Linear Algebra
Linear Algebra:
Gespeichert in:
Bibliographische Detailangaben
1. Verfasser: Smith, Larry (VerfasserIn)
Format: Elektronisch E-Book
Sprache:English
Veröffentlicht: New York, NY Springer New York 1998
Ausgabe:Third Edition
Schriftenreihe:Undergraduate Texts in Mathematics
Schlagworte:
Mathematics
Matrix theory
Linear and Multilinear Algebras, Matrix Theory
Mathematik
Numerische Mathematik
Numerisches Verfahren
ALGOL
Matrizengleichung
Eigenwertproblem
Lineare Algebra
Einführung
Aufsatzsammlung
Lehrbuch
Online-Zugang:Volltext
Beschreibung:This text was originally written for a one-semester-course in linear algebra at the (U. S.) sophomore-undergraduate level, preferably directly following a one variable calculus course, so that linear algebra could be used in a course on multidimensional calculus and/or differential equations. Students at this level generally have had little contact with complex numbers or abstract mathematics, so the book deals almost exclusively with real finite-dimensional vector spaces, but in a setting and formulation that permits easy generalization to abstract vector spaces. The parallel complex theory is developed in part in the exercises. The goal of the first two editions was the principal axis theorem for real symmetric linear transformations. Twenty years of teaching in Germany, where linear algebra is a one-year course taken in the first year of study at the university, has modified that goal. The principal axis theorem becomes the first of two goals, and to be achieved as originally planned in one semester, a more or less direct path is followed to its proof.
As a consequence there are many subjects that are not developed, and this is intentional: this is only an introduction to linear algebra. As compensation, a wide selection of examples of vectorspaces and linear transformations is presented, to serve as a testing ground for the theory. Students with a need to learn more linear algebra can do so in a course in abstract algebra, which is the appropriate setting. Through this book they will be taken on an excursion to the algebraic/analytic zoo, and introduced to some of the animals for the first time. Further excursions can teach them more about the curious habits of some of these remarkable creatures. In these condedition of the book I added, among other things, a safari into the wilderness of canonical forms, where the hardy student could pursue the Jordanform, which has become the second goal of this book, with the tools developed in the preceding chapters. In this edition I have added the tip of the iceberg of invariant theory to show that linear algebra alone is not capable of solving these canonical forms problems, even in the simplest case of 2x2 complexmatrices.
Gottingen, Germany, February 1998 Larry Smith Contents vii viii Preface Preface vii 1. Vectors in the Plane and in Space 1 1. 1 FirstSteps 1 1. 2 Exercises 12 2. Vector Spaces 15 2. 1 Axioms for Vector Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2. 2 Cartesian (or Euclidean) Spaces. . . . . . . . . . . . . . . . . . . . . . . 18 2. 3 Some Rules for Vector Algebra 21 2. 4 Exercises 22 3. Examples of Vector Spaces 25 3. 1 Three Basic Examples 25 3. 2 Further Examples of Vector Spaces. . . . . . . . . . . . . . . . . . . 27 3. 3 Exercises . . . . . . . . . . 30 4. Subspaces 35 4. 1 Basic Properties of Vector Subspaces 35 4. 2 Examples of Subspaces 41 4. 3 Exercises 42 5. Linear Independence and Dependence 47 5. 1 Basic Definitions and Examples. . . . . . . . . . . . . . . . . . . . . . . 47 5. 2 Properties of Independent and Dependent Sets 50 5. 3 Exercises 53 ix x Contents 6. Finite-Dimensional Vector Spaces and Bases 57 6. 1 Finite-Dimensional Vector Spaces. . . . . . . . . . . . . . . . . . . . . 57 6. 2 Properties of Bases. . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 6. 3 Using Bases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 6. 4 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 7. The Elements of Vector Spaces: A Summing Up 75 7. 1 Numerical Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 7. 2 Exercises 82 8. Linear Transformations 85 8. 1 Definition of Linear Transformations 85 8. 2 Examples of Linear Transformations. . . . . . . . . . . . . . . . . . 89 8. 3 Properties of Linear Transformations 91 8. 4 Images and Kernels of Linear Transformations 94 8. 5 Some Fundamental Constructions 98 8. 6 IsomorphismofVectorSpaces 102 8. 7 Exercises 109 9. Linear Transformations: Examples and Applications 113 9. 1 Numerical Examples 113 9. 2 Some Applications 123 9. 3 Exercises 124 10. Linear Transformations and Matrices 129 3 10. 1 Linear Transformations and Matrices in m. 129 10.
2 Some Numerical Examples 134 10. 3 Matrices and Their Algebra 136 10. 4 Special Types of Matrices 141 10. 5 Exercises 151 11. Representing Linear Transformations by Matrices 159 11. 1 Representing a Linear Transformation by a Matrix. . 159 11. 2 Basic Theorems 165 11. 3 Change of Bases 174 11. 4 Exercises 178 12
Beschreibung:1 Online-Ressource (XII, 454 p)
ISBN:9781461216704
9781461272380
ISSN:0172-6056
DOI:10.1007/978-1-4612-1670-4

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