERATIONS ON MATRICES
3. ROW ECHELON FORM
5. THE UNRESTRICTED CASE: A CONSISTENCY CONDITION
6. THE UNRESTRICTED CASE: A GENERAL SOLUTION
7. INVERSES OF NONSINGULAR MATRICES
CHAPTER 3 VECTOR SPACES
1. VECTORS AND VECTOR SPACES
2. SUBSPACES AND LINEAR COMBINATIONS
3. LINEAR DEPENDENCE AND LINEAR INDEPENDENCE
6. ROW SPACES OF MATRICES
9. EQUIVALENCE RELATIONS AND CANONICAL FORMS OF MATRICES
CHAPTER 4 DETERMINANTS
1. INTRODUCTION AS A VOLUME FUNCTION
2. PERMUTATIONS AND PERMUTATION MATRICES
4. PRACTICAL EVALUATION AND TRANSPOSES OF DETERMINANTS
6. DETERMINANTS AND RANKS
CHAPTER 5 LINEAR TRANSFORMATIONS
1. DEFINITIONS
2. REPRESENTATION OF LINEAR TRANSFORMATIONS
3. REPRESENTATIONS UNDER CHANGE OF BASES
CHAPTER 6 EIGENVALUES AND EIGENVECTORS
1. INTRODUCTION
2. RELATION BETWEEN EIGENVALUES AND MINORS
3. SIMILARITY
4. ALGEBRAIC AND GEOMETRIC MULTIPLICITIES
5. JORDAN CANONICAL FORM
6. FUNCTIONS OF MATRICES
7. APPLICATION: MARKOV CHAINS
CHAPTER 7 INNER PRODUCT 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
CHAPTER 8 APPLICATIONS TO DIFFERENTIAL EQUATIONS
1. INTRODUCTION
2. HOMOGENEOUS DIFFERENTIAL EQUATIONS
3. LINEAR DIFFERENTIAL EQUATIONS: THE UNRESTRICTED CASE
4. LINEAR OPERATORS: THE GLOBAL VIEW
Answers
Symbols
Index

✦ Subjects

Matrix, Linear Algebra