2012/05/27 by Eric Chisolm, Eric D. Chisolm, Chisolm, Eric · 1 voice
Mathematics · Physics and Astronomy · #Algebra over a field #Algebra representation #Algebraic and Geometric Analysis #Cellular algebra #Clifford algebra #Conformal geometric algebra #Dimension (graph theory) #Exterior algebra #Filtered algebra #Geometric algebra #Geometry #Linear subspace #Mathematics #Multivector #Product (mathematics) #Pure mathematics #Universal geometric algebra #math-ph #math.MP
paper · pdf · doi:10.48550/arxiv.1205.5935
published in arXiv (Cornell University) (Cornell University) · 92 pages and 1 figure
arxiv created 2012/05/27 · openalex publication_date 2012/05/27 · arxiv updated 2012/05/29 · openalex created_date 2022/09/10 · openalex updated_date 2026/07/28
This is an introduction to geometric algebra, an alternative to traditional vector algebra that expands on it in two ways: 1. In addition to scalars and vectors, it defines new objects representing subspaces of any dimension. 2. It defines a product that's strongly motivated by geometry and can be taken between any two objects. For example, the product of two vectors taken in a certain way represents their common plane. This system was invented by William Clifford and is more commonly known as Clifford algebra. It's actually older than the vector algebra that we use today (due to Gibbs) and includes it as a subset. Over the years, various parts of Clifford algebra have been reinvented independently by many people who found they needed it, often not realizing that all those parts belonged in one system. This suggests that Clifford had the right idea, and that geometric algebra, not the reduced version we use today, deserves to be the standard "vector algebra." My goal in these notes is to describe geometric algebra from that standpoint and illustrate its usefulness. The notes are work in progress; I'll keep adding new topics as I learn them myself.