Fall 2004

# Mathematics Math21b Fall 2004

## 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 taught in 3 sections. - Instructors: Oliver Knill, SC-434, knill@math Janet Chen, SC 321g, jjchen@math Section page of Janet - Course assistants: Philip Powell ppowell@fas Tien Anh Nguyen tanguyen@fas Goutham Seshadri seshadri@fas Azra Pravdic pravdic@fas - Lectures: Mo-We-Fr 10-11 Mo-We-Fr 11-12 Tu-Th 10-11:30 - Problem Sections: Tue, in SciC 111 7:00-8:00pm (Goutham) Wed, in SciC 101B 8:30-10:00pm (Philip) Thu, in SciC 116 8:30-9:30pm (Tien Anh) Sun, in SciC 222 8:30-9:30pm (Aki) - Office hours: Oliver: Mo,We,Fr 15-16 Janet: Mo,Th 15-16 - Website: http://www.courses.fas.harvard.edu/~math21b/ Section page of Janet - Text: Otto Bretscher, Linear Algebra with Applications, third 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. Example: 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. Tue-Thu section HW is splitted differently but the total homework is the same. - 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) -------------------------------------------------------- So Mo Tu We Th Fr Sa -------------------------------------------------------- 19 20 21 22 23 24 25 20. September, Orientation +-----+-----+ | | | 26 27 28 29 30 1 2 1 27. Start of classes 3 4 5 6 7 8 9 2 10 11 12 13 14 15 16 3 17 18 19 20 21 22 23 4 24 25 26 27 28 29 30 5 27. Oct, 1. Hourly 6:30 SciC D 31 1 2 3 4 5 6 6 November 7 8 9 10 11 12 13 7 11. Columbus day 14 15 16 17 18 19 20 8 21 22 23 24 25 26 27 9 25-27. Thanksgiving, no class 28 29 30 1 2 3 4 10 1. Dec. 2. Hourly 6:00 SciC D 5 6 7 8 9 10 11 11 12 13 14 15 16 17 18 12 19 20 21 22 23 24 25 13 winter break 22. - 3. Jan | | | +------+----+ 26 27 28 29 30 31 1 2 3 4 5 6 7 8 4. Jan -14. Jan Reading period 9 10 11 12 13 14 15 16 17 18 19 20 21 22 20. January Final: 9:15AM 23 24 25 26 27 28 29 Boylson Hall 110 30 31 --------------------------------------------------------- - Day to day syllabus: Lecture Date Book Topic 1. Week: Systems of linear equations Lect 1 9/27 1.1 introduction to linear systems Lect 2 9/29 1.2 matrices and Gauss-Jordan elimination Lect 3 10/1 1.3 on solutions of linear systems 2. Week: Linear transformations Lect 4 10/4 2.1 linear transformations and their inverses Lect 5 10/6 2.2 linear transformations in geometry Lect 6 10/8 2.3-4 matrix algebra (product and inverse) 3. Week: Linear subspaces Lect 7 10/11 Columbus day, no class Lect 8 10/13 3.1 image and kernel Lect 9 10/15 3.2 subspaces, bases and linear independence 4. Week: Dimension Lect 10 10/18 3.3 dimension Lect 11 10/20 3.4 coordinates Lect 12 10/22 4.1 linear spaces 5. Week: Orthogonality Lect 13 10/25 4.1 linear spaces II and review Lect 14 10/27 *** Review for first midterm Midterm Lect 15 10/29 5.1 orthonormal bases and orthogonal projections 6. Week: Datafitting Lect 16 11/1 5.2 Gram-Schmidt and QR factorization Lect 17 11/3 5.3 orthogonal transformations Lect 18 11/5 5.4 least squares and data fitting 7. Week: Determinants Lect 19 11/8 6.1 determinants 1 Lect 20 11/10 6.2 determinants 2 Lect 21 11/12 7.1-2 eigenvalues 8. Week: Diagonalization Lect 22 11/15 7.3 eigenvectors Lect 23 11/17 7.4 diagonalization Lect 24 11/19 7.5 complex eigenvalues 9. Week: Stability and symmetric matrices Lect 25 11/22 7.6 stability Lect 26 11/24 8.1 symmetric matrices 11/26 THANKSGIVING, no class 10. Week: Differential equations Lect 27 11/29 9.1 Differential equations I Lect 28 12/1 *** Review for second midterm Lect 29 12/3 9.2 Differential equations II 11. Week: Function spaces Lect 30 12/6 9.4 Nonlinear systems Lect 31 12/8 4.2 Function spaces Lect 32 12/10 9.3 Linear differential operators 12. Week: Partial differential equations Lect 33 12/13 5.5 Inner product spaces Lect 34 12/15 5.5 Fourier theory Lect 35 12/17 (handout) Partial differential equations 13. Week: Review and Vacation Lect 36 12/20 Review Last homework due ```