Elements of linear algebra:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
London u.a.
Chapman & Hall
1994
|
| Edition: | 1. ed. |
| Series: | Chapman & Hall mathematics
|
| Subjects: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006525453&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Physical Description: | XIII, 226 S. graph. Darst. |
| ISBN: | 0412552809 |
Staff View
MARC
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| 245 | 1 | 0 | |a Elements of linear algebra |c P. M. Cohn |
| 250 | |a 1. ed. | ||
| 264 | 1 | |a London u.a. |b Chapman & Hall |c 1994 | |
| 300 | |a XIII, 226 S. |b graph. Darst. | ||
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Record in the Search Index
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|---|---|
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| adam_text | Contents
Preface xi
Note to the reader xiv
Introduction 1
1 Vectors 5
1.1 Notation 5
1.2 Definition of vectors 6
1.3 Addition of vectors 7
1.4 Multiplication by a scalar 7
1.5 Geometrical interpretation 9
1.6 Linear dependence of vectors 10
1.7 Subspaces of a vector space 11
1.8 A basis for the space of n vectors 12
1.9 The construction of a basis 13
1.10 General vector spaces 14
Exercises 17
2 The solution of a system of equations: the regular case 19
2.1 The use of vector notation 19
2.2 Definition of a regular system and statement of the results 20
2.3 Elementary operations 21
2.4 Proof of the main theorem (Theorem 2.1) 22
2.5 Gaussian elimination and the Gauss Jordan reduction 24
2.6 Illustrations 25
2.7 The choice of pivot 28
2.8 The linear dependence of n + 1 vectors in n dimensions 29
2.9 The uniqueness of the dimension 29
Exercises 30
3 Matrices 33
3.1 Definition of a matrix 33
3.2 The matrix as a linear operator 34
viii Contents
3.3 Equality of matrices 35
3.4 The vector space of matrices 36
3.5 Multiplication of square matrices 36
3.6 The zero matrix and the unit matrix 38
3.7 Multiplication of general matrices 39
3.8 Block multiplication of matrices 40
3.9 The inverse of a matrix 41
3.10 Calculation of the inverse 43
3.11 The transpose of a matrix 46
3.12 An application to graph theory 47
Exercises 49
4 The solution of a system of equations: the general case 52
4.1 The general linear system and the associated homogeneous
system 52
4.2 The rank of a system 53
4.3 The solution of homogeneous systems 53
4.4 Illustrations (homogeneous case) 55
4.5 The solution of general systems 57
4.6 Illustrations (general case) 58
4.7 The row echelon form 59
4.8 Constructing a basis: a practical method 60
4.9 The PAQ reduction of matrices 61
4.10 Computations of rank 62
4.11 Sylvester s law of nullity 63
Exercises 64
5 Determinants 67
5.1 Motivation 67
5.2 The 2 dimensional case 67
5.3 The 3 dimensional case 69
5.4 The rule of signs in the 3 dimensional case 69
5.5 Permutations 70
5.6 The Kronecker e function 71
5.7 The determinant of an n x n matrix 72
5.8 Cofactors and expansions 73
5.9 Properties of determinants 74
5.10 An expression for the cofactors 76
5.11 Evaluation of determinants 76
5.12 A formula for the inverse matrix 79
5.13 Cramer s rule 82
5.14 A determinantal criterion for linear dependence 83
5.15 A determinantal expression for the rank 85
Exercises 85
Contents ix
6 Coordinate geometry 88
6.1 The geometric interpretation of vectors 88
6.2 Coordinate systems 88
6.3 The length of a vector 90
6.4 The scalar product 91
6.5 Vectors in n dimensions 91
6.6 The construction of orthonormal bases 93
6.7 The Cauchy Schwarz inequality 94
6.8 The length of a vector in general coordinates 95
6.9 The vector product in 3 space 96
6.10 Lines in 3 space 98
6.11 The equation of a line in space 100
6.12 The equation of a plane in space 102
6.13 Geometrical interpretation of the solution of linear
equations 105
6.14 Pairs of lines 107
6.15 Line and plane 107
6.16 Two planes in space 108
6.17 Pairs of skew lines 110
Exercises 113
7 Coordinate transformations and linear mappings 116
7.1 The matrix of a transformation 116
7.2 Orthogonal matrices 118
7.3 Rotations and reflexions 120
7.4 The triple product 125
7.5 Linear mappings 127
7.6 A normal form for the matrix of a linear mapping 132
7.7 The image and kernel of a linear mapping 134
Exercises 135
8 Normal forms of matrices 138
8.1 Similarity of matrices 138
8.2 The characteristic equation; eigenvalues 141
8.3 Similarity reduction to triangular form 144
8.4 The Cayley Hamilton theorem 147
8.5 The reduction of real quadratic forms 148
8.6 Quadratic forms on a metric space 151
8.7 The Jordan normal form 156
Exercises 162
9 Applications I. Algebra and geometry 164
9.1 Quadrics in space 164
9.2 The classification of central quadrics 168
x Contents
9.3 Positivity criteria 170
9.4 Simultaneous reduction of two quadratic forms 173
9.5 The polar form 176
9.6 Linear programming 179
9.7 The method of least squares 184
Exercises 186
10 Applications II. Calculus, mechanics, economics 189
10.1 Functional dependence and Jacobians 189
10.2 Extrema, Hessians and the Morse lemma 191
10.3 Normal modes of vibration 192
10.4 Linear differential equations with constant coefficients and
economic models 196
10.5 Homogeneous systems of linear differential equations and
the Wronskian 202
10.6 Inversion by iteration 204
10.7 Linear difference equations 208
Exercises 210
Answers to the Exercises 212
Notation and symbols used 219
Bibliography 221
Index 223
|
| any_adam_object | 1 |
| author | Cohn, Paul M. 1924-2006 |
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| dewey-full | 551.48/9 512 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 551 - Geology, hydrology, meteorology 512 - Algebra |
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| dewey-search | 551.48/9 512 |
| dewey-sort | 3551.48 19 |
| dewey-tens | 550 - Earth sciences 510 - Mathematics |
| discipline | Geologie / Paläontologie Mathematik |
| edition | 1. ed. |
| format | Book |
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| id | DE-604.BV009856929 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T09:43:21Z |
| institution | BVB |
| isbn | 0412552809 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-006525453 |
| oclc_num | 31078076 |
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| owner_facet | DE-20 DE-91 DE-BY-TUM DE-824 DE-703 |
| physical | XIII, 226 S. graph. Darst. |
| publishDate | 1994 |
| publishDateSearch | 1994 |
| publishDateSort | 1994 |
| publisher | Chapman & Hall |
| record_format | marc |
| series2 | Chapman & Hall mathematics |
| spellingShingle | Cohn, Paul M. 1924-2006 Elements of linear algebra Algebras, Linear Lineare Algebra (DE-588)4035811-2 gnd |
| subject_GND | (DE-588)4035811-2 |
| title | Elements of linear algebra |
| title_auth | Elements of linear algebra |
| title_exact_search | Elements of linear algebra |
| title_full | Elements of linear algebra P. M. Cohn |
| title_fullStr | Elements of linear algebra P. M. Cohn |
| title_full_unstemmed | Elements of linear algebra P. M. Cohn |
| title_short | Elements of linear algebra |
| title_sort | elements of linear algebra |
| topic | Algebras, Linear Lineare Algebra (DE-588)4035811-2 gnd |
| topic_facet | Algebras, Linear Lineare Algebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006525453&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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