21-241 Matrix Algebra

Description

Linear algebra covers material which is essential to anyone who does any mathematical computation in engineering and the sciences. In application and in class, the subject divides naturally into two parts: computation and formal structure.

These are intimately related, but operationally distinct: on the one hand, computations with matrices and linear equations can be made into efficient algorithms, in mental code or in computer code, and, once created, these can be carried out with little attention to the theory. On the other band, in order to understand, to choose and then correctly optimize the applications of linear algebra, it is necessary to see the underlying formal algebraic structure.

What does this mean for this course? Your challenge will be to master the algorithmic aspects of the subject, without thinking that this is all that there is to the subject, and to deal with the underlying formal structure by using the concrete model of matrices and vectors as a guide and as a tool.

Learning Objectives

After completing this course, you should be able to

1. Solve linear systems of equations Ax=b using Gaussian elimination.
2. Understand the terminology relating to Gaussian elimination (pivots, multipliers, back substitution, invertibility of A, factorization into A = LU).
3. Understand the definitions of Vector Space, Linear Independence, Basis and Dimension.
4. Identify the four fundamental subspaces of a matrix, find a basis for each and determine their dimension.
5. Understand the concept of Orthogonality and determine the Orthogonal Projection onto a subspace.
6. Find least-squares approximations for an overdetermined system.
7. Apply the Gram-Schmidt process to find an orthogonal basis for a subspace (factorization into A = QR).
8. Understand the properties of determinants.
9. Apply the formulas for computing determinants.
10. Understand the connection among determinants, invertability and volumes.
11. Compute eigenvalues and eigenvectors of a matrix.
12. Use eigenvalues and eigenvectors to diagonalize, compute powers of or compute the exponential of a matrix.
13. Understand results relating to symmetric matrices and positive definite matrices (real eigenvalues and orthogonal eigenvectors, tests for x'Ax > 0, applications)
14. Compute the singular value decomposition of a matrix, and understand it's connection to linear transformations and change of basis.

How to use this site...

Home

The description gives a brief overview of the topics we will discuss this semester. The learning objectives give an itemized list of the skills you should be developing. The list of learning objectives may give you some useful direction in terms of studying for exams.

Course Information

Times and rooms for lecture and recitation sections.

Blog

Provides information about the course: posting of homework, changes to office hours, times and locations for review sessions, and general announcements.

Schedule

List of topics to be covered each week, with links to reading assignments and homework.

Policies

Information about the calculation of grades, dates for exams, policies for late assignments and other matters. Familiarize yourself with these policies early in the semester.