Fall 2003

# Mathematics Math21b Fall 2003

## Linear Algebra and Differential Equations

 ```- Math21b: Linear Algebra and Differential Equations This is an introduction to linear algebra, including linear transformations, determinants, eigenvectors, eigenvalues, inner products and linear spaces. As applications, the course introduces discrete dynamical systems, differential equations, Fourier series as well as some partial differential equations. This course is tought in 2 sections. - Instructors: Oliver Knill, SC-434, knill@math Izzett Coskun, SC 333b, coskun@math - Course assistants: Minhua Zhang, zhang18@fas Phillip Powell, ppowell@fas Jeff Amlin, amlin@fas - Lectures: Mo-We-Fr 10-11 109 Mo-We-Fr 11-12 309 - Problem Sections: Thursday 6:30-8:00 PM 101B Wednesday 3:00-4:30 PM 109 Sunday 7:00-8:30 PM 111 - Office hours: Oliver: Tuesday 11-12 and Thursday 11-12 Izzet: Tuesday 1-2 PM and Wednesday 3-4 PM - Website: http://www.courses.fas.harvard.edu/~math21b/ - Text: Otto Bretscher, Linear Algebra with Applications, second edition. Prentice-Hall, Upper Saddle River, NJ, 2001. - About this course: - teaches methods to solve systems of linear equations Ax = b, - allows you to analyze and solve systems of linear differential equations, - solve discrete linear dynamical systems. An example are Markov processes, - learn to do least square fit with arbitrary function sets and also know why it works, - you will learn the basics of Fourier series and how to use it to solve linear partial differential equations, - prepares you for the further study in other fields of mathematics and its applications, like for example quantum mechanics, combinatorics, - improves thinking skills, problem solving skills, algorithmic and the ability to use more abstract tools. - Homework: HW will be assigned in each class and is due the next lecture. - Exams: Two midterm exams and one final exam. - Grades: First and second hourly 20 % each Homework 20 % Final exam 40 % - Calendar: (12 weeks of class) Su Mo Tu We Th Fr Sa Week -------------------------------------------------------- 14 15 16 17 18 19 20 16. Sep: Orientation +------+----+ 21 22 23 24 25 26 27 1 22. Sep: First day of class 28 29 30 1 2 3 4 2 5 6 7 8 9 10 11 3 October 12 13 14 15 16 17 18 4 13. Oct: Columbus day 19 20 21 22 23 24 25 5 Oct 22: 1. Midterm 7:30-9 PM 26 27 28 29 30 31 1 6 2 3 4 5 6 7 8 7 November 9 10 11 12 13 14 15 8 11. Nov Veterans day 16 17 18 19 20 21 22 9 Nov 19: 2. Midterm 7:30-9 PM 23 24 25 26 27 28 29 10 28. Nov Thanksgiving 30 1 2 3 4 5 6 11 December 7 8 9 10 11 12 13 12 14 15 16 17 18 19 20 13 17. Dec -3. Jan. Recess +------+----+ 21 22 23 24 25 26 27 28 29 30 31 1 2 3 4 5 6 7 8 9 10 5. Jan -16. Jan Reading 11 12 13 14 15 16 17 18 19 20 21 22 23 24 17. Jan -27. Jan Exams 25 26 27 28 29 30 31 --------------------------------------------------------- - Day to day syllabus: Lecture Date Book Topic 1. Week: Systems of linear equations Lect 1 9/22 1.1 introduction to linear systems Lect 2 9/24 1.2 matrices and Gauss-Jordan elimination Lect 3 9/26 1.3 on solutions of linear systems 2. Week: Linear transformations Lect 4 9/29 2.1 linear transformations and their inverses Lect 5 10/1 2.2 linear transformations in geometry Lect 6 10/3 2.3 inverse of a linear transformation 3. Week: Linear subspaces Lect 7 10/6 2.4 matrix products Lect 8 10/8 3.1 image and kernel Lect 9 10/10 3.2 subspaces, bases and linear independence 4. Week: Dimension 10/13 COLUMBUS DAY, no class Lect 10 10/15 3.3 dimension Lect 11 10/17 3.4 coordinates 5. Week: Orthogonality Lect 12 10/20 5.1 orthonormal bases and orthogonal projections Lect 13 10/22 *** Review for first midterm Midterm Lect 14 10/24 5.2 Gram-Schmidt and QR factorization 6. Week: Determinants Lect 15 10/27 5.3 orthogonal transformations Lect 16 10/29 5.4 least squares and data fitting Lect 17 10/31 6.2 determinants I 7. Week: Eigensystems Lect 18 11/3 6.3 determinants II Cramer Lect 19 11/5 7.1 Eigenvalues introduction Lect 20 11/7 7.2 Eigenvalues 8. Week: Diagonalization Lect 21 11/10 7.3 Eigenvectors Lect 22 11/12 7.4 Diagonalization Lect 23 11/14 7.5 Complex eigenvalues 9. Week: Stability and symmetric matrices Lect 24 11/17 7.6 Stability Lect 25 11/19 *** Review for second midterm Lect 26 11/21 8.1 Symmetric matrices 10. Week: Differential equations Lect 27 11/24 9.1 Differential equations I Lect 28 11/26 9.2 Differential equations II 11/28 THANKSGIVING, no class 11. Week: Function spaces Lect 29 12/1 9.4 (Handout) Nonlinear systems Lect 30 12/3 10.1 Function spaces (compare also 4.1,4.2) Lect 31 12/5 10.1/9/3 Linear differential operators 12. Week: Partial differential equations Lect 32 12/8 10.2 Fourier series Lect 33 12/10 10.3 Partial differential equations I Lect 34 12/12 10.4 Partial differential equations II 13. Week: Review and Vacation Lect 35 12/15 Review Week 10-12 ```