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 lines and 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 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.