MATH
21 B
Mathematics Math21b Spring 2007
Linear Algebra and Differential Equations
Syllabus
Course Head: Oliver Knill
Office: SciCtr 434
A more detailed lecture plan
Math21b: Linear Algebra and Differential Equations 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 6 sections.
Instructors:
  • Samik Basu
  • Veronique Godin
  • David Helm
  • Thomas Lam
  • Matt Leingang
  • Oliver Knill
Course assistants: See the Section page
Lecture times:
  • Mo-We-Fr 9-10
  • Mo-We-Fr 10-11
  • Mo-We-Fr 11-12
  • Tu-Th 10-11:30
  • Tu-Th 11:30-1:00
Problem Sections: See the Sections page.
Website: http://www.courses.fas.harvard.edu/~math21b/
Text: We use Otto Bretscher, Linear Algebra with Applications, third edition. Prentice-Hall, Upper Saddle River, NJ, 2001. This great book has been used for many years here.
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 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, combinatorics
  • 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 splitted 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:Wed 3/7 7-8:30pmHall C 2. Midterm:Tue 4/10 7-8:30pmHall C
Grades:
                                          Grade1  Grade2
 First hourly                              20     20   
 Second hourly                             20     20
 Homework                                  20     20
 Lab                                        5
 Final exam                                35     40
 -------------------------------------------------------
 Total                                    100    100
 
 Grade = Max(Grade1,Grade2)
 
 Doing the mathematica project will soften a bit the final
 exam. 
 
Calendar:
 --------------------------------------------------------
 So Mo Tu We Th Fr Sa
 --------------------------------------------------------
  S  M  T  W  T  F  S
          31  1  2  3        31. Jan Plenary introduction
  4  5  6  7  8  9 10    1   5. Feb Lectures start
 11 12 13 14 15 16 17    2
 18 19 20 21 22 23 24    3
 25 26 27 28  1  2  3    4   March
  4  5  6  7  8  9 10    5   March 7. First midterm Hall C
 11 12 13 14 15 16 17    6
 18 19 20 21 22 23 24    7
 25 26 27 28 29 30 31        Spring recess
  1  2  3  4  5  6  7    8   April
  8  9 10 11 12 13 14    9   April 10  Second midterm Hall C
 15 16 17 18 19 20 21   10                               
 22 23 24 25 26 27 28   11
 29 30  1  2  3  4  5   12   May
  6  7  8  9 10 11 12
 13 14 15 16 17 18 19
 ---------------------------------------------------------
 
Day to day syllabus: A more detailed lecture plan.
    Lecture Date   Book Topic
 
 1. Week:  Systems of linear equations
 
    Lect 1   2/5  1.1   introduction to linear systems  
    Lect 2   2/7  1.2   matrices and Gauss-Jordan elimination
    Lect 3   2/9  1.3   on solutions of linear systems
 
 2. Week:  Linear transformations
 
    Lect 4   2/12  2.1   linear transformations and their inverses
    Lect 5   2/14  2.2   linear transformations in geometry 
    Lect 6   2/16  2.3-4 matrix product and inverse
 
 3. Week:  Linear subspaces
 
    Lect 7   2/19  Presidents day, no class
    Lect 8   2/21  3.1   image and kernel 
    Lect 9   2/23  3.2   bases and linear independence 
 
 4. Week:  Dimension and linear spaces
 
    Lect 10  2/26  3.3   dimension 
    Lect 11  2/28  3.4   coordinates
    Lect 12  3/2   4.1   linear spaces 
 
 5. Week:  Orthogonality
 
    Lect 13  3/5   review for first midterm        
    Lect 14  3/7   4.1  linear spaces II
    Lect 15  3/9   5.1  orthonormal bases and orthogonal projections
 
 6. Week:  Datafitting
 
    Lect 16  3/12  5.2  Gram-Schmidt and QR factorization 
    Lect 17  3/14  5.3  orthogonal transformations
    Lect 18  3/16  5.4  least squares and data fitting
 
 7. Week:  Determinants
 
    Lect 19  3/19  6.1   determinants 1
    Lect 20  3/21  6.2   determinants 2
    Lect 21  3/23  7.1-2 eigenvalues 
 
 Spring break
 
 8. Week:  Diagonalization
 
    Lect 22  4/2   7.3  eigenvectors
    Lect 23  4/4   7.4  diagonalization
    Lect 24  4/6   7.5  complex eigenvalues
 
 9. Week:  Stability and symmetric matrices
 
    Lect 25  4/9   Review for second midterm
    Lect 26  4/11  7.6  stability          
    Lect 27  4/13  8.1  symmetric matrices
 
 10. Week:  Differential equations
 
    Lect 27  4/16  9.1  differential equations I
    Lect 28  4/18  9.2  differential equations II
    Lect 29  4/20  9.4  nonlinear systems
 
 11. Week:  Function spaces
 
    Lect 30  4/23  4.2  function spacess
    Lect 31  4/25  9.3  linear differential operators
    Lect 32  4/27  5.5  inner product spaces           
 
 12. Week:  Partial differential equations
 
    Lect 33  4/30  5.5  Fourier theory I
    Lect 34  5/2   5.5  Fourier theory II
    Lect 35  5/4   Partial differential equations
 
Please send questions and comments to math21b@fas.harvard.edu
Math21b | Oliver Knill | Spring 2007 | Department of Mathematics | Faculty of Art and Sciences | Harvard University