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Algebraic Foundations I - Groups | Chapter 2: Morphisms of groups
When studying algebraic structures, we are interested in the maps between them that are "compatible" with their structure. In the case of groups, such maps are called homomorphisms, and here we introduce them, their basic properties and many examples. We go on to study the kernel of a group homomorphism, which is the subgroup that is mapped to the neutral element. It is an important tool in studying homomorphisms and groups. In the last part, we discover that the automorphisms of a group (bijective homomorphisms from the group to itself) form a group themselves.
Sections
00:00:00 Greetings
00:00:42 1. Homomorphisms of groups
00:03:30 Definition of homomorphism, isomorphism, endomorphism, automorphism
00:09:22 First properties and examples of homomorphisms
00:39:17 2. The kernel
00:39:28 Definition of the kernel of a homomorphism
00:40:02 Kernel theorem
00:48:36 Cosets of the kernel
01:00:43 Modular arithmetic as an example
01:09:20 3. Automorphism groups
01:16:22 Inner autormorphisms (conjugations)
01:30:46 The center of a group
This video was made using LaTeX Beamer, OmniGraffle, LaTeXit and SlidePilot.
#maths #algebra #groups #grouptheory #subgroups #symmetry #permutations #integers #homomorphism #kernels #automorphism
#modulararithmetic #isomorphism
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Red Panda Aurora in Budapest Zoo
Переглядів 89Місяць тому
Red Panda Aurora in Budapest Zoo
Generalized functions and weak derivatives - A very brief introduction
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This video gives a brief introduction into the ideas underlying generalized functions (aka distributions) and the weak derivatives we can take of them. The emphasis is on working out the ideas rather than mathematical rigor and generality. Sections 00:00 Greetings 03:06 1. Test functions and generalized functions 05:47 The space T* of generalized functions (distributions) 06:31 Embedding contin...
Algebraic Foundations I - Groups | Chapter 1: Definition and basic properties of groups
Переглядів 882 місяці тому
We compare algebraic properties of the integers, the permutations of a finite set, and the symmetries of a circle and those of a triangle. In doing so, we find they have certain properties in common, which lead us to the definition of a "group". Then find some basic properties of groups and study subgroups, which are subsets of groups that are groups in their own right. Sections 00:00:00 Greeti...
Algebraic Foundations - Introduction
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This course will teach the most fundamental algebraic structures on a university level: Groups, rings and fields, and then some others. These structures underly practically all pure mathematics research, but also appear in the laws of physics, or in engineering applications such as cryptography. #algebra #groups #rings #fields #lawsofphysics #cryptography
Linear Algebra in 2D - Chapter 8: Eigenvalues and eigenvectors
Переглядів 1613 місяці тому
Eigenvalues are a characteristic values of a matrix or linear map that determine its geometric properties to a great extent (though in general not completely). We learn how to compute eigenvalues and eigenvectors via the characteristic polynomial, and use this to classify linear maps of the plane. Finally, we take a brief look of applications of eigenvalues to practical problem solving. Section...
Linear Algebra in 2D - Homework 03
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Determinants and eigenvalues. Download the homework here: wglobke.github.io/exercises/linalg2D/problems03.pdf
Linear Algebra in 2D - Homework 02
Переглядів 603 місяці тому
Inverse matrices and change of basis. Download the homework here: wglobke.github.io/exercises/linalg2D/problems02.pdf
Linear Algebra in 2D - Chapter 7: Determinants
Переглядів 493 місяці тому
Determinants are a wonderful tool to study linear maps and matrices. We motivate them from geometric considerations and derive their basic properties. At the end, we see how determinants can be applied to yield closed form solutions for systems of linear equations. Sections 00:00:00 Greetings 00:00:36 Area and determinants 00:04:44 Area of a parallelogram 00:12:57 Sign as orientation 00:14:52 D...
