Editing Maths (section)

=Algebraic Structures=
==Properties==

* '''associative''' - 𝒶(𝒷𝒸) = (𝒶𝒷)𝒸. '''∀𝓍,𝓎,𝓏. 𝓍∘(𝓎∘𝓏)=(𝓍∘𝓎)∘𝓏'''
* '''commutative''' - 𝒶𝒷 = 𝒶𝒷.  '''∀𝓍,𝓎. 𝓍∘𝓎=𝓎∘𝓍'''
* '''Identity element''': There exists an element ℯ such that for each element 𝓍, ℯ∘𝓍 = 𝓍 = 𝓍∘ℯ ;   formally: '''∃ℯ ∀𝓍. ℯ∘𝓍=𝓍=𝓍∘ℯ'''. The identity element of multiplication is 1 as 1×𝓍=𝓍=𝓍×1. The identity of addition is 0 as 0+𝓍=𝓍=𝓍+0.
* '''Inverse element''': It can easily be seen that the identity element is unique. If this unique identity element is denoted by ℯ then for each 𝓍, there exists an element 𝒾 such that 𝓍∘𝒾=ℯ=𝒾∘𝓍;   formally: '''∀𝓍 ∃𝒾. 𝓍∘𝒾=ℯ=𝒾∘𝓍'''. The multiplicative inverse: '''𝒶𝒶⁻¹=1=𝒶⁻¹𝒶'''


* '''domain''' - The set of elements on which a function has a valid definition is it's domain. For '''𝒻: X→Y''', X is the domain.
* '''codomain / target''' - The possible set that a function outputs. For '''𝒻: X→Y''', X is the target. There may be things in the target that aren't actually reachable by the function.
* '''image / range''' - The set of elements that a function maps to and no more. This is like the '''target''' but only the possible elements are in the set. For example the '''cos''' function could be defined as having a target of the real numbers, but the range could only be between -1 and 1. It is not always possible to define the range. Concepts of Modern Mathematics (pg. ~67). Some ambiguity as sometimes 'range' refers to the codomain.


* '''homomorphism''' - A category of function. Preserves the structure. '''𝒻: X→Y'''. '''𝒻(a)𝒻(b)=𝒻(c)''' and '''𝒶𝒷=𝒸'''. A homomorphism of vector spaces is referred to as a '''linear map'''.
* '''injection''' - Each target element must be reachable from only one element in the domain. It does not need to be '''surjective''', meaning that it's possible to have elements in the target that aren't reachable at all. But you can not have an element in the domain that is reachable from more than one target element. cos is not an injection because an infinite number of inputs get mapped to values between -1 and 1.
* '''surjection (onto)''' - If every element of a target set T is reachable by a function, that function is a surjection '''onto''' T. This means the target is also the range. Does not need to be an injection. - Concepts of Modern Mathematics (pg. 70).
* '''bijection''' - A bijection is invertible. It is an '''injection''' and a '''surjection'''. A function which relates each member of a set S (the domain) to a separate and distinct member of another set T (the range), where each member in T also has a corresponding member in S. A mapping that is both one-to-one (an injection) and onto (a surjection).
* '''isomorphism'''  - A '''homomorphism''' that is also a '''bijection'''. An operation-preserving bijection. A homomorphism with an inverse that is also a homomorphism. '''𝒻: X→Y, ℊ: Y→X'''
* '''automorphism''' - Galios Groups related. An isomorphism from a structure to itself.
* '''Abelion''' - An abelian group/operation is '''commutative'''.

==Structures==
==='''Set'''===
* A collection of elements.

==='''Group'''===
An algebraic structure that is a set with a single binary operation.

An '''infinite group''' has an unlimited number of elements. (ℤ, +)

A '''finite groups''' has a finite number of elements. (ℤₘ, +) - The integers mod m.

The order of an element is the number of elements in the subgroup it generates. |2| = 3 

* '''(𝔾, ∗)''' - Group '''𝔾''' with operation '''∗'''.

