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 Algebras Explained: Commutators, SU(2), SO(3), Euc(n)

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A Lie algebra is the vector space of infinitesimal generators attached to a Lie group at its identity element, equipped with a bilinear antisymmetric commutator…
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 Guide: How to Learn Abstract Algebra Step by Step

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Self-studying abstract algebra requires only precalculus-level math and basic familiarity with mathematical proofs. A practical path runs from proof-writing…
computers

Ramsey Theory Explained: The Party Riddle and R(m,n)

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Ramsey's theorem guarantees that in any sufficiently large gathering, a specific pattern of mutual relationships must appear, no matter how the relationships…
linearalgebra_selfstudy

Learn Linear Algebra: Best Textbooks and Study Roadmap

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What Is the Best Way to Learn Linear Algebra? The most effective path through linear algebra emphasizes vector spaces and linear transformations rather…
complexexponents

Exponents Explained: Irrational and Imaginary Powers

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What Is the General Definition of Exponentiation? Exponentiation is defined for all real and imaginary exponents using the formula xy = ey ln(x), where…
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

Why Mathematics Defines Numbers and Functions as Sets

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Direct answer: Standard mathematics defines numbers, functions, and other objects as sets because the 19th-century collapse of certainty in Euclidean geometry…
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…

The Concept of Zero: History, Rules, and Why 0/0 Fails

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Zero is neither positive nor negative, is defined as the additive identity so that x + 0 = x for every number x, and produces 0 when multiplied by any…
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

Well-Formed Formulas in Set Theory: WFFs Explained Clearly

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What Is a Well-Formed Formula in Set Theory? A well-formed formula (wff) in set theory is a string of symbols built strictly from the alphabet of set…