Elements of Abstract and Linear Algebra - E. H. Connell
Sets, Cartesian pro ducts 1
Relations, partial orderings, Hausdor® maximality principle, 3
equivalence relations
Functions, bijections, strips, solutions of equations, 5
right and left inverses, pro jections
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 mo d n 25
Homomorphisms 27
Permutations, the symmetric groups 31
Pro duct of groups 34
Chapter 3 Rings
Rings 37
Units, domains, ¯elds 38
The integers mo d n 40
Ideals and quotient rings 41
Homomorphisms 42
Polynomial rings 45
Pro duct of rings 49
The Chinese remainder theorem 50
Characteristic 50
Bo olean rings 51
Chapter 4 Matrices and Matrix Rings
Addition and multiplication of matrices, invertible matrices 53
Transp ose 55
Triangular, diagonal, and scalar matrices 56
Elementary op erations and elementary matrices 57
Systems of equations 59
vii
Determinants, the classical adjoint 60
Similarity, trace, and characteristic p olynomial 64
Chapter 5 Linear Algebra
Mo dules, submo dules 68
Homomorphisms 69
Homomorphisms on R n
71
Cosets and quotient mo dules 74
Pro ducts and copro ducts 75
Summands 77
Indep endence, generating sets, and free basis 78
Characterization of free mo dules 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 lo cally 91
The transp ose principle 92
Nilp otent homomorphisms 93
Eigenvalues, characteristic ro ots 94
Jordan canonical form 96
Inner pro duct spaces, Gram-Schmidt orthonormalization 98
Orthogonal matrices, the orthogonal group 102
Diagonalization of symmetric matrices 103
Chapter 6 App endix
The Chinese remainder theorem 108
Prime and maximal ideals and UFDs
109
Splitting short exact sequences 114
Euclidean domains 116
Jordan blo cks 122
Jordan canonical form 123
Determinants 128
Dual spaces 130
Outline
Chapter 1 Background and Fundamentals of MathematicsSets, Cartesian pro ducts 1
Relations, partial orderings, Hausdor® maximality principle, 3
equivalence relations
Functions, bijections, strips, solutions of equations, 5
right and left inverses, pro jections
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 mo d n 25
Homomorphisms 27
Permutations, the symmetric groups 31
Pro duct of groups 34
Chapter 3 Rings
Rings 37
Units, domains, ¯elds 38
The integers mo d n 40
Ideals and quotient rings 41
Homomorphisms 42
Polynomial rings 45
Pro duct of rings 49
The Chinese remainder theorem 50
Characteristic 50
Bo olean rings 51
Chapter 4 Matrices and Matrix Rings
Addition and multiplication of matrices, invertible matrices 53
Transp ose 55
Triangular, diagonal, and scalar matrices 56
Elementary op erations and elementary matrices 57
Systems of equations 59
vii
Determinants, the classical adjoint 60
Similarity, trace, and characteristic p olynomial 64
Chapter 5 Linear Algebra
Mo dules, submo dules 68
Homomorphisms 69
Homomorphisms on R n
71
Cosets and quotient mo dules 74
Pro ducts and copro ducts 75
Summands 77
Indep endence, generating sets, and free basis 78
Characterization of free mo dules 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 lo cally 91
The transp ose principle 92
Nilp otent homomorphisms 93
Eigenvalues, characteristic ro ots 94
Jordan canonical form 96
Inner pro duct spaces, Gram-Schmidt orthonormalization 98
Orthogonal matrices, the orthogonal group 102
Diagonalization of symmetric matrices 103
Chapter 6 App endix
The Chinese remainder theorem 108
Prime and maximal ideals and UFDs
109
Splitting short exact sequences 114
Euclidean domains 116
Jordan blo cks 122
Jordan canonical form 123
Determinants 128
Dual spaces 130