* Axioms
** The operation must be '''closed'''. - '''a,b ∈ 𝔾 → a∗b ∈ 𝔾'''
** The operation must be '''associative'''. - '''(a∗b)∗c = a∗(b∗c)'''
** Operation must have an '''identity''' element that has no effect on any other element under operation. '''∃e(a∗e = e∗a = a)'''.
** Every element must have an '''inverse'''. An element when combined with the original will return the identity. '''∀a∃a⁻¹(a∗a⁻¹=a⁻¹∗a=e)'''

* '''(ℤ, +)''' - The integers under addition.
** Closure: '''a,b ∈ ℤ → a+b ∈ ℤ'''
** Associativity: '''(a+b)+c = a+(b+c)'''
** Identity: '''a+0 = 0+a = a'''
** Inverse: '''a⁻¹ = -a'''

* '''(ℤ, ×)''' - The integers under multiplication it '''not a group'''
** Closure: '''a,b ∈ ℤ → a×b ∈ ℤ'''
** Associativity: '''(a×b)×c = a×(b×c)'''
** Identity: '''a×1 = 1×a = a'''
** '''NO''' Inverse: '''2x=1 has no solution in ℤ'''

* If the operation is also '''commutative''' then it is an '''Abelian''' group.
* [https://www.youtube.com/watch?v=WwndchnEDS4 Bill Shillito].

==='''Ring'''===
Similar to a field but multiplication doesn't require an inverse.

'''ℤ, ℚ, ℝ, ℂ, ℤₘ''' are all '''unital commutative rings.'''

'''𝕄₂(ℝ)''' the set of all '''2×2 real matrices''' is a non-commutative '''ring''' as matrix multiplication is not commutative.

* '''+''' must be '''abelian'''.
* '''×''' must be '''closed''' and '''associative'''.
* [https://www.youtube.com/watch?v=syHBApgJhsA Bill Shillito - Rings and Fields]

* '''(R, +, ∗)'''
** '''(R, +)''' is an '''abelian group''' - '''addition'''
*** '''Closed'''
*** '''Associative'''
*** Identity - '''Additive identity'''
*** Inverse - '''Additive inverse −a'''
*** '''Commutative'''
** '''(R, ∗)'''
*** Closed under '''×'''
*** Operation '''×''' must be '''associative'''.
** '''+''' and '''×''' must be linked by the '''distributive property'''. Multiplication distributes over addition.
*** a×(b+c) = a×b+a×c
*** (a+b)×c = a×c+b×c
**** 0×a = a×0 = 0 - Zero times anything is zero
**** a×−b = −a×b = −(a×b) - A positive times a negative is a negative.
**** −a×−b = a×b - A negative times a negative is a positive.

* '''×''' '''might''' not be '''commutative'''.  If it is then it is a '''commutative ring''' (not called abelian which is only for groups).
* If the ring has a '''multiplicative identity''' then it is a '''unital ring'''. For '''ℤ, ℚ, ℝ, ℂ, ℤₘ''' the '''multiplicative identity''' is 1. For '''𝕄ₘ(ℝ)''' it is the '''identity matrix'''.
* An element of a unital ring that has a multiplicative inverse is called a '''unit'''. Not every element is necessarily a unit. In ℤ only 1 and −1.
* If '''every element other than 0 is a unit'''. It is a '''division ring'''. ℚ is a division ring.
* A '''zero divisor''' of a ring is a nonzero element that can be multiplied by some other nonzero element to produce 0. In ℤ₆, 2, 3 and 4 are zero divisors. 2×3=0
* An element '''can not''' be a '''unit''' and a '''zero divisor'''.
* An '''integral domain''' is a '''commutative, unital ring''' with '''no zero divisors'''. '''ℤ, ℚ, ℝ, ℂ''' are '''integral domains'''.
** ℤₘ is not necessarily an integral domain. '''ℤₚ''' is an integral domain '''if p is a prime number'''.
** In an integral domain you can '''cancel factors''' from both sides of an equation. '''<strike>2</strike>×x=<strike>2</strike>×3'''

====Polynomial Ring====
A '''monomial''' is the product of a number and a non-negative integer power of a variable. eg: 2x³

A '''polynomial''' is a finite sum of monomials. a₀+a₁x+a₂x²+a₃x³+…+aₙxⁿ

n is the degree of the polynomial. It's highest power.

