Linear Algebra (eBook)

R. R. Stoll, E. T. Wong (Autoren)

eBook Download: PDF

2014 | 1. Auflage
336 Seiten
Elsevier Science (Verlag)
978-1-4832-6523-0 (ISBN)

Infos

Linear Algebra is intended to be used as a text for a one-semester course in linear algebra at the undergraduate level. The treatment of the subject will be both useful to students of mathematics and those interested primarily in applications of the theory. The major prerequisite for mastering the material is the readiness of the student to reason abstractly. Specifically, this calls for an understanding of the fact that axioms are assumptions and that theorems are logical consequences of one or more axioms. Familiarity with calculus and linear differential equations is required for understanding some of the examples and exercises. This book sets itself apart from other similar textbooks through its dedication to the principle that, whenever possible, definitions and theorems should be stated in a form which is independent of the notion of the dimension of a vector space. A second feature of this book which is worthy of mention is the early introduction of inner product spaces and the associated metric concepts. Students soon feel at ease with this class of spaces because they share so many properties with physical space when equipped with a rectangular coordinate system. Finally, the book includes a chapter concerned with several applications to other fields of the theory that have been developed.

Inhalt

Front Cover 1
Linear Algebra 4
Copyright Page 5
Table of Contents 8
Preface 6
Symbols 11
Chapter 1. VECTOR SPACES 12
1 VECTORS 13
2 DEFINITIONS OF A VECTOR SPACE 15
3 SUBSPACES AND THEIR ALGEBRA 22
4 VECTOR SPACES OVER ARBITRARY FIELDS 32
Chapter 2. FURTHER PROPERTIES OF VECTOR SPACES 35
1 BASES AND DIMENSION 35
2 ISOMORPHISM 46
3 CALCULATION METHODS 52
4 CHANGE OF BASIS 62
5 GEOMETRIC ASPECTS OF VECTOR SPACES 68
Chapter 3. INNER-PRODUCT SPACES 75
1 EUCLIDEAN SPACES 76
2 ORTHONORMAL BASES 81
3 DISTANCES AND NORMS 89
4 ORTHOGONAL COMPLEMENTS AND ORTHOGONAL PROJECTIONS 93
5 UNITARY SPACES 100
Chapter 4. LINEAR TRANSFORMATIONS 104
1 DEFINITION OF A LINEAR TRANSFORMATION 104
2 RANGE, NULL SPACE, RANK, AND NULLITY 109
3 THE VECTOR SPACES L (V, W) AND L (V, V) 113
4 LINEAR FUNCTIONALS AND DUAL SPACES 125
5 ANNIHILATORS 132
6 ADJOINTS 139
7 UNITARY AND ORTHOGONAL TRANSFORMATIONS 148
Chapter 5. MATRICES 156
1 RANK 156
2 SIMILAR LINEAR TRANSFORMATIONS AND MATRICES 161
3 ELEMENTARY MATRICES 167
4 TRIANGULAR MATRICES 171
5 DETERMINANTS 177
Chapter 6. ALGEBRAIC PROPERTIES OF LINEAR TRANSFORMATIONS 190
1 POLYNOMIAL RINGS 190
2 MINIMAL POLYNOMIALS 198
3 CHARACTERISTIC VALUES AND VECTORS 207
4 DLAGONALIZATION OF SELF-ADJOINT TRANSFORMATIONS 216
5 CHARACTERISTIC POLYNOMIALS 222
6 TRIANGULABLE LINEAR TRANSFORMATIONS 229
Chapter 7. BILINEAR FORMS AND QUADRATIC FORMS 242
1 BILINEAR FORMS 243
2 QUADRATIC FORMS 251
3 EXTERNAL PROPERTIES OF CHARACTERISTIC VALUES OF A SYMMETRIC MATRIX 266
Chapter 8. DECOMPOSITION THEOREMS FOR NORMAL TRANSFORMATIONS 272
1 DIRECT SUMS AND PROJECTIONS 272
2 A DECOMPOSITION THEOREM 277
3 NORMAL TRANSFORMATIONS 282
4 THE JORDAN NORMAL FORM 291
Chapter 9. SEVERAL APPLICATIONS OF LINEAR ALGEBRA 295
1 LINEAR DIFFERENTIAL EQUATIONS 295
2 ECONOMICS: INTERACTIONS AMONG INDUSTRIES AND CONSUMERS 303
3 CHEMISTRY: ANALYSIS OF MULTICOMPONENT MIXTURES 310
4 PHYSICS: COUPLED OSCILLATIONS AND NORMAL MODES 313
5 CHEMICAL PHYSICS : THE HARMONIC OSCILLATOR 320
Appendix: NOTIONS OF SET THEORY 327
INDEX 333

Produkt-Details

• Erscheint lt. Verlag: 12.5.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Technik
• ISBN-10: 1-4832-6523-4 / 1483265234
• ISBN-13: 978-1-4832-6523-0 / 9781483265230