Linear Algebra

Math V2010, Fall 1997

Syllabus

Teaching Assistant: Marie-Amelie Bertin. She will be grading most of the homework. Her e-mail is bertin@math.columbia.edu Her office hours are Monday 10AM to 1PM in the Columbia Mathematics Help Room, Room 406 Mathematics.

Prerequisite: Math V1106 (Calculus IIS) or Math V1201 (Calculus IIIA) or the equivalent. In fact all we will use from that course is the notion of vector in the plane and in space; the dot product of vectors; and the equations of lines and planes in space

Text: Introduction to Linear Algebra by Gilbert Strang, published by Wellesley-Cambridge Press. The book is available from the Columbia Bookstore. Please get a copy, as I will be assigning reading and homework out of it. I expect to cover chapters 1 through 7, plus applications if there is time.

Course Objectives: The first goal of the course is to teach students how to use linear algebra as a powerful tool for computation. The second goal is to show how these computations can be conceptualized in a geometric framework. The final goal is to give a gentle introduction to the theory of abstract vector spaces.

Student Population: Linear Algebra is the language needed to manipulate data involving more than one variable, so students from many different fields take this course. This is also a required course for Econ/Math and Mathematics Majors, although students who have taken Honors Math III-IV are exempted from this requirement. It is a prerequisite for many higher level math courses, for example:

Topics: Elementary matrix operations and systems of linear equations; Gaussian elimination; Vector spaces, subspaces; linear dependence/independence, bases and dimension; orthogonality, projections and least squares approximations; determinants and their properties; eigenvalues, eignevectors; diagonalizaton and characteristic polynomials; the Cayley-Hamilton theorem for matrices, invariant subspaces; applications as time permits.

Test dates:

Last day to drop classes: Thursday, November 13. Last day of class: Thursday, December 4

Grading: Midterm 2 is a "make-up test". I will drop the lower of your scores on the first two midterms; the higher score will count 20% of your grade. Midterm 3 will also count 20% of your grade. The final exam will be worth 50%. The final 10% will come from the homework. For the usual admonitions and threats, click here.

Homework: Assignments will be due in class every Thursday, except on exam weeks. Assignments will be handed out a week in advance. No late homework will be accepted. Several of the assigned problems will be randomly chosen for grading; written solutions to most of the assigned problems will be handed out. Homework will be returned in class. Homework is essential in learning linear algebra. They are not a test and you are encouraged to talk to other students about difficult problems - after you have struggled with them on your own. MAPLE, MATHEMATICA and MATLAB are excellent software packages for learning about and solving problems in linear algebra. They will be available for your use on most of the computer systems in the Mathematics Department. You should attempt all the suggested homework problems. If you are having trouble with the homework, you should first use the Mathematics Help Room in Room 406 Mathematics. I will also answer specific questions by e-mail.

Other Texts: An interesting, slightly more elementary text with a much more geometric point of view is Linear Algebra Through Geometry by Thomas Banchoff and John Wermer, published by Springer-Verlag. An exhaustive and yet quite readable account of matrix theory is given in The Theory of Matrices , by F.R. Gantmacher (Chelsea).

Computer Code (courtesy of Professor S. Popescu):

Related and/or interesting links:

* The Java Script Linear Algebra Calculator by Marcus Kazmierczak is able to calculate some basic Linear Algebra functions such as the trace, determinant, adjoint and inverse of a matrix, and addition and multiplication of matrices.
* Other Java Scripts by Marcus Kazmierczak:
* Linear Algebra Web Notes: a complete undergraduate Linear Algebra course including the text of lectures, homework assignments and tests with solutions, a discussion page, and Maple examples by Mark V. Sapir.
* From the MacTutor History of Mathematics archive: