Amir-Moez / Fass / Sneddon Elements of Linear Space

1. Auflage 2014
ISBN: 978-1-4832-7909-1
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band Volume 26, 160 Seiten, Web PDF

Reihe: International Series in Pure and Applied Mathematics

Elements of Linear Space is a detailed treatment of the elements of linear spaces, including real spaces with no more than three dimensions and complex n-dimensional spaces. The geometry of conic sections and quadric surfaces is considered, along with algebraic structures, especially vector spaces and transformations. Problems drawn from various branches of geometry are given. Comprised of 12 chapters, this volume begins with an introduction to real Euclidean space, followed by a discussion on linear transformations and matrices. The addition and multiplication of transformations and matrices are given emphasis. Subsequent chapters focus on some properties of determinants and systems of linear equations; special transformations and their matrices; unitary spaces; and some algebraic structures. Quadratic forms and their applications to geometry are also examined, together with linear transformations in general vector spaces. The book concludes with an evaluation of singular values and estimates of proper values of matrices, paying particular attention to linear transformations always on a unitary space of dimension n over the complex field. This book will be of interest to both undergraduate and more advanced students of mathematics.

Autoren/Hrsg.

Amir-Moez, A. R.

Fass, A. L.

Sneddon, I. N.

Ulam, S.

Stark, M.

