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Math

18 Jul 20259 min read

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Number Theory

A lot of the substantive number theory material is relevant to or even linked in Cryptography and Cybersecurity

Groups, Fields and Rings

A group is a set with a binary operation that satisfies the following constraints: the operation is associative, it has an identity element, and every element of the set has an inverse element.

every ring is a group, and every field is a ring.

A ring is an abelian group with an additional operation, where the second operation is associative and the distributive property make the two operations “compatible”.

A field is a ring such that the second operation also satisfies all the properties of an abelian group (after throwing out the additive identity), i.e. it has multiplicative inverses, multiplicative identity, and is commutative.

Pieter adds: note that the multiplicative group of a field (obviously) does not contain the zero from the field in question as there is no inverse

Marc van Leeuwen adds: The phrase “i.e.,… is commutative” is somewhat confusing, as it suggests that being commutative is a group property. Commutativity is actually an additional property to the one requiring nonzero elements to form a multiplicative group; without it one has a division ring.

— Source: Answer by BBischof to What are the differences between rings, groups, and fields?

Modular Arithmetic

More Number Theory

Some more random (bonus) number theory concepts

Complex Numbers

Calculus (Single & Multivariable)

Linear Algebra

Gilbert Strang lectures on Linear Algebra (MIT)

Lectures 1-3 from old version:

  1. Lec 1 | MIT 18.06 Linear Algebra, Spring 2005
  2. 2. Elimination with Matrices. (2025-02-25)
  3. 3. Multiplication and Inverse Matrices (2025-02-25)

New:

  1. 1. The Geometry of Linear Equations - watched previously
  2. 2. Elimination with Matrices. - watched (above)
  3. 3. Multiplication and Inverse Matrices - watched (above)
  4. 4. Factorization into A = LU
  5. 5. Transposes, Permutations, Spaces R^n
  6. 6. Column Space and Nullspace
  7. 7. Solving Ax = 0: Pivot Variables, Special Solutions
  8. 8. Solving Ax = b: Row Reduced Form R
  9. 9. Independence, Basis, and Dimension
  10. 10. The Four Fundamental Subspaces
  11. 11. Matrix Spaces; Rank 1; Small World Graphs
  12. 12. Graphs, Networks, Incidence Matrices
  13. 13. Quiz 1 Review
  14. 14. Orthogonal Vectors and Subspaces
  15. 15. Projections onto Subspaces
  16. 16. Projection Matrices and Least Squares
  17. 17. Orthogonal Matrices and Gram-Schmidt
  18. 18. Properties of Determinants
  19. 19. Determinant Formulas and Cofactors
  20. 20. Cramer’s Rule, Inverse Matrix, and Volume
  21. 21. Eigenvalues and Eigenvectors
  22. 22. Diagonalization and Powers of A
  23. 23. Differential Equations and exp(At)
  24. 24. Markov Matrices; Fourier Series
  25. 24b. Quiz 2 Review
  26. 25. Symmetric Matrices and Positive Definiteness
  27. 26. Complex Matrices; Fast Fourier Transform
  28. 27. Positive Definite Matrices and Minima
  29. 28. Similar Matrices and Jordan Form
  30. 29. Singular Value Decomposition
  31. 30. Linear Transformations and Their Matrices
  32. 31. Change of Basis; Image Compression
  33. 32. Quiz 3 Review
  34. 33. Left and Right Inverses; Pseudoinverse
  35. 34. Final Course Review

Additional:

Concepts (Linear Algebra)

Real Analysis

Functional Analysis

See these Textbook and Lecture Notes Recommendations from Perplexity

Fourier

History of Mathematics

Graph View

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