Grassmann Algebra Volume 1: Foundations: Exploring extended vector algebra with Mathematica

587 Pages·2012·29.22 MB·English

by  John Browne

Jllf
gebta
~tttssmann
ume 1:
~of ~ounbattons
Grassmann algebra extends vector algebra by introducing the exterior product to algebraicize the
notion of linear dependence. With it, vectors may be extended to higher-grade entities: bivectors,
trivectors, ... multivectors. The extensive exterior product also has a regressive dual: the regres
sive product. The pair behaves a little like the Boolean duals of union and intersection. By
interpreting one of the elements of the vector space as an origin point, points can be defined, and
the exterior product can extend points into higher-grade located entities from which lines, planes
and multiplanes can be defined. Theorems of Projective Geometry are simply formulae involv
ing these entities and the dual products. By introducing the (orthogonal) complement operation,
the scalar product of vectors may be extended to the interior product of multivectors, which in
this more general case may no longer result in a scalar. The notion of the magnitude of vectors is
extended to the magnitude ofmultivectors: for example, the magnitude of the exterior product of
two vectors (a bivector) is the area of the parallelogram formed by them. To develop these
foundational concepts, we need only consider entities which are the sums of elements of the
same grade. This is the focus of this volume.
But the entities of Grassmann algebra need not be of the same grade, and the possible product
types need not be constricted to just the exterior, regressive and interior products. For example
quaternion algebra is simply the Grassmann algebra of scalars and bivectors under a new prod
uct operation. Clifford, geometric and higher order hypercomplex algebras, for example the
octonions, may be defined similarly. If to these we introduce Clifford's invention of a scalar
which squares to zero, we can define entities (for example dual quaternions) with which we can
perform elaborate transformations. Exploration of these entities, operations and algebras will be
the focus of the volume to follow this.
There is something fascinating about the beauty with which the mathematical structures that
Hermann Grassmann discovered describe the physical world, and something also fascinating
about how these beautiful structures have been largely lost to the mainstreams of mathematics
and science. He wrote his seminal Ausdehnungslehre (Die Ausdehnungslehre. Vollstiindig und in
strenger Form) in 1862. But it was not until the latter part of his life that he received any signifi
cant recognition for it, most notably by Gibbs and Clifford. In recent times David Hestenes'
Geometric Algebra must be ·given the credit for much of the emerging awareness of Grass
mann' s innovation.
In the hope that the book be accessible to scientists and engineers, students and professionals
alike, the text attempts to avoid any terminology which does not make an essential contribution
to an understanding of the basic concepts. Some familiarity with basic linear algebra may
however be useful.
I hope you enjoy exploring this beautiful mathematical system.

