MATH 12a Fall 2003
Linear Algebra & Discrete Dynamical Systems, Part I
Professor Gu

 Lecture Monday Tuesday Wednesday Friday week of 9/01 course introduction, vectors in R^2, R^3 and R^n, vector space R^n, thinking in R^n, dot product, angle, length, orthogonality and projection lines and planes, motivational examples for linear system, systems of equations, Gaussian elimination method, Geometric meaning of solution sets Gauss-Jordan reduction, echelon forms, examples, rank of a matrix, describe existence and uniqueness of solution(s) of a linear system in terms of rank, homogenous/non-homog eqns week of 9/08 matrices, identity, special type of matrices, transpose, matrix arithmetic, and matrix properties inverse matrices, Uniqueness, finding inverse matrices, applications to cryptography and graph theory vector space R^n, linear combonations, spanning set, linear independence/dependence subspaces of R^n, def'ns, spans are subspaces, null space, row space, col space, basis week of 9/15 unique rep'n of bases, dimension, bases for row and col space, row rank=col rank, rank+nullity theorem linear transformations, matrix of, equivalent ways to say a linear transformation being one-one or onto determinants and their calculations, cofactor expansions application of determinants, Cramer's rule, volume, invertible matrix theorem week of 9/22 eigenvalues and eigenvectors char eqn, properties finding e-values and e-vectors, lots of examples diagonalization and its applications to real world problems Review & Take-home Midterm handed out week of 9/29 Collect Exams what is a dynamical system, def'n, examples (like logistic map), basic questions graphical analysis, 1-d linear dynamics, show Discrete Tool finding fixed points, attracting/repelling, mean value theorem 1-d nonlinear case: the multiplier theorem, applied to logistic map week of 10/05 periodic points, multiplier theorem for periodic orbits, bifurcation bifurcation diagram, kinds of bifurcations, period-doubling universal behavior, Feigenbaum's constant, def'n of chaos, s.d.i.c., Sarkovskii's theorem week of 10/13 high-d linear dynamics, eigenvalues, eigenvectors, foxes/rabbits more on foxes/rabbits/tribbles, (if time permits, fun example: Merlin) Review for final exam Take-home Final Exam