Book:Richard Kaye/Linear Algebra

Subject Matter

 * Linear Algebra

Contents

 * Preface


 * 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 $\C$


 * 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 $\C$


 * 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 $\C$
 * 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



Source work progress
* : Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.5$ Row and column operations: Exercise $1.3 \ \text {(e)}$