Quaternions | Finite groups | Group theory
In group theory, the quaternion group Q8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset of the quaternions under multiplication. It is given by the group presentation where e is the identity element and e commutes with the other elements of the group. Another presentation of Q8 is (Wikipedia).
This is a video I have been wanting to make for some time, in which I discuss what the quaternions are, as mathematical objects, and how we do calculations with them. In particular, we will see how the fundamental equation of the quaternions i^2=j^2=k^2=ijk=-1 easily generates the rule for
From playlist Quaternions
This is lecture 9 of an online mathematics course on groups theory. It covers the quaternions group and its realtion to the ring of quaternions.
From playlist Group theory
The Quaternion Symmetry Group – Vi Hart
From playlist G4G11 Videos
Quaternions as 4x4 Matrices - Connections to Linear Algebra
In math, it's usually possible to view an object or concept from many different (but equivalent) angles. In this video, we will see that the quaternions may be viewed as 4x4 real-valued matrices of a special form. What is interesting here is that if you know how to multiply matrices, you a
From playlist Quaternions
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
Lie Groups and Lie Algebras: Lesson 2 - Quaternions
This video is about Lie Groups and Lie Algebras: Lesson 2 - Quaternions We study the algebraic nature of quaternions and cover the ideas of an algebra and a field. Later we will discover how quaternions fit into the description of the classical Lie Groups. NOTE: An astute viewer noted th
From playlist Lie Groups and Lie Algebras
Quaternion algebras via their Mat2x2(F) representations
In this video we talk about general quaternion algebras over a field, their most important properties and how to think about them. The exponential map into unitary groups are covered. I emphasize the Hamiltionion quaternions and motivate their relation to the complex numbers. I conclude wi
From playlist Algebra
Abstract Algebra | The quaternion group
We present the quaternion group. This is an important example of a non-abelian group of small order. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/
From playlist Abstract Algebra
Abstract Algebra | Subgroups and quotient groups of the quaternions.
We present a description of all subgroups and quotient groups of the quaternions. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/
From playlist Abstract Algebra
Lie Groups and Lie Algebras: Lesson 12 - The Classical Groups Part X (redux)
Lie Groups and Lie Algebras: Lesson 12 - The Classical Groups Part X (redux) We name the classical groups, finally! This video ended a bit short, I added the missing part in the "redux" version of this lesson. Please consider supporting this channel via Patreon: https://www.patreon.com/
From playlist Lie Groups and Lie Algebras
Lie Groups and Lie Algebras: Lesson 30 - SL(1,Q) from sl(1,Q)
Lie Groups and Lie Algebras: Lesson 30 - SL(1,Q) from sl(1,Q)' I this lecture we examine the lesser known member of the three Lie groups that share the "angular momentum" algebra: The Special Linear Group of transformations of a one dimensional quaternionic vector space. This is an exampl
From playlist Lie Groups and Lie Algebras
LieGroups and Lie Algebras: Lesson 4 - The Classical Groups Part II
Lie Groups and Lie Algebras: Lesson 4 - The Classical Groups Part II We introduce the idea of the classical matrix groups and their associated carrier spaces. In this video we discuss the representation of complex numbers and quaternions as matrices and then we discuss the idea of a metri
From playlist Lie Groups and Lie Algebras
Set Theory (Part 14c): More on the Quaternions
No background in sets required for this video. In this video, we will learn how the quaternions can be thought of as pairings of complex numbers. We also will show how the quaternions can be written as a 2x2 complex matrix as opposed to a 4x4 real matrix and how the unit quaternions form t
From playlist Set Theory
Representation theory: The Schur indicator
This is about the Schur indicator of a complex representation. It can be used to check whether an irreducible representation has in invariant bilinear form, and if so whether the form is symmetric or antisymmetric. As examples we check which representations of the dihedral group D8, the
From playlist Representation theory
Lie Groups and Lie Algebras: Lesson 3 - Classical Groups Part I
Lie Groups and Lie Algebras: Lesson 3 - Classical Groups Part I We introduce the idea of the classical matrix groups and their associated carrier spaces. Please consider supporting this channel via Patreon: https://www.patreon.com/XYLYXYLYX
From playlist Lie Groups and Lie Algebras
Set Theory (Part 14b): Quaternions and 3D Rotations
No background in sets needed for this video - learn about the foundations of quaternions, derivation of the Hamilton product, and their application to 3D rotations. We will also see how dot and cross products are related to quaternion math. This video will be of particular interest to comp
From playlist Set Theory by Mathoma
Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys The Quaternion Group
From playlist Abstract Algebra
