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Outline
Chapter 1 Background and Fundamentals of Mathematics
Sets, Cartesian products 1
Relations, partial orderings, Hausdor® maximality principle, 3
equivalence relations
Functions, bijections, strips, solutions of equations, 5
right and left inverses, projections
Notation for the logic of mathematics 13
Integers, subgroups, unique factorization 14
Chapter 2 Groups
Groups, scalar multiplication for additive groups 19
Subgroups, order, cosets 21
Normal subgroups, quotient groups, the integers mod n 25
Homomorphisms 27
Permutations, the symmetric groups 31
Product of groups 34
Chapter 3 Rings
Rings 37
Units, domains, ¯elds 38
The integers mod n 40
Ideals and quotient rings 41
Homomorphisms 42
Polynomial rings 45
Product of rings 49
The Chinese remainder theorem 50
Characteristic 50
Boolean rings 51
Chapter 4 Matrices and Matrix Rings
Addition and multiplication of matrices, invertible matrices 53
Transpose 55
Triangular, diagonal, and scalar matrices 56
Elementary operations and elementary matrices 57
Systems of equations 59
vii
Determinants, the classical adjoint 60
Similarity, trace, and characteristic polynomial 64
Chapter 5 Linear Algebra
Modules, submodules 68
Homomorphisms 69
Homomorphisms on Rn 71
Cosets and quotient modules 74
Products and coproducts 75
Summands 77
Independence, generating sets, and free basis 78
Characterization of free modules 79
Uniqueness of dimension 82
Change of basis 83
Vector spaces, square matrices over ¯elds, rank of a matrix 85
Geometric interpretation of determinant 90
Linear functions approximate di®erentiable functions locally 91
The transpose principle 92
Nilpotent homomorphisms 93
Eigenvalues, characteristic roots 94
Jordan canonical form 96
Inner product spaces, Gram-Schmidt orthonormalization 98
Orthogonal matrices, the orthogonal group 102
Diagonalization of symmetric matrices 103
Chapter 6 Appendix
The Chinese remainder theorem 108
Prime and maximal ideals and UFDs 109
Splitting short exact sequences 114
Euclidean domains 116
Jordan blocks 122
Jordan canonical form 123
Determinants 128
Dual spaces 130
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