Inhaltsverzeichnis

1;Front Cover;1
2;Elements of Linear Spaces;4
3;Copyright Page;5
4;Table of Contents;6
5;PREFACE;10
6;PART I;12
6.1;CHAPTER
1. REAL EUCLIDEAN SPACE;12
6.1.1;1.1 Seal ars and vectors;12
6.1.2;1.2 Sums and scalar multiples of vectors;12
6.1.3;1.3 Linear independence;13
6.1.4;1.4 Theorem;13
6.1.5;1.5 Theorem;13
6.1.6;1.6 Theorem;13
6.1.7;1.7 Base (Co-ordinate system);14
6.1.8;1.8 Theorem;15
6.1.9;1.9 Inner product of two vectors;16
6.1.10;1.10 Projection of a vector on an axis;16
6.1.11;1.11 Theorem;17
6.1.12;1.12 Theorem;17
6.1.13;1.13 Theorem;18
6.1.14;1.14 Orthonormal base;18
6.1.15;1.15 Norm of a vector and angle between two vectors in terms of components;18
6.1.16;1.16 Orthonormalization of a base;19
6.1.17;1.17 Subspaces;20
6.1.18;1.18 Straight line;21
6.1.19;1.19 Plane;22
6.1.20;EXERCISES 1;24
6.1.21;ADDITIONAL PROBLEMS;26
6.2;CHAPTER
2. LINEAR TRANSFORMATIONS AND MATRICES;27
6.2.1;2.1 Definition;27
6.2.2;2.2 Addition and Multiplication of Transformations;27
6.2.3;2.3 Theorem;27
6.2.4;2.4 Matrix of a Transformation A;27
6.2.5;2.5 Unit and zero transformation;30
6.2.6;2.6 Addition of Matrices;31
6.2.7;2.7 Product of Matrices;31
6.2.8;2.8 Rectangular matrices;32
6.2.9;2.9 Transform of a vector;32
6.2.10;EXERCISES 2;34
6.2.11;ADDITIONAL PROBLEMS 2;36
6.3;CHAPTER
3. DETERMINANTS AND LINEAR EQUATIONS;39
6.3.1;3.1 Definition;39
6.3.2;3.2 Some properties of determinants;40
6.3.3;3.3 Theorem;40
6.3.4;3.4 Systems of linear equations;40
6.3.5;EXERCISES 3;45
6.4;CHAPTER
4. SPECIAL TRANSFORMATIONS AND THEIR MATRICES;48
6.4.1;4.1 Inverse of a linear transformation;48
6.4.2;4.2 A practical way of getting the inverse;49
6.4.3;4.3 Theorem;49
6.4.4;4.4 Adjoint of a transformation;49
6.4.5;4.5 Theorem;49
6.4.6;4.6 Theorem;50
6.4.7;4.7 Theorem;50
6.4.8;4.8 Orthogonal (Unitary) transformations;50
6.4.9;4.9 Theorem;51
6.4.10;4.10 Change of Base;51
6.4.11;4.11 Theorem;52
6.4.12;EXERCISES 4;54
6.4.13;ADDITIONAL PROBLEMS 4;55
6.5;CHAPTER
5. CHARACTERISTIC EQUATION OF A TRANSFORMATION AND QUADRATIC FORMS;58
6.5.1;5.1 Characteristic values and characteristic vectors of a transformation;58
6.5.2;5.2 Theorem;58
6.5.3;5.3 Definition;59
6.5.4;5.4 Theorem;59
6.5.5;5.5 Theorem;59
6.5.6;5.6 Special transformations;59
6.5.7;5.7 Change of a matrix to diagonal form;60
6.5.8;5.8 Theorem;61
6.5.9;5.9 Definition;62
6.5.10;5.10 Theorem;62
6.5.11;5.11 Quadratic forms and their reduction to canonical form;63
6.5.12;5.12 Reduction to sum or differences of squares;65
6.5.13;5.13 Simultaneous reduction of two quadratic forms;65
6.5.14;EXERCISES 5;68
6.5.15;ADDITIONAL PROBLEMS 5;69
7;PART II;72
7.1;CHAPTER
6. UNITARY SPACES;72
7.1.1;Introduction;72
7.1.2;6.1 Scalars, Vectors and vector spaces;72
7.1.3;6.2 Subspaces;72
7.1.4;6.3 Linear independence;72
7.1.5;6.4 Theorem;72
7.1.6;6.5 Base;73
7.1.7;6.6 Theorem;73
7.1.8;6.7 Dimension theorem;74
7.1.9;6.8 Inner Product;74
7.1.10;6.9 Unitary spaces;74
7.1.11;6.10 Definition;74
7.1.12;6.11 Theorem;74
7.1.13;6.12 Definition;74
7.1.14;6.13 Theorem;74
7.1.15;6.14 Definition;75
7.1.16;6.15 Orthonormalization of a set of vectors;75
7.1.17;6.16 Orthonormal base;75
7.1.18;6.17 Theorem;75
7.1.19;EXERCISES 6;76
7.2;CHAPTER
7. LINEAR TRANSFORMATIONS, MATRICES AND DETERMINANTS;78
7.2.1;7.1 Definition;78
7.2.2;7.2 Matrix of a Transformation A;78
7.2.3;7.3 Addition and Multiplication of Matrices;78
7.2.4;7.4 Rectangular matrices;79
7.2.5;7.5 Determinants;79
7.2.6;7.6 Rank of a matrix;80
7.2.7;7.7 Systems of linear equations;81
7.2.8;7.8 Inverse of a linear transformation;83
7.2.9;7.9 Adjoint of a transformation;84
7.2.10;7.10 Unitary Transformation;84
7.2.11;7.11 Change of Base;85
7.2.12;7.12 Characteristic values and Characteristic vectors of a transformation;85
7.2.13;7.13 Definition;85
