A Vector Space Approach to Geometry

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 as well as its value in understanding probability, determinant theory, and function spaces.

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.

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