Math 211

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Section   Title
Chapter 0: Introduction
What is Algebra?
Al-Khwarizmi 780-850 (St. Andrew's U.)
Arabic Numerals (St. Andrew's U.)
Indian Numerals (St. Andrew's U.)
Number Systems
Leopold Kronecker 1823-1891 (St. Andrew's U.)

Chapter 1: The Integers
1.1 Introduction
1.2 Divisibility
1.3 The Euclidean Algorithm, First Version
Euclid 325-265 B.C. (St. Andrew's U.)
1.4 The Greatest Common Divisor
1.5 The Division Algorithm
1.6 The Euclidean Algorithm, Second Version
1.7 The Extended Euclidean Algorithm
Indian Mathematics (Overview) (St. Andrew's U.)
1.8 Linear Diophantine Equations
Diophantus 200-284 (St. Andrew's U.)
Pierre Fermat 1601-1665 (St. Andrew's U.)
Plimpton 322 Tablet (St. Andrew's U.)
Fermat's Last Theorem (St. Andrew's U.)
Fermat's Last Theorem (Queen's U.)
1.9 Unique Factorization
Eratosthenes of Cyrene 276-194 B.C. (St. Andrew's U.)
Prime Numbers (St. Andrew's U.)

Chapter 2: Modular Arithmetic
2.1 Introduction
2.2 The Calculus of Remainders
Carl Friederich Gauss 1777-1855 (St. Andrew's U.)
A computer game based on the "Casting-out Nines" rule
2.3 The Cancellation Law
2.4 Solving Congruence Equations
2.5 The Ring Z/mZ and the Field F_p
2.6 The Chinese Remainder Theorem
Indian Mathematics (Overview) (St. Andrew's U.)
2.7 Fermat's Theorem
Pierre Fermat 1601-1665 (St. Andrew's U.)
Father Marin Mersenne 1588-1641 (St. Andrew's U.)
Prime Numbers (and Fermat's Theorem) (St. Andrew's U.)
Blaise Pascal 1623-1665 (St. Andrew's U.)
2.8 Public Key Cryptography
Introduction to Cryptography (PGP)
RSA on the internet

Chapter 3: Polynomials
3.1 Introduction
Scipione del Ferro 1465-1526 (St. Andrew's U.)
Nicolo Tartaglia 1499-1557 (St. Andrew's U.)
Girolamo Cardano 1501-1576 (St. Andrew's U.)
Ludovico Ferrari 1522-1565 (St. Andrew's U.)
Paolo Ruffini 1765-1822 (St. Andrew's U.)
Abel 1802-1829 (St. Andrew's U.)
Evariste Galois 1811-1832 (St. Andrew's U.)
The Fundamental Theorem of Algebra (St. Andrew's U.)
Quadratic, Cubic and Quartic Equations (St. Andrew's U.)
3.2 Review of Complex Numbers
Rafael Bombelli 1526-1572 (St. Andrew's U.)
Jacob Benoulli 1654-1705 (St. Andrew's U.)
Albert Girard 1595-1632 (St. Andrew's U.)
Rene Descartes 1596-1650 (St. Andrew's U.)
Abraham De Moivre 1667-1754 (St. Andrew's U.)
Leonard Euler 1707-1783 (St. Andrew's U.)
Caspar Wessel 1745-1818 (St. Andrew's U.)
Jean Robert Argand 1768-1822 (St. Andrew's U.)
Gauss 1777-1855 (St. Andrew's U.)
3.3 Polynomials: First Properties
3.4 The Division Algorithm
3.5 The Remainder Theorem
Rene Descartes 1596-1650 (St. Andrew's U.)
Jean Le Ronde Alembert 1717-1783 (St. Andrew's U.)
3.6 The Euclidean Algorithm
3.7 Unique Factorization
3.8 Factoring Methods
Abel 1802-1829 (St. Andrew's U.)
Evariste Galois 1811-1832 (St. Andrew's U.)
The Fundamental Theorem of Algebra (St. Andrew's U.)
Quadratic, Cubic and Quartic Equations (St. Andrew's U.)

Chapter 4: Interpolation and Approximation and the Geometry of R^n
4.1 Introduction
4.2 The Lagrange Interpolation Polynomial
Jean Louis Lagrange 1736-1813 (St. Andrew's U.)
Alexandre-Theophile Vandermonde 1735-1796 (St. Andrew's U.)
4.3 The Least Square Method
Adrien-Marie Legendre 1752-1833 (St. Andrew's U.)
4.4 The Geometry of R^n
Augustin Louis Cauchy 1759-1857 (St. Andrew's U.)
Hermann Amandus Schwarz 1843-1921 (St. Andrew's U.)
Pythagoras 1736-1813 (St. Andrew's U.)
4.5 A Distance Problem
4.6 Orthogonal vectors and the Gram-Schmidt Procedure
Jorgen Pedersen Gram 1850-1916 (St. Andrew's U.)
Erhardt Schmidt 1876-1959 (St. Andrew's U.)
4.7 Function Spaces and Fourier Approximation
Jean Baptiste Joseph Fourier 1768-1830 (St. Andrew's U.)

Chapter 5: Matrix Polynomials and Discrete Linear Systems
5.1 Introduction
5.2 Matrix Polynomials - First Properties
5.3 Evaluating Matrix Polynomials - Method I (Diagonable Case)
5.4 Review of Diagonalization
5.5 Evaluating Matrix Polynomials - Method I (Non-diagonable Case)
Camille Jordan 1838-1922 (St. Andrew's U.)
5.6 Evaluating Matrix Polynomials - Method II
Arthur Cayley 1821-1895 (St. Andrew's U.)
William Rowan Hamilton 1805-1865 (St. Andrew's U.)
Adolf Hurwitz 1859-1919 (St. Andrew's U.)
5.7 The Generalized Remainder Formula
James Joseph Sylvester 1814-1897 (St. Andrew's U.)
5.8 Evaluating Matrix Polynomials - Method III
5.9 Application to Discrete Linear Systems
Leonardo da Pisa (Fibonacci) 1170?-1250? (St. Andrew's U.)
Leonardo da Vinci 1452-1519 (St. Andrew's U.)
Fra Luca Pacioli 1445-1519 (St. Andrew's U.)
Andrei Andeyevich Markov 1859-1922 (St. Andrew's U.)

Chapter 6: The Jordan Canonical Form
6.1 Introduction
6.2 Algebraic and Geometric Multiplicities of Eigenvalues
6.3 How to find P (for m < 4)
6.4 Generalized eigenvectors and the JCF
6.5 A procedure for finding P
6.6 A Proof of the Cayley-Hamilton Theorem
6.7 Appendix: Proof of the Jordan Canonical Form

Chapter 7: Powers of Matrices
7.1 Introduction
7.2 Powers of Numbers
7.3 Sequences of powers of matrices
7.4 Finding lim A^n (when A is power convergent)
7.5 The spectral radius and Gersgorin's theorem
7.6 Geometric series
7.7 Stochastic matrices and Markov chains
Andrei Andeyevich Markov 1859-1922 (St. Andrew's U.)
Oskar Perron 1880-1975 (St. Andrew's U.)
7.8 Application: Finding approximate eigenvalues of a matrix




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