This course is a one semester introduction to Linear Algebra intended for engineering students but appropriate for anyone interested in the subject. The course is designed to help students master the fundamental algorithms of linear algebra and achieve a sound understanding of the mathematical issues underlying these techniques.

Linear algebra is a simple but powerful tool for organizing and understanding the world around us. As the name suggests, linear algebra combines techniques from algebra and from geometry. It is this interplay of algebra and geometry that gives linear algebra its power. From a purely algebraic viewpoint, linear algebra is the theory of matrices---solving systems of linear equations, matrix addition, multiplication, row and column reduction, inner products, and determinants. From the geometric point of view, linear algebra may be seen as the theory of linear maps---e.g. rotations and dilations of lines, planes or higher dimensional spaces, and geometric notions like distance, angle and volume in these spaces. The algebraic methods give rise to simple and widely applicable algorithms while the geometric ideas strengthen our intuition and provide links to the many applications of the subject. An important goal of this course is to help students develop facility at making the translation back and forth between algebra and geometry.

If linear algebra is the theory of straight and flat planes, then why should the field have such wide application for understanding the world around us? After all, in the real world there are no perfectly straight lines and no perfectly flat planes. One answer is simply that our idealization of things in the small is linear. Thus, before Christopher Columbus, many people believed the earth to be flat. Indeed, if we magnify the surface of a sphere so that we see an enlarged view of only a small piece of that surface, then it will appear to be flat. In calculus, we learn that the tangent line to the graph of a differentiable function at a point is a ``good approximation" to the graph near that point. These are examples of the method of linearization in which one studies first-order approximations (also called linear approximations) to functions.

These ideas will lead us naturally to the study of discrete linear dynamical systems and later to continuous linear dynamical systems and linear systems of differential equations. As we will see, the structure of the trajectories of such a system can be determined from the eigenvalues and eigenvectors of the linear map determining the system. We will develop the tools required for calculating and understanding the eigenvalues and eigenvectors of a linear map and also give a number of other applications (e.g. quadratic forms).

Problem solving is an essential part of the course. Homework problems will be assigned at each regular class meeting (i.e. Mondays and Wednesdays). These problems should be done before the next class meeting so that we can discuss them. Your work on the problems from the week will be collected in class on Monday morning of the following week. Late homeworks will not be accepted.

Sections:

Friday class meetings will be devoted to discussion of problems assigned during the week. Be sure to prepare for sections so that you can ask lots of questions. These discussion meetings are intended to provide you with opportunities to go over problems and discuss more examples and techniques that you might not have understood from the regular class meetings or on the homeworks.

Exams:

There will be two in-class hour exams during the semester and a final exam at the end of the semester. The dates of the two hour exams will be discussed together as a class and will be scheduled by general agreement. Thus the hour exam dates given below are only tentative. The correct dates will be announced in class.

Make-up Exams:

You are expected to take all exams at the scheduled times. No make-up exams will be given except in extreme cases of illness. A note from a physician stating that you could not take the regularly scheduled exam due to medical reasons is required before any make-up exam can be given.

Your final grade for the course will be determined according to the following scale:

Academic Honesty Policy:

Given the sterling qualities of character to be found in each and every student at Boston University, it is certainly unnecessary to mention that plagiarism and cheating are not only dishonest and immoral, but are also against the policies of Boston University. Please keep in mind that in the highly unlikely event that you do choose to plagiarize or cheat, you will be referred immediately to the University Academic Standards Committee for disciplinary action.