An Introduction to Linear Algebra

PART I "DETERMINANTS, VECTORS, MATRICES, AND LINEAR EQUATIONS" I. DETERMINANTS 1.1. Arrangements and the Î-symbol 1.2. Elementary properties of determinants 1.3. Multiplication of determinants 1.4. Expansion theorems 1.5. Jacobi's theorem 1.6. Two special theorems on linear equations II. VECTOR SPACES AND LINEAR MANIFOLDS 2.1. The algebra of vectors 2.2. Linear manifolds 2.3. Linear dependence and bases 2.4. Vector representation of linear manifolds 2.5. Inner products and orthonormal bases III. THE ALGEBRA OF MATRICES 3.1. Elementary algebra 3.2. Preliminary notions concerning matrices 3.3. Addition and multiplication of matrices 3.4. Application of matrix technique to linear substitutions 3.5. Adjugate matrices 3.6. Inverse matrices 3.7. Rational functions of a square matrix 3.8. Partitioned matrices IV. LINEAR OPERATIONS 4.1. Change of basis in a linear manifold 4.2. Linear operators and their representations 4.3. Isomorphisms and automorphisms of linear manifolds 4.4. Further instances of linear operators V. SYSTEMS OF LINEAR EQUATIONS AND RANK OF MATRICES 5.1. Preliminary results 5.2. The rank theorem 5.3. The general theory of linear equations 5.4. Systems of homogeneous linear equations 5.5. Miscellaneous applications 5.6. Further theorems on rank of matrices VI. ELEMENTARY OPERATIONS AND THE CONCEPT OF EQUIVALENCE 6.1. E-operations and E-matrices 6.2. Equivalent matrices 6.3. Applications of the preceding theory 6.4. Congruence transformations 6.5. The general concept of equivalence 6.6. Axiomatic characterization of determinants PART II FURTHER DEVELOPMENT OF MATRIX THEORY VII. THE CHARACTERISTIC EQUATION 7.1. The coefficients of the characteristic polynomial 7.2. Characteristic polynomials and similarity transformations 7.3. Characteristic roots of rational functions of matrices 7.4. The minimum polynomial and the theorem of Cayley and Hamilton 7.5. Estimates of characteristic roots 7.6. Characteristic vectors VIII. ORTHOGONAL AND UNITARY MATRICES 8.1. Orthogonal matrices 8.2. Unitary matrices 8.3. Rotations in the plane 8.4. Rotations in space IX. GROUPS 9.1. The axioms of group theory 9.2. Matrix groups and operator groups 9.3. Representation of groups by matrices 9.4. Groups of singular matrices 9.5. Invariant spaces and groups of linear transformations X. CANONICAL FORMS 10.1. The idea of a canonical form 10.2. Diagonal canonical forms under the similarity group 10.3. Diagonal canonical forms under the orthogonal similarity group and the unitary similarity group 10.4. Triangular canonical forms 10.5. An intermediate canonical form 10.6. Simultaneous similarity transformations XI. MATRIX ANALYSIS 11.1 Convergent matrix sequences 11.2 Power series and matrix functions 11.3 The relation between matrix functions and matrix polynomials 11.4 Systems of linear differential equations PART III QUADRIATIC FORMS XII. "BILINEAR, QUADRATIC, AND HERMITIAN FORMS" 12.1 Operators and forms of the bilinear and quadratic types 12.2 Orthogonal reduction to diagonal form 12.3 General reduction to diagonal form 12.4 The problem of equivalence. Rank and signature 12.5 Classification of quadrics 12.6 Hermitian forms XIII. DEFINITE AND INDEFINITE FORMS 13.1 The value classes 13.2 Transformations of positive definite forms 13.3 Determinantal criteria 13.4 Simultaneous reduction of two quadratic forms 13.5 "The inequalities of Hadamard, Minkowski, Fischer, and Oppenheim" MISCELLANEOUS PROBLEMS BIBLIOGRAPHY INDEX