Vector Space Approach to Geometry (eBook)

Melvin Hausner (Autor)

eBook Download: EPUB

2018
416 Seiten
Dover Publications (Verlag)
978-0-486-83539-6 (ISBN)

Infos

A fascinating exploration of the correlation between geometry and linear algebra, this text also offers elementary explanations of the role of geometry in other branches of math and science. 1965 edition.

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A fascinating exploration of the correlation between geometry and linear algebra, this text portrays the former as a subject better understood by the use and development of the latter rather than as an independent field. The treatment offers elementary explanations of the role of geometry in other branches of math and science — including physics, analysis, and group theory — as well as its value in understanding probability, determinant theory, and function spaces.Outstanding features of this volume include discussions 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. Students and other mathematically inclined readers will find that this inquiry into the interplay between geometry and other areas offers an enriched appreciation of both subjects.

Autor

Melvin Hausner is Emeritus Professor at New York University's Courant Institute of Mathematical Sciences. His other books include Elementary Probability Theory and Lie Groups, Lie Algebras.

Inhalt

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 Geometry2. 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 Mass3. 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 Dimension4. 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 Coordinates5. 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 Spaces6. 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 Areas7. 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 Figures8. 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 Implications9. 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-Space10. 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 Reflections11. 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

Produkt-Details

• Erscheint lt. Verlag: 17.10.2018
• Reihe/Serie: Dover Books on Mathematics
• Reihe/Serie: Dover Books on Mathematics
• Sprache: englisch
• Gewicht: 27 g
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Schlagworte: Algebra • Axioms • barycentric • Barycentric coordinates • basis vectors • determinants • determinant theory • elementary explanations • experiments • function spaces • Geometry • linear algebra • linear transformations • Math • Mathematics • Physics • Probability • Pure Mathematics • Science • science and math • tangent spaces • textbooks • Vector Space
• ISBN-10: 0-486-83539-1 / 0486835391
• ISBN-13: 978-0-486-83539-6 / 9780486835396