## Math 211

### List of Overheads

List of all handouts     For further reading
01. Algebraic Methods (2 pages)
02. Number Systems
03. Divisibility
03a. The Principle of Induction
04. The Euclidean Algorithm
05. The greatest common divisor
06. The division algorithm
07. The Euclidean Algorithm: 1st/2nd Version (2 pages)
08. The Euclidean Algorithm: 2nd Version
09. The extended Euclidean Algorithm - Examples (2pp)
10. The extended Euclidean Algorithm
11. Diophantine equations (2 pages)
11a. The Plimpton 322 Clay Tablet
12. The GCD-criterion and its consequences
13. The general solution of the diophantine equation mx+ny=c
14. How to solve mx+ny=c
15. Proof of the Formula (2pp.)
16. How to solve mx+ny+kz=c
16a. Prime numbers
17. Unsolved conjectures about primes
18. The Fundamental Theorem of Arithmetic
19. The GCD-formula
20. The GCD-formula vs. Euclidean algorithm
21. The Calculus of Remainders
22. Computing a^n efficiently
23. The Cancellation Law
24. Solving the congruence ax=b (mod m)
25. The Ring Z/mZ and the Field F_p
26. The Wheels Problem
27. The Chinese Remainder Theorem
28. Fermat's Theorem
29. Mersenne numbers
29a. The Binomial Theorem (2pp.)
30. Public Key Cryptography (2pp.)
31. The dancing men
32. The RSA method (2pp.)
32a. The RSA-155 Challenge (3pp.)
32b. The History of Algebra
33. Complex Numbers (History)
33a. Complex Numbers (3pp.)
34. Arctan and Argument
35. Solutions of z^n=a
36. Solutions of z^6=1+sqrt(-3)
36a. The sixth roots of a = 1+i
37. Polynomials
37a. The Degree of a Polynomial
38. The Division Algorithm (for polynomials)
39. The Remainder Theorem (2pp.)
40. The Euclidean Algorithm (for polynomials)
41. The GCD-criterion for polynomials
42. Irreducible polynomials
43. The Quadratic Formula
43a. Irreducible Quadratic Polynomials over Fp for p le 5
44. Unique Factorization for polynomials
45. The Multiplicity of a root
46. Rules for factoring over Q (3pp.)
47. The Method of Comparing Coefficients (2pp.) (not presented)
48. The Fundamental Theorem of Algebra (2pp.)
49. The Factorization Theorem in R[x]
49a. Solutions of Polynomial Equations (2pp.)
49b. Term 1 Review

2nd Term:

50. Review/Preview (2pp)
51. The Lagrange Interpolation Formula (2pp)
52. The Lagrange Interpolation Polynomial (matrix method)
53. The Least Square Method
54. The Geometry of R^n (2pp.)
55. Two Distance Problems
56. Orthogonal Projection (graph)
57. Solution of the Two Distance Problems
58. Orthogonal Projection Examples
58a. Review of Linear Independence (3pp.)
59. Orthogonal Vectors and Orthogonal Projection
59a. The Gram-Schmidt Orthogonalization Procedure
60. Orthogonal Matrices
61. Fourier Approximation (not on exam)
62. The Fourier Aproximation of f(x) = 1 - x/Pi (graph)
63. The Rabbit Problem
64. Matrix Polynomials
64a. Review of Dimensions of Linear Sets
65. Review of Diagonalization
66. Diagonalization Theorems (2pp.)
67. Evaluating Matrix Polynomials - Method I (Diag. Case)
68. Jordan Blocks
69. The Jordan Canonical Form
70. Evaluating Matrix Polynomials - Method I
71. Evaluating Matrix Polynomials - Method II
72. The Generalized Remainder Formula
72a. Finding rem(f,g): an example
73. The Lagrange-Sylvester Interpolation Formula (not on exam)
74. Evaluating Matrix Polynomials - Method III
75. Comparison of the Three Methods
76. Discrete Linear Systems
77. The Cost of Breeding Rabbits
78. Fibonacci Numbers
79. Phyllotaxis
80. Difference Equations as Discrete Linear Systems
81. The Golden Section
82. Markov Chains as Discrete Linear Systems
83. A Rat Maze (3 pp.)
84. Solving Discrete Linear Systems (for info only)
85. Equilibrium Points (for info only)
86. Algebraic and Geometric Multiplicities
86a. The Direct Sum of Subspaces
87. The Invariance Property
88. The Jordan Canonical Form
89. The Jordan Canonical Form: Examples
90. Generalized Eigenvectors (5pp.)
91. Powers of Complex Numbers (2pp.)
91a. Limits of Matrix Sequences
92. Power Convergent Matrices (Special Cases)
93. Power Convergent Matrices
93a. Properties of Constituent Matrices (2pp.)
94. Simple Eigenvalues
97. Stochastic Matrices and Markov Chains
98. Primitive Stochastic Matrices
99. Shipping of Commodities II (2pp.)
100. A Rat Maze II (2pp.)