Preface 1 Matrices 1.1 The Basic Operations 1.2 Row Reduction 1.3 The Matrix Transpose 1.4 Deternunants 1.5 Permutations 1.6 Other Formulas for the Determinant Exercises 2 Groups 2.1 Laws ofComposition 2.2 Groups and Subgroups 2.3　Subgroups of the Additive Group of Intege 2.4 Cyclic Groups 2.5　Homomorphisms 2.6　Isomorphisms 2.7 Equivalence Relations and Partitions 2.8 Cosets 2.9 Modular Arithmetic 2.10 The Correspondence Theorem 2.11 Ptoduct Groups 2.12 Quotient Groups Exercises 3 VectorSpaces 3.1 SubspacesoflRn 3.2 Fields 3.3 Vector Spaces 3.4 Bases and Dimension 3.5 Computing with Bases 3.6 DirectSums 3.7 Infinite-DimensionalSpaces Exercises 4 LinearOperators 4.1 The Dimension Formula 4.2 The Matrix of a Linear Transformation 4.3 Linear Operators 4.4 Eigenvectors 4.5 The Characteristic Polynomial 4.6 Triangular and DiagonaIForms 4.7 JordanForm Exercises 5 Applications ofLinear Operators 5.1 OrthogonaIMatrices and Rotations 5.2 Using Continuity 5.3 Systems ofDifferentialEquations 5.4 The Matrix Exponential Exercises 6 Symmetry 6.1 Symmetry ofPlane Figures 6.2 Isometries 6.3 Isometries ofthe Plane 6.4 Finite Groups of Orthogonal Operators on the Pl 6.5 Discrete Groups oflsometries 6.6 Plane Crystallographic Groups 6.7 Abstract Symmetry: Group Operations 6.8 The Operation on Cosets 6.9 The Counting Formula 6.10 Operations on Subsets 6.11 Permutation Representations 6.12 Finite Subgroups ofthe Rotation Group Exercises 7 More Group Theory 7.1 Cayley's Theorem 7.2 The Class Equation 7.3 Groups 7.4 The Class Equation of the IcosahedraIGroup 7.5 Conjugationin the Symmetric Group 7.6 Normalizers 7.7 The Sylow Theorems 7.8 Groups ofOrder12 7.9 TheFreeGroup 7.10 Generators and Relations 7.11 The Todd-Coxeter Algorithm Exercises 8 BilinearForms 8.1 BilinearForms 8.2 SymmetricForms …… 9 Linear Groups 10 Group Representations 11 Rings 12 Factoring 13 Quadratic Number Fields 14 Linear Algebra in a Ring 15 Fields 16 Galois theory