Algebra

vector spaces

Vector Spaces: Concepts, History, and Applications Guide

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The Concept A vector space is an additively written abelian group together with a field that operates on it.Intuitive picture vs abstract definition Vector…

Groups, The Path from a Simple Concept to Mysterious Results

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Introduction The concept of a group is as simple as it gets: a set with a binary operation like addition and a couple of natural laws like the requirement…
complex numbers views

Views On Complex Numbers

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Abstract Why do we need yet another article about complex numbers? This is a valid question and I have asked it myself. I could mention that I wanted…
infinity

Why Division by Zero Is Impossible: 10 Mathematical Reasons

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A division by zero is primarily an algebraic question. The reasoning therefore follows the indirect pattern of most algebraic proofs:What if it was…
world of algebras

Introduction to the World of Algebras

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Abstract Richard Pierce describes the intention of his book [2] about associative algebras as his attempt to prove that there is algebra after Galois…
lie group physics

When Lie Groups Became Physics

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Abstract We explain by simple examples (one-parameter Lie groups), partly in the original language, and along the historical papers of Sophus Lie, Abraham…
evariste galois

Évariste Galois and His Theory

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 * Oct. 25th, 1811  † May 31st, 1832 ... or why squaring the circle is doomed. Galois died in a duel at the age of twenty. Yet, he gave…
What is a linear equation

First-Order Linear Equation: Definition & Solutions

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Definition / Summary This article summarizes the first-order (linear) polynomial equation in one variable, its solution, and natural extensions to matrices…
Lie Algebra Intro

Lie Algebra Basics: Definitions, Equations & Examples

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Definition / Summary A Lie algebra (pronounced "Lee") is the tangent-space algebra of a Lie group at its identity element. Concretely, it is a vector…
lie algebra representations

Learn Lie Algebras: A Walkthrough – The Representations

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  Part III: Representations  Sums and Products. Frobenius began in ##1896## to generalize Weber's group characters and soon investigated…
Lie Algebra Structure

Learn Lie Algebras: A Walkthrough – The Structures

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  Part II: StructuresDecompositions.Lie algebra theory is to a large extend the classification of the semisimple Lie algebras which…
lie algebra basics

Learn Lie Algebras: A Walkthrough – The Basics

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  Part I: Basics Introduction. This article is meant to provide a quick reference guide to Lie algebras: the terminology, important theorems,…
manifold2

A Journey to The Manifold SU(2): Representations

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Part 1 Image source: [23]  Some useful bases of ##\mathfrak{su}(2,\mathbb{C})## Notations can differ from author to author: the…
mathoperators

How to Tell Operations, Operators, Functionals, and Representations Apart

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24 Comments
 All these concepts belong to the toolbox of physicists. I read them quite often on our forum and their usage is sometimes a bit confusing.…
whatisatensor

Tensors Explained: Scalars, Vectors, Matrices & Math

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Introduction Let me start with a counter-question. What is a number? Before you laugh, there is more to this question than one might think. A number can…
representations

Linear Representations and Why Precision is Important in Math

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First of all: What is a representation? It is the description of a mathematical object like a Lie group or a Lie algebra by its actions on another space…
cubicfunction

Learn How to Solve the Cubic Equation for Dummies

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Everybody learns the "quadratic formula" for solving equations of the form [itex]A x^2 + B x + C = 0[/itex], even though you don't really need such a formula,…
selfstudy

Self-Study Roadmap: Abstract Algebra, Groups to Galois

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Overview Roadmap There are three major areas of mathematics: geometry, analysis, and algebra. This insight gives a roadmap for learning basic abstract…
computers

Ramsey Theory: Foundations, Generalizations, Key Results

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Ramsey theory has its origins in a very nice riddle Consider a party of 6 people. Any two of these six will either be meeting each other for the first…
linearalgebra_selfstudy

Linear Algebra Roadmap: Prereqs, Books & Topics Guide

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Overview This article gives a roadmap for students to learn the basics of linear algebra. Aside from calculus, linear algebra is one of the most applicable…
complexexponents

Understanding Exponents: Real, Irrational & Imaginary

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Introduction This is how both you and I learned exponents back in elementary school: 53 = 5 × 5 × 5 Using repeated multiplication you can raise anything…
Eigenvectors

What Are Eigenvectors and Eigenvalues in Math?

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Two important concepts in Linear Algebra are eigenvectors and eigenvalues for a linear transformation that is represented by a square matrix. Besides…
partialfractions

Introduction to Partial Fractions Decomposition

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Partial fractions decomposition is an algebraic technique that can be used to decompose (break down) a product of rational expressions into a sum…
micro3

Set-Theoretic Foundations of Numbers and Functions

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Set-Theoretic Foundations of Mathematics It is important to realize that in standard mathematics we attempt to characterize everything in terms of sets.…
lineartransformations

Learn About Matrix Representations of Linear Transformations

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Let X and Y be finite-dimensional vector spaces. Let ##T:X\to Y## be a linear transformation. Let ##A=(e_1,\dots,e_n)## and ##B=(f_1,\dots,f_m)## be ordered…

Understanding Zero: History, Division, Exponents, 0!

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The goal of this FAQ is to clarify the concept of 0, and specifically the operations that are allowed with it.The best way to start this FAQ is to…
variables

Some Conceptual Difficulties in the Roles of Variables and Constants

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1. Variables and Constants When is a constant not a constant? When it varies.In the standard equation ##y = a x + b##, we are used to thinking of…
micro1

Hrbacek–Jech 1.2: Formal Language and WFFs Explained

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Hrbacek–Jech Chapter 1.2: Formal Language and WFFs Introduction Hrbacek and Jech do not go into full detail about what a property is formally. This…