Math21b, Spring 2010, Linear Algebra and Differential Equations,

Math21b: Linear Algebra and Differential Equations

is an introduction to linear algebra, including linear transformations, determinants, eigenvectors, eigenvalues, inner products and linear spaces. As for applications, the course introduces discrete dynamical systems, differential equations, Fourier series as well as some partial differential equations. Other highlights are applications in statistics like Markov chains or data fitting with arbitrary functions. The course is taught in 6 sections.

Instructors:

Course assistants:

Head CA: Jenny Wang: wang24\@fas.harvard.edu See the Section page

Lecture times:

Mo-We-Fr 10-11
Mo-We-Fr 11-12
Mo-We-Fr 12-1
Tu-Th 10-11:30
Tu-Th 11:30-1:00

Problem Sections:

See the Sections page. MQC:

Website:

http://www.courses.fas.harvard.edu/~math21b/

Text:

We use Otto Bretscher, Linear Algebra with Applications, fourth edition. Prentice-Hall, Upper Saddle River, NJ, 2008. This is the latest edition.

About this course:

teaches methods to solve systems of linear equations Ax = b,
allows you to analyze and solve systems of linear differential equations,
you learn to solve discrete linear dynamical systems like discrete Markov processes.
you will master the technique of least square fit with arbitrary function sets and 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 scientific fields like for example quantum mechanics or combinatorics or statistics
it 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 split usually 1/3 from Tue to Thu and 2/3 from Thu to Tue.

Exams:

We have two midterm exams and one final exam. Here are the midterm exam dates:

1. Midterm:

Tue 3/2

7-8:30pm

Hall D

2. Midterm:

Tue 4/6

7-8:30pm

Hall D

Grades:

                                          Grade1  Grade2 
 First hourly                              20     20   
 Second hourly                             20     20
 Homework                                  20     20
 Lab                                        5
 Final exam                                35     40      
 -------------------------------------------------------------
 Total                                    100    100

Calendar:

 --------------------------------------------------------
 So Mo Tu We Th Fr Sa
 --------------------------------------------------------
 24 25 26 27 28 29 30    0   Jan 25 intro meeting in Hall B
 31  1  2  3  4  5  6    1   Feb 1 first day of class
  7  8  9 10 11 12 13    2
 14 15 16 17 18 19 20    3   Feb 15, Presidents day
 21 22 23 24 25 26 27    4
 28  1  2  3  4  5  6    5   March 2. First midterm
  7  8  9 10 11 12 13    6
 14 15 16 17 18 19 20    a   March 13-March 21 Spring break
 21 22 23 24 25 26 27    8
 28 29 30 31  1  2  3    9
  4  5  6  7  8  9 10   10   April 6. Second midterm
 11 12 13 14 15 16 17   11
 18 19 20 21 22 23 24   12
 25 26 27 28 29 30  1   13   
  2  3  4  5  6  7  8   14   April 29-May 6, Reading period
  9 10 11 12 13 14 15   15   May 7 - May 15 Exam period
 ---------------------------------------------------------

Day to day syllabus:

    Lecture Date   Book Topic
 
 1. Week:  Systems of linear equations
 
    Lect 1   2/1  1.1   introduction to linear systems  
    Lect 2   2/3  1.2   matrices and Gauss-Jordan elimination
    Lect 3   2/5  1.3   on solutions of linear systems
 
 2. Week:  Linear transformations
 
    Lect 4   2/8   2.1   linear transformations and their inverses
    Lect 5   2/10  2.2   linear transformations in geometry 
    Lect 6   2/12  2.3-4 matrix product and inverse
 
 3. Week:  Linear subspaces
 
    Lect 7   2/15  Presidents day, no class
    Lect 8   2/17  3.1   image and kernel 
    Lect 9   2/19  3.2   bases and linear independence 
 
 4. Week:  Dimension and linear spaces
 
    Lect 10  2/22  3.3   dimension 
    Lect 11  2/24  3.4   coordinates
    Lect 12  2/26  4.1   linear spaces 
 
 5. Week:  Orthogonality
 
    Lect 13  3/1   review for first midterm        
    Lect 14  3/3   4.1  linear spaces II
    Lect 15  3/5   5.1  orthonormal bases and orthogonal projections
 
 6. Week:  Datafitting
 
    Lect 16  3/8   5.2  Gram-Schmidt and QR factorization 
    Lect 17  3/10  5.3  orthogonal transformations
    Lect 18  3/12  5.4  least squares and data fitting
 
 Spring break 3/13-3/21
 
 7. Week:  Determinants
 
    Lect 19  3/22  6.1   determinants 1
    Lect 20  3/24  6.2-3 determinants 2
    Lect 21  3/26  7.1-2 eigenvalues 
 
 8. Week:  Diagonalization
 
    Lect 22  3/29  7.3  eigenvectors
    Lect 23  4/31  7.4  diagonalization
    Lect 24  4/2   7.5  complex eigenvalues
 
 9. Week:  Stability and symmetric matrices
 
    Lect 25  4/5   Review for second midterm
    Lect 26  4/7   7.6  stability          
    Lect 27  4/9   8.1  symmetric matrices
 
 10. Week:  Differential equations
 
    Lect 28  4/12  9.1  differential equations I
    Lect 29  4/14  9.2  differential equations II
    Lect 30  4/16  9.4  nonlinear systems
 
 11. Week:  Function spaces
 
    Lect 31  4/19  4.2  linear trafos on function spaces
    Lect 32  4/21  9.3  inhomogeneous differential equations
    Lect 33  4/23  5.5  inner product spaces           
 
 12. Week:  Partial differential equations
 
    Lect 34  4/26  5.5  Fourier theory
    Lect 35  4/28  Partial differential equations
 
 Spring reading period 4/29-5/6
 Spring exam    period 5/7-5/15