Some polynomial rings: ℤ[𝓍], ℚ[𝓍], ℝ[𝓍], ℂ[𝓍], ℤₘ[𝓍]

In '''ℝ[x]''' we can take congruents to the modulus '''𝓍²+1'''. Two polynomials are congruent if their difference is divisible by '''𝓍²+1'''. This makes the polynomial ring behave like complex numbers. When using a ''''prime' polynomial''' as modulus, the '''polynomial ring''' is a '''field'''.

'''𝓍³+𝓍²-2𝓍+3 ≡ -3𝓍+2   (mod 𝓍²+1)'''

Mentioned in a concrete guide to modern mathematics, page 90.

==='''Field'''===
Like a ring with stricter multiplication axioms.

A '''field''' is a division ring where '''×''' is '''commutative, unital''' and has '''no zero divisors''' and '''every nonzero element is a unit'''.

A field forms an abelian group under both addition and multiplication.

* A set with 2 binary operations (+, ×)
* Both must be '''commutative''' and '''associative'''
* Operations must have '''identity elements''' ('''+''' is '''0''' and '''×''' is '''1''')
* Every element must have an '''inverse''' (x = -x, x = 1/x)
* Must follow the '''distributive law'''.
* Must be '''closed under addition, multiplication, taking of inverses''' (Results must in in the same set).

'''ℚ, ℝ, ℂ''' are '''fields'''.

'''ℤₚ''' is a field '''if p is prime'''.

'''ℤ''' is '''not a field'''.

'''ℝ''' is the '''completion''' of ℚ. It allows for '''calculus'''.

'''ℂ''' is '''algebraically closed'''. Every polynomial equation in ℂ[x] has solutions in ℂ.

But '''ℂ''' is '''not ordered'''.

==='''Vector Space'''===
* Like a 2D or 3D plane. Can be built from 'unit vectors' or '[https://www.youtube.com/watch?v=P2LTAUO1TdA basis]'. for example (1,0), (0,1).
* There was a Youtube on the topic (Blue2Brown1).
* '''Scalar''' - A number used to multiply a vector. Infinite dimensional vector spaces exist such as when vectors are functions.
* ℝⁿ - A vector space of n-dimensions over the field ℝ. ℝ is the scalar type. ℝ², ℝ³, etc...

