Umfang:
Online-Ressource (XXI, 526 p. 70 illus, digital)
ISBN:
9783642309946
Serie:
SpringerLink
Inhalt:
Preface -- Preliminaries -- 1. Linear Equations -- 2. Matrices and Determinants -- 3. Vector Spaces -- 4. Linear Transformations of a Vector Space to Itself -- 5. Jordan Normal Form -- 6. Quadratic and Bilinear Forms -- 7. Euclidean Spaces -- 8. Affine Spaces -- 9. Projective Spaces -- 10. The Exterior Product and Exterior Algebras -- 11. Quadrics -- 12. Hyperbolic Geometry -- 13. Groups, Rings, and Modules -- 14. Elements of Representation Theory -- Historical Note -- References -- Index.
Inhalt:
This book on linear algebra and geometry is based on a course given by renowned academician I.R. Shafarevich at Moscow State University. The book begins with the theory of linear algebraic equations and the basic elements of matrix theory and continues with vector spaces, linear transformations, inner product spaces, and the theory of affine and projective spaces. The book also includes some subjects that are naturally related to linear algebra but are usually not covered in such courses: exterior algebras, non-Euclidean geometry, topological properties of projective spaces, theory of quadrics (in affine and projective spaces), decomposition of finite abelian groups, and finitely generated periodic modules (similar to Jordan normal forms of linear operators). Mathematical reasoning, theorems, and concepts are illustrated with numerous examples from various fields of mathematics, including differential equations and differential geometry, as well as from mechanics and physics.
Anmerkung:
Description based upon print version of record
,
Linear Algebra and Geometry; Preface; Acknowledgements; Contents; Preliminaries; Sets and Mappings; Some Topological Notions; Chapter 1: Linear Equations; 1.1 Linear Equations and Functions; 1.2 Gaussian Elimination; 1.3 Examples*; Chapter 2: Matrices and Determinants; 2.1 Determinants of Orders 2 and 3; 2.2 Determinants of Arbitrary Order; 2.3 Properties that Characterize Determinants; 2.4 Expansion of a Determinant Along Its Columns; 2.5 Cramer's Rule; 2.6 Permutations, Symmetric and Antisymmetric Functions; 2.7 Explicit Formula for the Determinant; 2.8 The Rank of a Matrix
,
2.9 Operations on Matrices2.10 Inverse Matrices; Chapter 3: Vector Spaces; 3.1 The Definition of a Vector Space; 3.2 Dimension and Basis; 3.3 Linear Transformations of Vector Spaces; 3.4 Change of Coordinates; 3.5 Isomorphisms of Vector Spaces; 3.6 The Rank of a Linear Transformation; 3.7 Dual Spaces; 3.8 Forms and Polynomials in Vectors; Chapter 4: Linear Transformations of a Vector Space to Itself; 4.1 Eigenvectors and Invariant Subspaces; 4.2 Complex and Real Vector Spaces; 4.3 Complexification; 4.4 Orientation of a Real Vector Space; Chapter 5: Jordan Normal Form
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5.1 Principal Vectors and Cyclic Subspaces5.2 Jordan Normal Form (Decomposition); 5.3 Jordan Normal Form (Uniqueness); 5.4 Real Vector Spaces; 5.5 Applications*; Chapter 6: Quadratic and Bilinear Forms; 6.1 Basic Definitions; 6.2 Reduction to Canonical Form; 6.3 Complex, Real, and Hermitian Forms; Chapter 7: Euclidean Spaces; 7.1 The Definition of a Euclidean Space; 7.2 Orthogonal Transformations; 7.3 Orientation of a Euclidean Space*; 7.4 Examples*; 7.5 Symmetric Transformations; 7.6 Applications to Mechanics and Geometry*; 7.7 Pseudo-Euclidean Spaces; 7.8 Lorentz Transformations
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Chapter 8: Affine Spaces8.1 The Definition of an Affine Space; 8.2 Affine Spaces; 8.3 Affine Transformations; 8.4 Affine Euclidean Spaces and Motions; Chapter 9: Projective Spaces; 9.1 Definition of a Projective Space; 9.2 Projective Transformations; 9.3 The Cross Ratio; 9.4 Topological Properties of Projective Spaces*; Chapter 10: The Exterior Product and Exterior Algebras; 10.1 Plücker Coordinates of a Subspace; 10.2 The Plücker Relations and the Grassmannian; 10.3 The Exterior Product; 10.4 Exterior Algebras*; 10.5 Appendix*; Chapter 11: Quadrics; 11.1 Quadrics in Projective Space
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11.2 Quadrics in Complex Projective Space11.3 Isotropic Subspaces; 11.4 Quadrics in a Real Projective Space; 11.5 Quadrics in a Real Affine Space; 11.6 Quadrics in an Affine Euclidean Space; 11.7 Quadrics in the Real Plane*; Chapter 12: Hyperbolic Geometry; 12.1 Hyperbolic Space*; 12.2 The Axioms of Plane Geometry*; 12.3 Some Formulas of Hyperbolic Geometry*; Chapter 13: Groups, Rings, and Modules; 13.1 Groups and Homomorphisms; 13.2 Decomposition of Finite Abelian Groups; 13.3 The Uniqueness of the Decomposition; 13.4 Finitely Generated Torsion Modules over a Euclidean Ring*
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Chapter 14: Elements of Representation Theory
Weitere Ausg.:
ISBN 9783642309939
Weitere Ausg.:
Buchausg. u.d.T. Šafarevič, Igorʹ R., 1923 - 2017 Linear algebra and geometry Berlin : Springer, 2013 ISBN 9783642309939
Weitere Ausg.:
ISBN 3642309933
Weitere Ausg.:
ISBN 9783642434099
Sprache:
Englisch
Fachgebiete:
Mathematik
Schlagwort(e):
Lineare Algebra
;
Geometrie
;
Lineare Algebra
;
Geometrie
DOI:
10.1007/978-3-642-30994-6
URL:
Volltext
(lizenzpflichtig)
URL:
Volltext
(lizenzpflichtig)
Mehr zum Autor:
Šafarevič, Igorʹ R. 1923-2017
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