Jlf
ge6ta
~tassmann
1:
~ofume ~ounbttfions
Exploring extended vector algebra with Mathematica
John Browne
l
[i
Barnard Pnblisbing
To discuss anything in this book or Grassmann algebra contact the author
John Browne
[email protected]
For more information about the book or the GrassmannAlgebra software visit
http:// sites .google .com/site/g rassmannalgebra
Keywords: Grassmann, Grassmann algebra, GrassmannAlgebra, Ausdehnungslehre, exterior
algebra, exterior product, regressive product, interior product, inner product, scalar product, dot
product, cross product, complement operation, duality, projective geometry, geometric algebra,
vector algebra, linear space, vector space, vector, point, bivector, bound vector, weighted point,
grade, determinant, multilinear, common factor, multiplanes, orthogonality, metric, invariance.
Published by Barnard Publishing
180 Laughing Waters Road
Eltham VIC 3095 Australia
Printed by the print-on-demand process
and distributed world-wide by local distributors
© Copyright John Browne 2012
This book is protected by copyright. The book or parts of it may not be copied or disseminated
in any way without the permission of the copyright owner. You may copy, reference or quote
small sections of the work as long as due acknowledgment is made.
ISBN: 978-1479197637
First printing October 2012
First edition
Contents
Preface 1
1 Introduction 5
1.1 Background 5
The mathematical representation of physical entities 5
The central concept of the Ausdehnungslehre 5
Comparison with the vector and tensor algebras 6
Algebraicizing the notion of linear dependence 6
Grassmann algebra as a geometric calculus 7
1.2 The Exterior Product 8
The anti-symmetry of the exterior product 8
Exterior products of vectors in a three-dimensional space 9
Terminology: elements and entities 10
The grade of an element 11
Interchanging the order of the factors in an exterior product 12
A brief summary of the properties of the exterior product 12
1.3 The Regressive Product 13
The regressive product as a dual product to the exterior product 13
Unions and intersections of spaces 14
A brief summary of the properties of the regressive product 14
The Common Factor Axiom 15
The intersection of two bi vectors in a three-dimensional space 17
1.4 Geometric Interpretations 17
Points and vectors 17
Sums and differences of points 18
Determining a mass-centre 20
Lines and planes 21
The intersection of two lines 22
1.5 The Complement 23
The complement as a correspondence between spaces 23
The Euclidean complement 24
The complement of a complement 26
The Complement Axiom 27
1.6 The Interior Product 28
The definition of the interior product 28
Inner products and scalar products 29
ii
-
-
Sequential interior products 29
Orthogonality 30
Measure and magnitude 30
Calculating interior products from their definition 31
Expanding interior products 32
The interior product of a bivector and a vector 32
The cross product 33
1.7 Summary 34
Summary of operations 34
Summary of objects 35
2 The Exterior Product 37
2.1 Introduction 37
2.2 The Exterior Product 38
Basic properties of the exterior product 38
Declaring scalar and vector symbols in GrassmannAlgebra 40
Entering exterior products 40
2.3 Exterior Linear Spaces 40
Composing m-elements 40
Composing elements automatically 41
Spaces and congruence 42
The associativity of the exter.ior product 42
Transforming exterior products 43
2.4 Axioms for Exterior Linear Spaces 44
Summary of axioms 44
Grassmann algebras 46
On the nature of scalar multiplication 46
Factoring scalars 47
Grassmann expressions 4 7
Calculating the grade of a Grassmann expression 48
2.5 Bases 49
Bases for exterior linear spaces 49
Declaring a basis in GrassmannAlgebra 49
Composing bases of exterior linear spaces 50
Composing palettes of basis elements 50
Standard ordering 51
Indexing basis elements of exterior linear spaces 52
2.6 Cobases 52
Definition of a cobasis 52
The cobasis ofunity 53
Composing palettes of cobasis elements 54
iii
The cobasis of a cobasis 54
2.7 Determinants 55
Determinants from exterior products 55
Properties of determinants 56
The Laplace expansion technique 56
Calculating determinants 57
2.8 Cofactors 58
Cofactors from exterior products 58
The Laplace expansion in cofactor form 59
Transformations of cobases 60
2.9 Solution of Linear Equations 61
Grassmann's approach to solving linear equations 61
Example solution: 3 equations in 4 unknowns 62
Example solution: 4 equations in 4 unknowns 62
2.10 Simplicity 63
The concept of simplicity 63
All (n-1)-elements are simple 63
Conditions for simplicity of a 2-element in a 4-space 64
Conditions for simplicity of a 2-element in a 5-space 64
Factorizing simple elements from first principles 65
2.11 Exterior Division 67
The definition of an exterior quotient 67
Division by a I-element 67
Division by a k-element 68
Automating the division process 69
2.12 Multilinear Forms 69
The span of a simple element 69
Composing spans 70
Example: Refactorizations 72
Multilinear forms 73
Defining m:k-forms 74
Composing m:k-forms 75
Expanding and simplifying m:k-forms 76
Developing invariant forms 76
The invariance ofm:k-forms 77
The complete span of a simple element 78
The Zero Form Theorem 81
Zero Form formulae 82
2.13 Unions and Intersections 85
Union and intersection as a multilinear form 85
Where the intersection is evident 86
iv
Where the intersections is not evident 88
Intersection with a non-simple element 89
Factorizing simple elements 90
2.14 Summary 92
3 The Regressive Product 93
3.1 Introduction 93
3.2 Duality 93
The notion of duality 93
Examples: Obtaining the dual of an axiom 94
Summary: The duality transformation algorithm 96
3.3 Properties of the Regressive Product 96
Axioms for the regressive product 96
The unit n-element 97
The inverse of an n-element 99
Grassmann' s notation for the regressive product 100
3.4 The Grassmann Duality Principle 101
The dual of a dual 101
The Grassmann Duality Principle 101
Using the GrassmannAlgebra function Dual 102
3.5 The Common Factor Axiom 104
Motivation 104
The Common Factor Axiom 105
Extension of the Common Factor Axiom to general elements 106
Special cases of the Common Factor Axiom 107
Dual versions of the Common Factor Axiom 107
Application of the Common Factor Axiom 108
When the common factor is not simple 110
3.6 The Common Factor Theorem 110
Development of the Common Factor Theorem 110
Proof of the Common Factor Theorem 113
The A and B forms of the Common Factor Theorem 115
Example: The decomposition of a 1-element 116
Example: Applying the Common Factor Theorem 117
Automating the application of the Common Factor Theorem 118
A special form of the Common Factor Theorem 120
3.7 The Regressive Product of Simple Elements 122
The regressive product of simple elements 122
The regressive product of (n-1)-elements 122
Regressive products leading to scalar results 122
v
The cobasis form of the Common Factor Axiom 123
The regressive product of cobasis elements 124
3.8 Expressing an Element in another Basis 125
Expressing an element in terms of another basis 125
Using the computable form of the Common Factor Theorem 126
Automating the process 127
The symmetric expansion ofa I-element in terms of another basis 128
3.9 Factorization of Simple Elements 129
Factorization using the regressive product 129
Factorizing elements expressed in terms of basis elements 131
The factorization algorithm 133
Factorization of (n-1)-elements 135
Factorizing simple m-elements 136
Factorizing contingently simple m-elements 138
Determining if an element is simple 140
3.10 Product Formulae for Regressive Products 141
TheProductFormula 141
Deriving Product Formulae 143
Deriving Product Formulae automatically 143
Computing the General Product Formula 145
Comparing the two forms of the Product Formula 149
The invariance of the General Product Formula 150
Alternative forms for the General Product Formula 150
The Decomposition Formula 151
Exploration: Dual forms of the General Product Formulae 153
The double sum form of the General Product Formula 154
3.11 Summary 156
4 Geometric Interpretations 157
4.1 Introduction 157
4.2 Geometrically Interpreted 1-Elements 158
Vectors 158
Points 160
Declaring a basis for a bound vector space 163
Composing vectors and points 164
Example: Calculation of the centre of mass 165
4.3 Geometrically Interpreted 2-Elements 166
Simple geometrically interpreted 2-elements 166
Bivectors 167
Bound vectors 169
Composing bivectors and bound vectors 171