Linear Algebra has become a staple of the undergraduate curriculum. It's importance has been ever increasing with applications in optimization and computation. It's prevalence in both theoretical and applied mathematics means that a linear algebra course can be taught from many differnt directions. This has given me the opportunity to craft the course towards the student's interests. To the left, you will find some of the resources to the course. Below you will find some of my thoughts on the points-of-view taken, while teaching the course.
Seminar Style: The course is modeled after what one might experience when taking an undergraduate math seminar. We meet three days a week, homework is due every two or three weeks, and we discuss our topic for the day during class meetings. If you were to visit us, you would see students working problems at the board 85% of the time. Having students acitvely engaged, in the classroom, gives them an opportunity to show what they know, work through items that they may not fully grasp, and have practice communicating mathematical ideas. As the course is a year long, I have broken it into three parts:
What is Linear Algebra? Any attempt to answer this question will inevitably leave out someone's favorite application. It's study began with the desire to solve systems of linear equations. This dates back to the work of Leibniz, Cramer, and Gauss in the 18th century. It was through Sylvester, in the 19th century, that the idea of a matrix became formalized and the notion of a linear may first came up. Grassman introduced the notion of a Grassman algebra, which really wasn't appreciated at the time. Through the works of Cayley and Frobenius, the subject found firm footing, in the realm of matrix equations and the early notions of representation theory. The idea of dimension we find intuitive is that used in linear algebra.
As the 20th century dawned, there was a realization that the ideas of linearity could be found in other spaces as well. Lebesgue introduced the concept of measure, revolutionary in itself, and eventually spawned the study of function spaces. Function spaces are examples of infinite-dimensional vector spaces. Two special cases are that of the Banach space and the Hilber space. The Hilbert space makes its appearance in Quantum mechanics. As one would expect, having infinite dimensions can make life quite complicated.
Applications: It is through the growth of computers and computational mathematics that linear algebra has found a home in applied mathematics. There is a whole topic of Numerical Linear Algebra, where one can study the nuances and difficulties of computing the objects in linear algebra. Such items include, but certainly aren't limited to: LU, QR, and SVD decompositions, the eigenvalue problem, discrete Fourier transform, and wavelets. Each of these items holds an important place in applied mathematics. For example, the Fast Fourier Transform makes its appearance in signal processing, the Discrete Fourier Transform in image compressions, and solutions to partial differential equations. For students who are interested in computer programming and computations, these topics give them the opportunity to learn linear algebra, implement algorithms, and have concrete results for their work.