Gratis un libro di Algebra lineare di E. H. Connell

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L’autore del libro è E. H. Connell, del Dipartimento
di Matematica dell’Università di Miami; il titolo è
Elements of Abstract and Linear Algebra; data dell’ultima
revisione: 02/12/01

          Indice
dei contenuti:

Chapter 1 Background and Fundamentals of Mathematics

Sets, Cartesian products 1

Relations, partial orderings, Hausdor_ maximality principle,
equivalence relations 3

Functions, bijections, strips, solutions of equations, right
and left inverses, projections 5

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, fields 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 56

Triangular, diagonal, and scalar matrices 56

Elementary operations and elementary matrices 57

Systems of equations 59

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 fields, rank of a matrix
85

Geometric interpretation of determinant 90

Linear functions approximate differentiable functions locally
91

The transpose principle 92

Nilpotent homomorphisms 93

Eigenvalues, characteristic roots 95

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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