The first half of this tutorial will consist of a rapid tour of the key elements of linear algebra. We will cover vector and matrix algebra, linear vector spaces and subspaces, basis and dimension, the determinant, and eigenvalues and eigenvectors. We will then consider first-order linear differential equations---first scalar equations and then systems of equations. We will see that linear algebra is an indispensable tool for solving systems of linear differential equations and understanding the geometric structure of their solutions. In addition to covering basic theory, we will consider a handful of case studies that illustrate the power of the mathematical techniques we are learning. These case studies will be drawn from across the sciences. The particular case studies we examine will depend on student interests.
Students who successfully complete this course will gain a solid introduction to the calculational techniques and key constructions and ideas of linear algebra. Students will also learn techniques for solving and understanding systems of linear differential equations. Additionally, this course serves as an introduction to formal mathematics; students will learn to work at a level of generality and abstraction a bit above that encountered in a typical introductory calculus sequence.
Students will be required to read sections of the text in advance of each class. Class sessions will be devoted primarily to discussion and problem solving. Evaluation will be based on class participation and weekly problem sets. This is a fast-paced tutorial for students with a strong preparation in mathematics. Pre-requisites: Calculus III. Differential Equations strongly recommended.
The first half of the course will cover basic linear algebra, including:
This will be followed by a treatment of linear first-order differential equations, including:
In the last part of the course we will consider systems of coupled linear differential equations. To do so, we will write systems in matrix form and show how the eigenvectors and eigenvalues of the resulting matrix can be used to form general solutions.
For most classes students will be assigned a few problems to discuss and/or present. There will also be several short cases studies applying course content that will be written up and handed in.