 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, SC434, knill@math
Izzett Coskun, SC 333b, coskun@math
 Course assistants:
Minhua Zhang, zhang18@fas
Phillip Powell, ppowell@fas
Jeff Amlin, amlin@fas
 Lectures:
MoWeFr 1011 109
MoWeFr 1112 309
 Problem Sections:
Thursday 6:308:00 PM 101B
Wednesday 3:004:30 PM 109
Sunday 7:008:30 PM 111
 Office hours:
Oliver: Tuesday 1112 and Thursday 1112
Izzet: Tuesday 12 PM and Wednesday 34 PM
 Website: http://www.courses.fas.harvard.edu/~math21b/
 Text:
Otto Bretscher, Linear Algebra with Applications,
second edition. PrenticeHall, 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 7:309 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 7:309 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 GaussJordan 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 GramSchmidt 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 1012
