In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the geometric product. Multiplication of vectors results in higher-dimensional objects called multivectors. Compared to other formalisms for manipulating geometric objects, geometric algebra is noteworthy for supporting vector division and addition of objects of different dimensions. The geometric product was first briefly mentioned by Hermann Grassmann, who was chiefly interested in developing the closely related exterior algebra. In 1878, William Kingdon Clifford greatly expanded on Grassmann's work to form what are now usually called Clifford algebras in his honor (although Clifford himself chose to call them "geometric algebras"). Clifford defined the Clifford algebra and its product as a unification of the Grassmann algebra and Hamilton's quaternion algebra. Adding the dual of the Grassmann exterior product (the "meet") allows the use of the Grassmann–Cayley algebra, and a conformal version of the latter together with a conformal Clifford algebra yields a conformal geometric algebra providing a framework for classical geometries. In practice, these and several derived operations allow a correspondence of elements, subspaces and operations of the algebra with geometric interpretations. For several decades, geometric algebras went somewhat ignored, greatly eclipsed by the vector calculus then newly developed to describe electromagnetism. The term "geometric algebra" was repopularized in the 1960s by Hestenes, who advocated its importance to relativistic physics. The scalars and vectors have their usual interpretation, and make up distinct subspaces of a geometric algebra. Bivectors provide a more natural representation of the pseudovector quantities in vector algebra such as oriented area, oriented angle of rotation, torque, angular momentum and the electromagnetic field. A trivector can represent an oriented volume, and so on. An element called a blade may be used to represent a subspace of and orthogonal projections onto that subspace. Rotations and reflections are represented as elements. Unlike a vector algebra, a geometric algebra naturally accommodates any number of dimensions and any quadratic form such as in relativity. Examples of geometric algebras applied in physics include the spacetime algebra (and the less common algebra of physical space) and the conformal geometric algebra. Geometric calculus, an extension of GA that incorporates differentiation and integration, can be used to formulate other theories such as complex analysis and differential geometry, e.g. by using the Clifford algebra instead of differential forms. Geometric algebra has been advocated, most notably by David Hestenes and Chris Doran, as the preferred mathematical framework for physics. Proponents claim that it provides compact and intuitive descriptions in many areas including classical and quantum mechanics, electromagnetic theory and relativity. GA has also found use as a computational tool in computer graphics and robotics. (Wikipedia).
Geometric Algebra - The Matrix Representation of a Linear Transformation
In this video, we will show how matrices as computational tools may conveniently represent the action of a linear transformation upon a given basis. We will prove that conventional matrix operations, particularly matrix multiplication, conform to the composition of linear transformations.
From playlist Geometric Algebra
Geometric Algebra - Linear Transformations, Outermorphism, and the Determinant
In this video, we will review some basic concepts from linear algebra, such as the linear transformation, prove important theorems which ground matrix operations, extend the linear transformation on vectors to higher-graded elements to bivectors and trivectors, and define the determinant o
From playlist Geometric Algebra
Geometric Algebra, First Course, Episode 08: The Geometric Product.
We finally arrive at the ability to multiply our Geometric numbers together. We see where the geometric product comes from, leading to the definition for vector multiplication, and we add some definitions that allow us to multiply all elements of our algebra. We also use automated testing
From playlist Geometric Algebra, First Course, in STEMCstudio
Linear Algebra 2e: Confirming All the 'Tivities
https://bit.ly/PavelPatreon https://lem.ma/LA - Linear Algebra on Lemma http://bit.ly/ITCYTNew - Dr. Grinfeld's Tensor Calculus textbook https://lem.ma/prep - Complete SAT Math Prep
From playlist Part 1 Linear Algebra: An In-Depth Introduction with a Focus on Applications
Geometric Algebra - Duality and the Cross Product
In this video, we will introduce the concept of duality, involving a multiplication by the pseudoscalar. We will observe the geometric meaning of duality and also see that the cross product and wedge product are dual to one another, which means that the cross product is already contained w
From playlist Geometric Algebra
Linear Algebra 2d: Addition of Geometric Vectors
https://bit.ly/PavelPatreon https://lem.ma/LA - Linear Algebra on Lemma http://bit.ly/ITCYTNew - Dr. Grinfeld's Tensor Calculus textbook https://lem.ma/prep - Complete SAT Math Prep
From playlist Part 1 Linear Algebra: An In-Depth Introduction with a Focus on Applications
Geometric Algebra, First Course, Episode 00: High Level Overview.
