Abstract Algebra with Applications

Vector Spaces First Introduction: Affine Geometry Second Introduction: Linear Equations Vector Spaces Linear and Affine Mappings Abstract Affine Geometry Representations of Linear Mappings by Matrices Determinants Volume Functions Eigenvalues and Eigenvectors Classification of Endomorphisms Up to Similarity Tensor Products and Base-Field Extension Metric Geometry Euclidean Spaces Linear Mappings Between Euclidean Spaces Bilinear Forms Groups of Automorphisms Application: Markov Chains Application: Matrix Calculus and Differential Equations Groups Introduction: Symmetries of Geometric Figures Groups Subgroups and Cosets Symmetric and Alternating Groups Group Homomorphisms Normal Subgroups and Factor Groups Free Groups: Generators and Relations Group Actions Group-Theoretical Applications of Group Actions Nilpotent and Solvable Groups Topological Methods in Group Theory Analytical Methods in Group Theory Groups in Topology Appendix Bibliography