A Vector Space Approach to Geometry - Brossura

Hausner, Melvin

Brossura

ISBN 10: 0486404528 ISBN 13: 9780486404523

Casa editrice: Dover Pubns, 2003

Riassunto

The effects of geometry and linear algebra on each other receive close attention in this examination of geometry’s correlation with other branches of math and science. In-depth discussions include a review of systematic geometric motivations in vector space theory and matrix theory; the use of the center of mass in geometry, with an introduction to barycentric coordinates; axiomatic development of determinants in a chapter dealing with area and volume; and a careful consideration of the particle problem. 1965 edition.

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Contenuti

1. The Center of Mass 1.1 Introduction 1.2 Some Physical Assumptions and Conventions 1.3 Physical Motivations in Geometry 1.4 Further Physical Motivations 1.5 An Axiomatic characterization of Center of Mass 1.6 An Algebraic Attack on Geometry 1.7 Painting a Triangle 1.8 Barycentric Coordinates 1.9 Some Algebraic Anticipation 1.10 Affine Geometry 2. Vector Algebra 2.1 Introduction 2.2 The Definition of Vector 2.3 Vector Addition 2.4 Scalar Multiplication 2.5 Physical and Other Applications 2.6 Geometric Applications 2.7 A Vector Approach to the Center of Mass 3. Vector Spaces and Subspaces 3.1 Introduction 3.2 Vector Spaces 3.3 Independence and Dimension 3.4 Some Examples of Vector Spaces: Coordinate Geometry 3.5 Further Examples 3.6 Affine Subspaces 3.7 Some Separation Theorems 3.8 Some Collinearity and Concurrence Theorems 3.9 The Invariance of Dimension 4. Length and Angle 4.1 Introduction 4.2 Geometric Definition of the Inner Product 4.3 Proofs Involving the Inner Product 4.4 The Metrix Axioms 4.5 Some Analytic Geometry 4.6 Orthogonal Subspaces 4.7 Skew Coordinates 5. Miscellaneous Applications 5.1 Introduction 5.2 The Method of Orthogonal Projections 5.3 Linear Equations: Three Views 5.4 A Useful Formula 5.5 Motion 5.6 A Minimum Principle 5.7 Function Spaces 6. Area and Volume 6.1 Introduction 6.2 Area in the Plane: An Axiom System 6.3 Area in the Plane: A Vector Formulation 6.4 Area of Polygons 6.5 Further Examples 6.6 Volumes in 3-Space 6.7 Area Equals Base Times Height 6.8 The Vector Product 6.9 Vector Areas 7. Further Generalizations 7.1 Introduction 7.2 Determinants 7.3 Some Theorems on Determinants 7.4 Even and Odd Permutations 7.5 Outer Products in n-Space 7.6 Some Topology 7.7 Areas of Curved Figures 8. Matrices and Linear Transformations 8.1 Introduction 8.2 Some Examples 8.3 Affine and Linear Transformations 8.4 The Matrix of a Linear Transformation 8.5 The Matrix of an Affine Transformation 8.6 Translations and Dilatations 8.7 The Reduction of an Affine Transformation to a Linear One 8.8 A Fixed Point Theorem with Probabilistic Implications 9. Area and Metric Considerations 9.1 Introduction 9.2 Determinants 9.3 Applications to Analytic Geometry 9.4 Orthogonal and Euclidean Transformations 9.5 Classification of Motions of the Plane 9.6 Classification of Motions of 3-Space 10. The Algebra of Matrices 10.1 Introduction 10.2 Multiplication of Matrices 10.3 Inverses 10.4 The Algebra of Matrices 10.5 Eigenvalues and Eigenvectors 10.6 Some Applications 10.7 Projections and Reflections 11. Groups 11.1 Introduction 11.2 Definitions and Examples 11.3 The "Erlangen Program" 11.4 Symmetry 11.5 Physical Applications of Symmetry 11.6 Abstract Groups Index

Product Description

This examination of geometry's correlation with other branches of math and science features a review of systematic geometric motivations in vector space theory and matrix theory; more. 1965 edition.

Dettagli dell?edizione

• Editore: Dover Pubns
• Data di pubblicazione: 2003
• Lingua: Inglese
• ISBN 10: 0486404528
• ISBN 13: 9780486404523
• Rilegatura: Copertina flessibile
• Numero di pagine: 397
• Contatto del produttore: non disponibile