==='''Common Groups'''===
* '''Aₙ - Alternating Group''' - The group of even permutations of a finite set. A₄ is symmetries of a Tetrahedron. - "Fearless Symmetry"
* '''Cₙ - Cyclic Group''' - 
* '''Dₙ - Dihedral Groups''' - A '''dihedral group''' is the group of '''symmetries''' of a regular polygon, which includes '''rotations and reflections'''. '''D₃''' is the '''dihedral group''' of the '''triangle'''. '''D₁≅Z₂''', '''D₂≅K₄'''.
* '''E⁺(n)''' -  The '''direct isometries''', i.e., isometries '''preserving orientation''', also called '''rigid motions'''; they are the moves of a rigid body in n-dimensional space. These include the translations, and the rotations, which together generate '''E⁺(n)'''. Also called a '''special Euclidean group''', and denoted '''SE(n)'''.
* '''T - Translation Group''' - T is a normal subgroup of E(n). T ◅ E(n).
* '''Sₙ - Symmetry Groups''' - '''S₃''' the '''symmetric group''' of 3 elements. '''S₃''' is '''isomorphic''' to '''D₃'''. ('''S₃ ≅ D₃''')
* '''O(n) - Orthogonal group''' - Reflections & rotations. The group of distance-preserving transformations of a Euclidean space of dimension n that preserve a fixed point, where the group operation is given by composing transformations ('''∘'''). It is equivalent to the group of n×n orthogonal matrices, where the group operation is given by matrix multiplication; an orthogonal matrix is a real matrix whose inverse equals its transpose. O(n, ℝ) is a subgroup of E(n), contains those that leave the origin fixed.
* '''SO(n) - Special Orthogonal''' - Rotation group. '''SO(2)''' is the [https://en.wikipedia.org/wiki/Circle_group circle group]. '''SO(3)''' is the group of all rotations about the origin of three-dimensional Euclidean space '''ℝ³''' under the operation of composition ('''∘'''). eg '''Roll, Pitch, Yaw'''.   '''Non-commutative'''. '''SO(n) ≤ O(n)'''.
* '''POₙ(ℝ)''' - '''Projective orthogonal group'''. '''PSO(V)''' for '''projective special orthogonal group'''.
* '''GLₙ(𝔽) - General Linear''' - '''Linear''' means '''matrix'''. The group of all n×n matrices with non-zero determinants under matrix multiplication. '''nonabelian''' group because matrix operations are non-commutative. The set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. The group '''GLₙ(ℝ)''' over the field of real numbers is a real '''Lie group''' of dimension '''n²''' as the set of all n×n real matrices, '''Mₙ(ℝ)''', forms a real vector space of dimension '''n²'''.
* '''Special Linear: SL(n)''' - Is the subgroup of '''GL(n, 𝔽)''' consisting of '''matrices with a determinant of 1'''. These form a group because the product of two matrices with determinant 1 again has determinant 1.
* '''ΓLₙ(ℝ)''' - General semilinear group. Contains '''GL'''.
* '''Sp(n)''' - The '''compact symplectic''' group '''Sp(n)''' is often written as '''USp(2n)''', indicating the fact that it is '''isomorphic''' to the '''group of unitary symplectic matrices'''.
* '''Sp(2n, 𝔽)''' - The '''symplectic group''' of degree '''2n''' over a field '''𝔽''' is the group of '''2n × 2n symplectic matrices''' with entries in '''𝔽''', and with the group operation that of matrix multiplication. Since all symplectic matrices have determinant 1, the symplectic group is a subgroup of the special linear group '''SL(2n, 𝔽)'''.
* '''Uₙ(ℝ) - The unitary group''' - A group preserving a sesquilinear form on a module. There is a subgroup, the special unitary group '''SUₙ(ℝ)'''.
* <math>\Sigma_A</math> - The permutation group of set A.
* ℚᵃˡᵍ - The solutions of polynomials. Is a '''field'''. Its Elements are called "'''algebraic numbers'''". It is a subset of ℂ. - Fearless Symmetry p48


* [https://en.wikipedia.org/wiki/Classical_group Wikipedia: Classical group]
* [https://en.wikipedia.org/wiki/List_of_finite_simple_groups Wikipedia: List of finite simple groups]
* [https://en.wikipedia.org/wiki/List_of_simple_Lie_groups Wikipedia: List of simple Lie groups]

==='''Common Lie Algebras'''===
* '''𝔰𝔬(3)''' - The '''Lie algebra''' of '''SO(n)''' and would be the '''rate of change''' for Roll, Pitch and Yaw. '''𝔰𝔬(n)''' is equal to '''𝔬(n)''' - Princeton Mathematics (p234)
* '''𝔤𝔩ₙ(ℂ)''' is the '''Lie algebra''' for '''GLₙ(ℂ)''', the space of all '''n×n complex matrices'''.
* '''𝔰𝔩ₙ(ℂ)''' - The '''Lie algebra''' of the special linear group '''SLₙ(n)'''. "Is the subspace of all matrices with '''trace zero'''" - Princeton Mathematics (p234)

==='''Cayley table'''===
A table showing all the results of all possible operations on a finite group.

==='''Eignvectors'''===
* '''Eigenvector''' - A vector in a vector space doesn't get rotated by a linear transformation. It stays on it's 'span'. It's the axis of rotation.
* '''Eigenvalue''' - The value an Eigenvector is scaled by. 𝛢𝓋⃗=λ𝓋⃗. 𝛢 is transformation matrix. 𝓋⃗ is an Eigenvector. λ is the eigenvalue.


* [https://www.youtube.com/watch?v=8F0gdO643Tc Khutoryansky]
* [https://www.youtube.com/watch?v=ue3yoeZvt8E LeiosOS]
* [https://www.youtube.com/watch?v=PFDu9oVAE-g 3Blue1Brown]
* [https://www.youtube.com/watch?v=RW-Seu-yenQ MathTheBeautiful: Linear Algebra 15n: Why Eigenvalues and Eigenvectors Are So Important!]
* [https://www.youtube.com/watch?v=bOreOaAjDno Eigenvalues in under 6 minutes]- meh