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Geometric Algebra for Mathematicians Geometric Algebra for Mathematicians Kalle Rutanen 09.04.2013 (this is an incomplete draft) 1 Contents 1 Preface 4 1.1 Relation to other texts . 4 1.2 Acknowledgements . 6 2 Preliminaries 11 2.1 Basic definitions . 11 2.2 Vector spaces . 12 2.3 Bilinear spaces . 21 2.4 Finite-dimensional bilinear spaces . 27 2.5 Real bilinear spaces . 29 2.6 Tensor product . 32 2.7 Algebras . 33 2.8 Tensor algebra . 35 2.9 Magnitude . 35 2.10 Topological spaces . 37 2.11 Topological vector spaces . 39 2.12 Norm spaces . 40 2.13 Total derivative . 44 3 Symmetry groups 50 3.1 General linear group . 50 3.2 Scaling linear group . 51 3.3 Orthogonal linear group . 52 3.4 Conformal linear group . 55 3.5 Affine groups . 55 4 Algebraic structure 56 4.1 Clifford algebra . 56 4.2 Morphisms . 58 4.3 Inverse . 59 4.4 Grading . 60 4.5 Exterior product . 64 4.6 Blades . 70 4.7 Scalar product . 72 4.8 Contraction . 74 4.9 Interplay . 80 4.10 Span . 81 4.11 Dual . 87 4.12 Commutator product . 89 4.13 Orthogonality . 90 4.14 Magnitude . 93 4.15 Exponential and trigonometric functions . 94 2 5 Linear transformations 97 5.1 Determinant . 97 5.2 Pinors and unit-versors . 98 5.3 Spinors and rotors . 104 5.4 Projections . 105 5.5 Reflections . 105 5.6 Adjoint . 107 6 Geometric applications 107 6.1 Cramer’s rule . 108 6.2 Gram-Schmidt orthogonalization . 108 6.3 Alternating forms and determinant . 109 6.4 Reciprocal bases . 109 3 1 Preface A real vector space V allows the representation of 1-dimensional, weighted, oriented subspaces. A given vector v 2 V represents the subspace span(fvg) = fαv : α 2 Rg, called the span of v. Inside this subspace, vectors can be compared with respect to v; if w = αv, for some α 2 R, then its magnitude (relative to v) is jαj and its orientation (relative to v) is sgn(α). To compare vectors between different subspaces requires a way to measure magnitudes. This is made possible by introducing a symmetric bilinear form in V , turning V into a symmetric bilinear space. An exterior algebra on V extends V to a unital associative algebra G(V ), which allows the representation of subspaces of V as elements of G(V ). Each k-dimensional subspace of V can be represented in G(V ) as an exterior product of k vectors, and vice versa. Each element representing the same subspace in G(V ), the span of the element, can be compared to a baseline element to obtain its relative magnitude and relative orientation. A symmetric bilinear form in V induces a symmetric bilinear form in G(V ) (the scalar product), so that G(V ) also becomes a symmetric bilinear space. Given a symmetric bilinear form in G(V ), one may measure magnitudes of subspaces, and define the contraction in G(V ), which is the algebraic representation of the orthogonal complement of a subspace A ⊂ V on a subspace B ⊂ V . A Clifford algebra on V introduces a single product, the geometric product, in G(V ), which encompasses both the spanning properties required to form subspaces, and the metric properties required to measure magnitudes. A special property of this product over the more specific products is that it is associative. This brings in the structure of a unital associative algebra, which has the desirable properties of possessing a unique identity and unique inverses. Applying the geometric product in a versor product provides a structure- preserving way to perform reflections in V , which extends to orthogonal transformations in V by the Cartan-Dieudonné theorem. 1.1 Relation to other texts This paper can be thought to complement the book Geometric Algebra for Computer Science [3], which we think is an excellent exposition of the state-of-the-art in geometric algebra, both in content and writing. A mathematician, however, desires for greater succinctness and detail; we aim to provide that in this paper. Ideally, one would alternate between this paper, for increased mathematical intuition, and that book, for increased geometric intuition. The organization of this paper is inspired by the Schaum’s Outlines series. In partic- ular, we begin each section by giving the definitions without additional prose, and then follow up with a set of theorems, remarks, and examples concerning those definitions. In addition, we give many of the theorems and remarks a short summary; we hope this makes it more efficient to skip familiar parts later when recalling the subject matter. In addition to [3], we merge together ideas from several other texts on geometric algebra, notably [4] and [2]. Here we briefly summarize the key points in which we have either adopted, or diverged from, an idea. A comparison between terms in different texts is given in Table 1. Remark 1.1.1 (Coordinate-free proofs). We prove the results in Clifford algebra 4 Table 1: Comparison of terms between texts