A Vector Space Approach to Geometry

by Hausner, Melvin

• ISBN13: 9780486404523
• ISBN10: 0486404528
• Format: Paperback
• Copyright: 2010-07-21
• Publisher: Dover Publications

Summary

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; the use of the center of mass in geometry, with an introduction to barycentric coordinates; consideration of the particle problem; more. 1965 edition.

Table of Contents

The Center of Mass

Introduction

Some Physical Assumptions and Conventions

Physical Motivations in Geometry

Further Physical Motivations

An Axiomatic characterization of Center of Mass

An Algebraic Attack on Geometry

Painting a Triangle

Barycentric Coordinates

Some Algebraic Anticipation

Affine Geometry

Vector Algebra

Introduction

The Definition of Vector

Vector Addition

Scalar Multiplication

Physical and Other Applications

Geometric Applications

A Vector Approach to the Center of Mass

Vector Spaces and Subspaces

Introduction

Vector Spaces

Independence and Dimension

Some Examples of Vector Spaces: Coordinate Geometry

Further Examples

Affine Subspaces

Some Separation Theorems

Some Collinearity and Concurrence Theorems

The Invariance of Dimension

Length and Angle

Introduction

Geometric Definition of the Inner Product

Proofs Involving the Inner Product

The Metrix Axioms

Some Analytic Geometry

Orthogonal Subspaces

Skew Coordinates

Miscellaneous Applications

Introduction

The Method of Orthogonal Projections

Linear Equations: Three Views

A Useful Formula

Motion

A Minimum Principle

Function Spaces

Area and Volume

Introduction

Area in the Plane: An Axiom System

Area in the Plane: A Vector Formulation

Area of Polygons

Further Examples

Volumes in 3-Space

Area Equals Base Times Height

The Vector Product

Vector Areas

Further Generalizations

Introduction

Determinants

Some Theorems on Determinants

Even and Odd Permutations

Outer Products in n-Space

Some Topology

Areas of Curved Figures

Matrices and Linear Transformations

Introduction

Some Examples

Affine and Linear Transformations

The Matrix of a Linear Transformation

The Matrix of an Affine Transformation

Translations and Dilatations

The Reduction of an Affine Transformation to a Linear One

A Fixed Point Theorem with Probabilistic Implications

Area and Metric Considerations

Introduction

Determinants

Applications to Analytic Geometry

Orthogonal and Euclidean Transformations

Classification of Motions of the Plane

Classification of Motions of 3-Space

The Algebra of Matrices

Introduction

Multiplication of Matrices

Inverses

The Algebra of Matrices

Eigenvalues and Eigenvectors

Some Applications

Projections and Reflections

Groups

Introduction

Definitions and Examples

The "Erlangen Program

Symmetry

Physical Applications of Symmetry

Abstract Groups

Index

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