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* '''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 refered to as a linear map.
* '''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 refered to as a linear map.
* '''bijection''' - (isomorphisms are bijections?)
* '''bijection''' - A bijection is invertable.
* '''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
* '''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.
* '''automorphism''' - Galios Groups related. An isomorphism from a structure to itself.

Revision as of 02:51, 18 February 2017

Misc

Unicode

  • Subscript: ₀₁₂₃₄₅₆₇₈₉₊₋₌₍₎ₐₑₒₓₔₕₖₗₘₙₚₛₜ
  • Superscript: ⁰¹²³⁴⁵⁶⁷⁸⁹⁺⁻⁼⁽⁾ᵃᵇᶜᵈᵉᶠᵍʰⁱʲᵏˡᵐⁿᵒᵖʳˢᵗᵘᵛʷˣʸᶻ
  • wikipedia

Glossary

Number Systems

  • ℕ - The set of all natural numbers. No zero. No negatives. Subtraction and division aren't always possible. ℕ = {1, 2, 3, ...}.
  • ℤ - The set of all integers. Subtraction is always possible. ℤ = {..., -3, -2, -1, 0, 1, 2, 3, ...}
  • ℚ - The set of all rational numbers (Quotient). Fractions. Division is always possible. ℚ is a field.
  • ℝ - The set of all real numbers. Includes irrational and transcendental numbers. √(2), π, e, φ. ℝ is an extension of ℚ to a larger field.
  • ℝ⁺ - The set of all positive reals.
  • ℂ - The set of all complex numbers. i²=-1. Also a field.
  • 𝔽ₚ - Set of integers modulo a prime p. Addition and multiplication also defined modulo p.
  • ℚ(γ) - γ is the root of the polynomial x³-x-1. When γ³ appears it is replaced with γ+1. See: 2.2 in princeton companion of mathematics. AUTOMORPHISM

Algebraic Structures

Properties

  • associative - a(bc) = (ab)c
  • commutative - ab = ba
  • 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 refered to as a linear map.
  • bijection - A bijection is invertable.
  • 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.

Structures

  • Set - A collection of elements.
  • Group - A set with a binary operation. The operation must be associative. If the operation is also commutative then it is an Abelian group. [TODO]: Get Axioms.
  • Field - A set with 2 binary operations (+, ×). Both must be commutative and associative. And have an 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).
  • Vector Space - Like a 2D or 3D plane. Can be built from 'unit vectors' or 'basis'. for example (1,0), (0,1). There was a Youtube on the topic. ℝ², ℝ³, ℝⁿ a vector space of n-dimensions over the field ℝ (ℝ is the scalar type).
  • Scalar - A number used to multiply a vector. Infinite dimensional vector spaces exist such as when vectors are functions.
  • Ring - Similar to a field but multiplication doesn't require an inverse. Multiplication might not be commutative. + must be abelian. × must be closed and associative. Bill Shillito - Rings and Fields

Groups

  • Dihedral Groups: Dₙ - D₃ is the dihedral group of the triangle.
  • Symmetry Groups: Sₙ - S₃ the symmetric group of 3 elements.
  • Special Orthogonal: SO(n) - special orthogonal group (rotation group?).
  • General Linear: GL(n)

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. v is Eigenvector. λ is the eigenvalue.


Analysis

Calculus

Gradients

Videos

MOOCs


YouTube

Unsorted

HOW TO LEARN ADVANCED MATHEMATICS WITHOUT HEADING TO UNIVERSITY - PART 1 PART 3

4chan - Math Textbook Recommendations

Harvard Course of Abstract Algebra (apparently goes well with the Artin book)

List of books


Order of abstract algebra


This guide on algebra

This guide to imaginary numbers

Math intuition

These guides


These notes are recommended

HOW TO LEARN MATH: FOR STUDENTS

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