Linear Algebra

Description

Cover Title page Preface Contents PART I MATRICES AND VECTOR SPACES 1 Matrices 1.1 Matrices 1.2 Addition and multiplication of matrices 1.3 The inverse of a matrix 1.4 The transpose of a matrix 1.5 Row and column operations 1.6 Determinant and trace 1.7 Minors and cofactors 2 Vector spaces 2.1 Examples and axioms 2.2 Subspaces 2.3 Linear independence 2.4 Bases 2.5 Coordinates 2.6 Vector spaces over other fields PART II BILINEAR AND SESQUILINEAR FORMS 3 Inner product spaces 3.1 The standard inner product 3.2 Inner products 3.3 Inner products over 4 Bilinear and sesquilinear forms 4.1 Bilinear forms 4.2 Representation by matrices 4.3 The base-change formula 4.4 Sesquilinear forms over 5 Orthogonal bases 5.1 Orthonormal bases 5.2 The Gram-Schmidt process 5.3 Properties of orthonormal bases 5.4 Orthogonal complements 6 When is a form definite 6.1 The Gram-Schmidt process revisited 6.2 The leading minor test 7 Quadratic forms and Sylvester's law of inertia 7.1 Quadratic forms 7.2 Sylvester's law of inertia 7.3 Examples 7.4 Applications to surfaces 7.5 Sesquilinear and Hermitian forms PART III LINEAR TRANSFORMATIONS 8 Linear transformations 8.1 Basics 8.2 Arithmetic operations on linear transformations 8.3 Representation by matrices 9 Polynomials 9.1 Polynomials 9.2 Evaluating polynomials 9.3 Roots of polynomials over 9.4 Roots of polynomials over other fields 10 Eigenvalues and eigenvectors 10.1 An example 10.2 Eigenvalues and eigenvectors 10.3 Upper triangular matrices 11 The minimum polynomial 11.1 The minimum polynomial 11.2 The characteristic polynomial 11.3 The Cayley-Hamilton theorem 12 Diagonalization 12.1 Diagonal matrices 12.2 A criterion for diagonalizability 12.3 Examples 13 Self-adjoint transformations 13.1 Orthogonal and unitary transformations 13.2 From forms to transformations 13.3 Eigenvalues and diagonalization 13.4 Applications 14 The Jordan normal form 14.1 Jordan normal form 14.2 Obtaining the Jordan normal form 14.3 Applications 14.4 Proof of the primary decomposition theorem Appendix A: A theorem of analysis Appendix B: Applications to quantum mechanics Index