INTRODUCTION TO VECTORS AND TENSORS
CONTENTS PART 1 BASIC MATHEMATICS SELECTED READINGS FOR PART I………………………………………………………… CHAPTER 0 ELEMENTARY MATRIX THEORY………………………………………… CHAPTER 1 SETS, RELATIONS, AND FUNCTIONS……………………………………… SECTION 1 SETS AND SET ALGEBRA……………………………………… SECTION 2 ORDERED PAIRS" CARTESIAN PRODUCTS" AND RELATIONS………… SECTION 3 FUNCTIONS…………………………………………………… CHAPTER 2 GROUPS, RINGS AND FIELDS…………………………………………… SECTION 4 THE AXIOMS FOR A GROUP…………………………………… SECTION 5 PROPERTIES OF A GROUP……………………………………… SECTION 6 GROUP HOMOMORPHISMS…………………………………… SECTION 7 RINGS AND FIELDS…………………………………………… PART I1 VECTOR AND TENSOR ALGEBRA SELECTED READINGS FOR PART II………………………………………………………… CHAPTER 3 VECTOR SPACES……………………………………………………… SECTION 8 THE AXIOMS FOR A VECTOR SPACE…………………………… SECTION 9 LINEAR INDEPENDENCE, DIMENSION AND BASIS…………… … SECTION 10 INTERSECTION, SUM AND DIRECT SUM OF SUBSPACES…………… SECTION 11 FACTOR SPACES……………………………………………… SECTION 12 INNER PRODUCT SPACES……………………………………… SECTION 13 ORTHOGONAL BASES AND ORTHOGONAL COMPLIMENTS…………… SECTION 14 RECIPROCAL BASIS AND CHANGE OF BASIS……………………… CHAPTER 4 LINEAR TRANSFORMATIONS……………………………………………… SECTION 15 DEFINITION OF A LINEAR TRANSFORMATION……………………… SECTION 16 SUMS AND PRODUCTS OF LINEAR TRANSFORMATIONS……………… SECTION 17 SPECIAL TYPES OF LINEAR TRANSFORMATIONS…………………… SECTION 18 THE ADJOINT OF A LINEAR TRANSFORMATION…………………… SECTION 19 COMPONENT FORMULAS……………………………………… CHAPTER 5 DETERMINANTS AND MATRICES…………………………………………… SECTION 20 THE GENERALIZED KRONECKER DELTAS AND THE SUMMATION CONVENTION……………………………… SECTION 21 DETERMINANTS………………………………………………… SECTION 22 THE MATRIX OF A LINEAR TRANSFORMATION……………………… SECTION SOLUTION OF SYSTEMS OF LINEAR EQUATIONS…………………… CHAPTER 6 SPECTRAL DECOMPOSITIONS…………………………………………… SECTION 24 DIRECT SUM OF ENDOMORPHISMS……………………………… SECTION 25 EIGENVECTORS AND EIGENVALUES……………………………… SECTION 26 THE CHARACTERISTIC POLYNOMIAL……………………………… SECTION 27 SPECTRAL DECOMPOSITION FOR HERMITIAN ENDOMORPHISMS…… SECTION 28 ILLUSTRATIVE EXAMPLES………………………………………… SECTION 29 THE MINIMAL POLYNOMIAL…………………………………… SECTION 30 SPECTRAL DECOMPOSITION FOR ARBITRARY ENDOMORPHISMS… … CHAPTER 7 TENSOR ALGEBRA……………………………………………………… SECTION 31 LINEAR FUNCTIONS, THE DUAL SPACE…………………………… SECTION 32 THE SECOND DUAL SPACE, CANONICAL ISOMORPHISMS………… SECTION 33 MULTILINEAR FUNCTIONS, TENSORS…………………………… SECTION 34 CONTRACTIONS… SECTION 35 TENSORS ON INNER PRODUCT SPACES…………………………… CHAPTER 8 EXTERIOR ALGEBRA…………………………………………………… SECTION 36 SKEW-SYMMETRIC TENSORS AND SYMMETRIC TENSORS………… SECTION 37 THE SKEW-SYMMETRIC OPERATOR……………………………… SECTION 38 THE WEDGE PRODUCT………………………………………… SECTION 39 PRODUCT BASES AND STRICT COMPONENTS……………………… SECTION 40 DETERMINANTS AND ORIENTATIONS……………………………… SECTION 41 DUALITY……………………………………………………… SECTION 42 TRANSFORMATION TO CONTRAVARIANT REPRESENTATION………… _______________________________________________________________________________ PART