Elements of Abstract and Linear Algebra

E. H. Connell

Table of contents :
Title Page……Page 1
Introduction……Page 3
Outline……Page 6
1. Background and Fundamentals of Mathematics……Page 9
Relations……Page 11
Functions……Page 13
Notation for the Logic of Mathematics……Page 21
The Integers……Page 22
2. Groups……Page 27
Subgroups……Page 29
Normal Subgroups……Page 33
Homomorphisms……Page 35
Permutations……Page 39
Products of Groups……Page 42
3. Rings……Page 45
Units……Page 46
The Integers Mod n……Page 48
Ideals and Quotient Rings……Page 49
Homomorphisms……Page 50
Polynomial Rings……Page 53
Product of Rings……Page 57
Characteristic……Page 58
Boolean Rings……Page 59
4. Matrices and Matrix Rings……Page 61
Triangular Matrices……Page 64
Elementary Operations and Elementary Matrices……Page 65
Systems of Equations……Page 67
Determinants……Page 68
Similarity……Page 72
5. Linear Algebra……Page 75
Homomorphisms……Page 77
Homomorphisms on Rn……Page 79
Cosets and Quotient Modules……Page 82
Products and Coproducts……Page 83
Summands……Page 85
Independence, Generating Sets, and Free Basis……Page 86
Characterization of Free Modules……Page 87
Uniqueness of Dimension……Page 90
Change of Basis……Page 91
Vector Spaces……Page 93
Geometric Interpretation of Determinant……Page 98
Linear Functions Approximate Differentiable Functions Locally……Page 99
The Transpose Principle……Page 100
Nilpotent Homomorphisms……Page 101
Eigenvalues……Page 103
Jordan Canonical Form……Page 104
Inner Product Spaces……Page 106
Orthogonal Matrices……Page 110
Diagonalization of Symmetric Matrices……Page 111
6. Appendix……Page 115
The Chinese Remainder Theorem……Page 116
Prime and Maximal Ideals and UFD……Page 117
Splitting Short Exact Sequences……Page 122
Euclidean Domains……Page 124
Jordan Blocks……Page 130
Jordan Canonical Form……Page 131
Determinants……Page 136
Dual Spaces……Page 138
Index……Page 143