Is this still Mathematics, or just shameless clickbait?! -- Elephants in Melbourne Zoo
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Please subscribe to my channel (even if you don't like math). 😀 For more cute animals, check out my other channel, ua-cam.com/video/ey46DQRTOMM/v-deo.html ua-cam.com/video/fqDsyqWtMwc/v-deo.html
Linear Algebra in 2D - Chapter 6: The Iwasawa decomposition
Переглядів 159Рік тому
This chapter tells you about the beautiful theorem stating that every invertible linear transformation can be uniquely composed of three types of geometric transformations: an orthogonal one (rotations/reflections), a scaling and a shearing. This theorem is named after Kenkichi Iwasawa, who proved a much more general version of it for topological spaces. mathshistory.st-andrews.ac.uk/Biographie...
Linear Algebra in 2D - Chapter 5: Change of coordinates
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Now that we know we can represent vectors as linear combinations of many different basis vectors, we explore how to change from one basis representation to another. Then we extend these ideas to affine changes of coordinates, where not just the axes, but also the origin of the coordinate system is changed. Also, we see how linear maps can be represented by matrices in different bases, and how t...
Linear Algebra in 2D - Chapter 4: Bases and inversion of matrices
Переглядів 57Рік тому
We learn that any pair of not mutually proportional vectors in the plane can give us a coordinate system with respect to which we can express any vector. Such a pair is called a basis, and we study some further properties of bases and linear independence. This relates to properties of matrices and linear maps as well, most importantly the question whether a matrix has an inverse matrix. We also...
Linear Algebra in 2D - Chapter 3: Linear equations
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We get acquainted with linear equations in two variables, how to interpret their solutions sets geometrically and how to solve them, in particular by casting them in matrix form and using row reduction (aka Gauss-Jordan elimination). Like & Subscribe! Sections 00:00:00 Introduction 00:01:00 Problems that lead to linear equations 00:08:11 Linear equations 00:10:27 One equation yields a line 00:1...
Linear Algebra in 2D - Homework 01
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Vectors, matrices, linear maps and linear equations. Download the homework here: wglobke.github.io/exercises/linalg2D/problems01.pdf
Linear Algebra in 2D - Chapter 2: Linear maps and matrices
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Linear Algebra in 2D - Chapter 2: Linear maps and matrices
Linear Algebra in 2D - Chapter 1: Coordinates and vectors
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Linear Algebra in 2D - Chapter 1: Coordinates and vectors
Linear Algebra in 2D - Chapter 0: Welcome
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Linear Algebra in 2D - Chapter 0: Welcome
Introduction to Mathematics - Chapter 7: Miscellaneous: Combinatorics, sums and numeral systems
Переглядів 672 роки тому
Introduction to Mathematics - Chapter 7: Miscellaneous: Combinatorics, sums and numeral systems
Introduction to Mathematics - Chapter 6: Equivalence relations
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Introduction to Mathematics - Chapter 6: Equivalence relations
Introduction to Mathematics - Homework 03
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Introduction to Mathematics - Homework 03
Introduction to Mathematics - Chapter 5: Relations, graphs and orders
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Introduction to Mathematics - Chapter 5: Relations, graphs and orders
Introduction to Mathematics - Chapter 4: Power sets and cardinalities of sets
Переглядів 1052 роки тому
Introduction to Mathematics - Chapter 4: Power sets and cardinalities of sets
Introduction to Mathematics - Homework 02
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Introduction to Mathematics - Homework 02
Introduction to Mathematics - Chapter 3: Sets and maps
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Introduction to Mathematics - Chapter 3: Sets and maps
Introduction to Mathematics - Chapter 2: Proofs
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Introduction to Mathematics - Chapter 2: Proofs
Introduction to Mathematics - Homework 01
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Introduction to Mathematics - Homework 01
Introduction to Mathematics - Chapter 1: Logic and propositions
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Introduction to Mathematics - Chapter 1: Logic and propositions
Introduction to Mathematics - Chapter 0: Welcome
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Introduction to Mathematics - Chapter 0: Welcome