Harvard University,FAS

  Fall 2001

Mathematics Math21b
Fall 2001

Linear Algebra
and Differential equations

Course Head: Dale Winter

Office: SciCtr 506
Email: amanita@math
Mainpage Syllabus Sections Calendar Homework Exams Supplements Links


Day-by-day summary of course

  • Unit 1: (9/17 to 10/12) An introduction to the objects, concepts and calculations of linear algebra in the context of real, finite-dimensional vector spaces.
  • Unit 2: (10/15 to 10/17) An introduction to the definitions of abstract vector spaces. The emphasis of this unit will be to show how abstract mathematical definitions are connected to the concepts and calculations introduced in Unit 1.
  • Unit 3: (10/19 to 10/29) An introduction to the ideas and applications of the concept of orthogonality in linear algebra and geometry. The main application that this section of the course develops is a computational procedure for calculating the line of best fit for a given set of data.
  • Unit 4: (10/31 to 11/5) An introduction to the properties and methods for calculating determinants of matrices.
  • Unit 5: (11/7 to 11/30) Methods for calculating the eigenvalues and eigenvectors of a matrix, with applications to the analysis of discrete and continuous dynamical systems. The main emphases of this unit will be on calculating the eigenvalues and eigenvectors of a given matrix, and using them to analyze the behavior of dynamical systems.
  • Unit 6: (12/5 to 12/14) An introduction to some of the basic techniques for finding symbolic solutions to ordinary and partial differential equations. The main technqiues employed here are separation of variables and Fourier series. The main emphasis of this section will be on gaining facility with the computational techniques used to obtain the symbolic solutions of the differential equations.

Day Date Topic Text
Monday 9/17/01 First day of class. Introduction to linear systems. 1.1
Wednesday 9/19/01 Matrices and Gauss-Jordan elimination. 1.2
Friday 9/21/01 The solutions of linear systems. 1.3

Monday 9/24/01 Linear transformations. 2.1
Wednesday 9/26/01 Geometrical interpretations of linear transformations. 2.2
Friday 9/28/01 The inverse of a linear transformation. 2.3

Monday 10/1/01 Composition of linear transformations. Matrix products. 2.4
Wednesday 10/3/01 The image and kernel of a linear transformation. 3.1
Friday 10/5/01 Subspaces, bases and linear independence. 3.2

Monday 10/8/01 NO CLASS - COLUMBUS DAY. No reading
Wednesday 10/10/01 The dimension of a subspace. 3.3
Friday 10/12/01 Coordinates of vectors. 3.4

Monday 10/15/01 Introduction to linear spaces. 4.1
Wednesday 10/17/01 Linear transformations and isomorphisms. 4.2
Friday 10/19/01 Orthonormal bases and orthogonal projections. 5.1

Monday 10/22/01 Review for Test 1. No reading
TUESDAY 10/23/01 EXAM 1: 7-9PM SCIENCE CENTER LECTURE HALL C Remember your calculator and one page of notes
Wednesday 10/24/01 The Gram-Schmidt process and QR factorization. 5.2
Friday 10/26/01 Orthogonal transformations and orthogonal matrices. 5.3

Monday 10/29/01 Least squares and data-fitting. 5.4
Wednesday 10/31/01 Introduction to determinants. 6.1
Friday 11/2/01 Properties of the determinant. 6.2

Monday 11/5/01 Interpretations of determinants, Cramer's rule. 6.3
Wednesday 11/7/01 Dynamical systems and eigenvectors. 7.1
Friday 11/9/01 Finding the eigenvalues of a matrix. 7.2

Monday 11/12/01 NO CLASS - VETERAN'S DAY. No reading
Wednesday 11/14/01 Finding the eigenvectors of a matrix. 7.3
Friday 11/16/01 Diagonalization. 7.4

Monday 11/19/01 Complex eigenvalues and rotations. 7.5
Wednesday 11/21/01 Stability. 7.6
Friday 11/23/01 NO CLASS - THANKSGIVING BREAK. No reading

Monday 11/26/01 An introduction to continuous dynamical systems. 9.1
Wednesday 11/28/01 The complex case: Euler's formula. 9.2
Friday 11/30/01 Non-linear systems of differential equations. Supplemental notes.

Monday 12/3/01 Review for Test 2. No reading
TUESDAY 12/4/01 EXAM 2: 7-9PM SCIENCE CENTER LECTURE HALL C Remember your calculator and one page of notes
Wednesday 12/5/01 Solving ordinary linear differential equations. Supplemental notes.
Friday 12/7/01 Calculating the Fourier series of a periodic function. Supplemental notes.

Monday 12/10/01 Solving the 1D heat equation by separation of variables. Supplemental notes.
Wednesday 12/12/01 Solving the 1D wave equation by separation of variables. Supplemental notes.
Friday 12/14/01 Summary of partial differential equations. Supplemental notes.

Please send comments to math21b@fas.harvard.edu.