Geometric Algebra is the 21st Century tool for Mathematical Physics. This video provides an introduction to the subject and announces an upcoming series of videos in which the viewer can construct their own Geometric Numbers (multivectors) in STEMCstudio, define various operators used for
From playlist Geometric Algebra, First Course, in STEMCstudio
Geometric Numbers (Geometric Algebra 1.1)
In the first video of the series we will discover new number systems, which turn out to be related to various geometrical operations. At the core is the equation x^2 = -1, which usually only leads to the discovery of the imaginary numbers. However, if we solve it very carefully there are t
From playlist Introductory
Geometric Algebra - Rotors and Quaternions
In this video, we will take note of the even subalgebra of G(3), see that it is isomorphic to the quaternions and, in particular, the set of rotors, themselves in the even subalgebra, correspond to the set of unit quaternions. This brings the entire subject of quaternions under the heading
From playlist Math
From Zero to Geo Introduction (Geometric Algebra Series)
This video is the introduction to my series on geometric algebra, From Zero to Geo. In this series, we will build geometric algebra from the ground up, starting from just high school algebra. My hope is that this series can be used by motivated high schoolers (or anybody above that level
From playlist From Zero to Geo
SIGGRAPH 2022 - Geometric Algebra
The SIGGRAPH 2022 course on Geometric Algebra. by Alyn Rockwood and Dietmar Hildenbrand
From playlist Introductory
Geometric Algebra 19 3D Geometric Algebra
In this lesson we examine the 3D geometric algebra for the Euclidean case. This is simpler than the spacetime algebra we have been considering so far. First, we construct the basis for the 3D algebra and then begin to make contact with the standard vector algebra of elementary physics. Aft
From playlist QED- Prerequisite Topics
A Swift Introduction to Geometric Algebra
This video is an introduction to geometric algebra, a severely underrated mathematical language that can be used to describe almost all of physics. This video was made as a presentation for my lab that I work in. While I had the people there foremost in my mind when making this, I realiz
From playlist Miscellaneous Math
GAME2020 - 1. Dr. Leo Dorst. Get Real! (new audio!)
Dr. Leo Dorst from the University of Amsterdam explains how Geometric Algebra subsumes/extends/invigorates Linear Algebra. More information at https://bivector.net This version has an updated audio track.
From playlist Bivector.net
Holomorphic rigid geometric structures on compact manifolds by Sorin Dumitrescu
Higgs bundles URL: http://www.icts.res.in/program/hb2016 DATES: Monday 21 Mar, 2016 - Friday 01 Apr, 2016 VENUE : Madhava Lecture Hall, ICTS Bangalore DESCRIPTION: Higgs bundles arise as solutions to noncompact analog of the Yang-Mills equation. Hitchin showed that irreducible solutio
From playlist Higgs Bundles
Geometric Algebra in 2D: complex numbers without the square root of minus one - Russell Goyder
Russell Goyder introduces geometric algebra from scratch, explaining how you can *multiply* vectors in a sensible way, that is deeply related to the geometry of rotations and reflections in space. After walking us through the basics, he shows how rotors represent rotations in the setting o
From playlist metauni festival 2023
QED Prerequisites Geometric Algebra: Introduction and Motivation
This lesson is the beginning of a significant diversion from QED prerequisites. No student needs to understand Geometric Algebra in order to begin the study of QED. However, since we have pushed the formal structure of Maxwell's Equations as far as I know how to go, I think it makes sense
From playlist QED- Prerequisite Topics
Addendum to A Swift Introduction to Geometric Algebra
This video is an addendum to my most popular video, A Swift Introduction to Geometric Algebra. It clears up some misunderstandings that have arisen from the original video, and then describes two useful ways of understanding the geometric product. This also leads to a discussion of the o
From playlist Miscellaneous Math
Algebra for Beginners | Basics of Algebra
#Algebra is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. Table of Conten
From playlist Linear Algebra