Concept This paper [4] [3] Product of k vectors Versor Versor - Product of k invertible vectors Invertible versor Invertible versor Versor Simple element of Cl(V ) k-blade simple k-vector k-blade Solutions to v ^ A = 0 Span ? Attitude Measure of size in Cl(V ) Magnitude Magnitude Weight without assuming a priviledged basis for the vectors. This reveals structure in proofs. The key theorem in this task is Theorem 4.5.14, which shows that vector-preserving homomorphisms preserve grade. Remark 1.1.2 (General form for the bilinear form). We do not assume any par- ticular form for the underlying symmetric bilinear form. This makes it explicit which theorems generalize to arbitrary symmetric bilinear forms, and which do not. In addi- tion, this reveals structure in proofs. Remark 1.1.3 (Contraction, not inner products). We adopt the definition of con- traction from [3]. This definition is different from [4] and [2]. See Section 4.8 for details. Remark 1.1.4 (Explicit notation for dual). We denote the dual of a k-blade Ak on Bl an invertible l-blade Bl by Ak . This notation is shown appropriate by Theorem 4.11.4, which says that Bl Bl span Ak = span(Ak) : (1.1.1) In addition, this makes the l-blade Bl explicit, which is important when taking duals in ∗ subspaces. We diverge from [3], which uses the notation Ak, and assumes the reader is −∗ aware of the l-blade Bl relative to which the dual is taken. The undual Ak in [3] is given −1 B by Ak l . Remark 1.1.5 (Minor corrections). To translate results between this paper and [3], it is useful to be aware of some minor errors in their text: • What they call a bilinear form is actually a symmetric bilinear form. • What they call a degenerate bilinear form is actually an indefinite non-degenerate symmetric bilinear form, so of signature (p; q; 0). • What they call a metric space is actually a norm space. • What they call a norm is correct, but only defined when the underlying symmetric bilinear form is positive-definite. These errors are minor, and in no way affect the excellent readability of [3]. 5 1.2 Acknowledgements My interest on geometric algebra grew from my interest in geometric algorithms and computer graphics. Specifically, I was led to consider the question of whether there was some good way to obtain the intersection of two affine flats. After having queried this in the then active comp.graphics.algorithms USENET newsgroup, I was replied by a knowledgeable person named Just’d’Faqs, who suggested that I take a look at geometric algebra. This got me interested in geometric algebra, but unfortunately I did not have the time to study it back then. In 2010, I started my doctoral studies on geometric algebra (although I eventually changed the subject). It is this way that I was led to my doctoral supervisor, professor Sirkka-Liisa Eriksson, and to work as an exercise assistant on her course on geometric algebra. This paper got its start during the Christmas holiday 2011, as an attempt to clarify things to myself, before the start of the geometric algebra course in Spring 2012. I found the process such a good learning experience that I decided to continue on the writing on my free time. However, it wasn’t until the end of 2012 that I would pick up on writing again, the motivation being the same geometric algebra course in Spring 2013. This paper, then, is the result of that process. I would like to thank Just’d’Faqs for guiding me to this direction, and Sirkka-Liisa Eriksson, for introducing me to the subject through her course. This paper wouldn’t exist without the excellent books of Hestenes et al. [4], Dorst et al. [3], and Doran et al. [2]; these are the texts from which I have absorbed my knowledge. 6 Table 2: Latin alphabet (small) Symbol Meaning Example a, b, c, d Vector in a vector space, 1-vector a 2 V e Exponential function eB2 f, g, h Function f(x), g ◦ h i, j Index i 2 I k, l, m Cardinality of a set, grade of a k-vector fa1; : : : ; akg, Ak n Dimension of a vector space o Avoided; resembles zero p, q, r A distinguished point limx!p f(x) = y s, t - u, v, w Vector in a vector space x, y, z Point in a topological space p, q, r Represented point in the conformal model [p] Point representative in the conformal model p 2 [p] 7 Table 3: Latin alphabet (capitals) Symbol Meaning Example Pn A, B, C Multi-vector A = k=0 Ak Ak, Bl, Cm k-vector B Basis of a vector space or of a topological space BX ⊂ TX C Closed sets of a topological space CX D, E - F Field α 2 F G, H Group I, J Index set faigi2I ⊂ V K - L Linear functions between vector spaces L(U; V ) M, N, O Neighborhood, open set Op 2 TX (p) P , Q - R Rotor S Subset, subspace S ⊂ V T Topology TX U, V , W Vector space v 2 V X, Y , Z Topological space x 2 X 8 Table 4: Greek alphabet (small) Symbol Meaning Example α, β, γ Scalar, scalar tuple α 2 F , β
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