256, Mathematics Final Exam 9:00 a.m.12:00 noon, December 18, 2008
The nal exam will be comprehensive. Approximately 40% of the exam will cover discrete mathematics and 60% will cover linear algebra. Your nal exam score may replace one of the two test scores (if that results in a higher average). You can nd copies of all handouts on the course website. R1. Logic and proofs. There will be no questions specically about 1.11.4, but the language and notation of those sections will be assumed on the exam. There will be proofs like those in 1.6 & 1.7. Prove 2 is irrational. R2. Sets and functions. Again there will be no exam questions specically about 2.12.3. There will be questions regarding cardinality; you are responsible for the topics listed in the handout for September 22 through Cantors Theorem (the diagonal argument). R3. Integers and division. (a) Division of integers. The division algorithm (know what it is) and the Euclidean algorithm (know how to use it to nd gcds). (b) Prime numbers. Fundamental Theorem of Arithmetic (statement only), Euclids Theorem ( primes; statement and proof), the Prime Number Theorem (statement only). Nothing on exam about Fermat primes or Mersenne primes. (c) The Extended Euclidean Algorithm. Know how to use it to nd gcd(a, b) and to write gcd(a, b) as a linear combination of a and b. (d) Modular arithmetic. Solve linear congruences and nd inverse modulo n. Solve systems of linear congruences (Chinese Remainder Theorem). Statement of Fermats Little Theorem and Corollary. (e) Public key cryptography. See pages 1 and 2 of October 6 handout. The algorithms are not covered on nal exam. R4. and Induction recursion. Proofs by mathematical induction and strong induction. Proofs of Fermats Little Theorem and existence part of Fundamental Theorem of Arithmetic. Fibonacci numbers; Lams Theorem and related e example relating fn to (statement and proof of each). S1. Vectors and matrices. Dot products and projection vectors. How to use Gaussian elimination to solve systems of linear equations (pivot variables and free variables). Constraint equations, rank, singular and nonsingular matrices. S2. Matrix algebra. Matrix multiplication and its properties (p. 102), column view and row view. Find the inverse of a matrix. Elementary matrices and Gaussian elimination viewed as matrix factorization. S3. Vector spaces. Denition of subspace. Cartesian description and parametric description; using Gaussian elimination to change description. Row space, column space, null space. Orthogonal complement. Theorems 1.4 and 1.5 (pp. 148, 149). Linear dependence and independence, basi...