7.2.14;7.14 Theorem;86
7.2.15;7.15 Theorem;86
7.2.16;EXCERCISES 7;87
7.3;CHAPTER
8. QUADRATIC FORMS AND APPLICATION TO GEOMETRY;90
7.3.1;8.1 Definition;90
7.3.2;8.2 Reduction of a quadratic form to canonical form;90
7.3.3;8.3 Reduction to Sum or difference of squares;91
7.3.4;8.4 Simultaneous reduction of two quadratic forms;91
7.3.5;8.5 Homogeneous Coordinates;91
7.3.6;8.6 Change of coordinate system;91
7.3.7;8.7 Invariance of rank;92
7.3.8;8.8 Second degree curves;93
7.3.9;8.9 Second degree Surfaces;95
7.3.10;8.10 Direction numbers and equations of straight lines and planes;100
7.3.11;8.11 Intersection of a straight line and a quadric;100
7.3.12;8.12 Theorem;101
7.3.13;8.13 A center of a quadric;102
7.3.14;8.14 Tangent plane to a quadric;103
7.3.15;8.15 Ruled surfaces;104
7.3.16;8.16 Theorem;106
7.3.17;EXERCISES 8;107
7.3.18;ADDITIONAL PROBLEMS 8;108
7.4;CHAPTER
9. APPLICATIONS AND PROBLEM SOLVING TECHNIQUES;111
7.4.1;9.1 A general projection;111
7.4.2;9.2 Intersection of planes;111
7.4.3;9-3 Sphere;112
7.4.4;9.4 A property of the sphere;112
7.4.5;9.5 Radical axis;113
7.4.6;9.6 Principal planes;114
7.4.7;9.7 Central quadric;115
7.4.8;9.8 Quadric of rank 2;116
7.4.9;9.9 Quadric of rank 1;117
7.4.10;9.10 Axis of rotation;118
7.4.11;9.11 Identification of a quadric;118
7.4.12;9.12 Rulings;119
7.4.13;9.13 Locus problems;119
7.4.14;9.14 Curves in space;120
7.4.15;9.15 Pole and polar;121
7.4.16;EXERCISES 9;122
8;PART III;126
8.1;CHAPTER
10. SOME ALGEBRAIC STRUCTURES;126
8.1.1;Introduction;126
8.1.2;10.1 Definition;126
8.1.3;10.2 Groups;126
8.1.4;10.3 Theorem;126
8.1.5;10.4 Corollary;126
8.1.6;10.5 Fields;126
8.1.7;10.6 Examples;127
8.1.8;10.7 Vector spaces;127
8.1.9;10.8 Subspaces;127
8.1.10;10.9 Examples of vector spaces;127
8.1.11;10.10 Linear independence;128
8.1.12;10.11 Base;128
8.1.13;10.12 Theorem;128
8.1.14;10.13 Corollary;128
8.1.15;10.14 Theorem;128
8.1.16;10.15 Theorem;129
8.1.17;10.16 Unitary spaces;129
8.1.18;10.17 Theorem;130
8.1.19;10.18 Orthogonality;131
8.1.20;10.19 Theorem;131
8.1.21;10.20 Theorem;132
8.1.22;10.21 Orthogonal complement of a subspace;132
8.1.23;EXERCISES 10;132
8.2;CHAPTER
11. LINEAR TRANSFORMATIONS IN GENERAL VECTOR SPACES;134
8.2.1;11.1 Definitions;134
8.2.2;11.2 Space of linear transformations;134
8.2.3;11.3 Algebra of linear transformations;134
8.2.4;11.4 Finite-dimensional vector spaces;135
8.2.5;11.5 Rectangular matrices;135
8.2.6;11.6 Rank and range of a transformation;135
8.2.7;11.7 Null space and nullity;136
8.2.8;11.8 Transform of a vector;136
8.2.9;11.9 Inverse of a transformation;136
8.2.10;11.10 Change of base;137
8.2.11;11.11 Characteristic equation of a transformation;137
8.2.12;11.12 Cayley-Hamilton Theorem;137
8.2.13;11.13 Unitary spaces and special transformations;138
8.2.14;11.14 Complementary subspaces;139
8.2.15;11.15 Projections;139
8.2.16;11.16 Algebra of projections;139
8.2.17;11.17 Matrix of a projection;140
8.2.18;11.18 Perpendicular projection;140
8.2.19;11.19 Decomposition of Hermitian transformations;140
8.2.20;EXERCISES 11;141
8.3;CHAPTER
12. SINGULAR VALUES AND ESTIMATES OF PROPER VALUES OF MATRICES;143
8.3.1;12.1 Proper values of a matrix;143
8.3.2;12.2 Theorem;143
8.3.3;12.3 Cartesian decomposition of a linear transformation;144
8.3.4;12.4 Singular values of a transformation;145
8.3.5;12.5 Theorem;145
8.3.6;12.6 Theorem;146
8.3.7;12.7 Theorem;146
8.3.8;12.8 Theorem;146
8.3.9;12.9 Theorem;147
8.3.10;12.10 Lemma;147
8.3.11;12.11 Theorem;148
8.3.12;12.12 The space of n-by-n matrices;148
8.3.13;12.13 Hilbert norm;148
8.3.14;12.14 Frobenius norm;149
8.3.15;12.15 Theorem;149
8.3.16;12.16 Theorem;150
8.3.17;12.17 Theorem;152
8.3.18;12.18 Real or imaginary singular values of a sum of transformations;152
8.3.19;EXERCISES 12;152
9;APPENDIX;154
10;INDEX;158