Linear Algebra stands at the crossroads between applied mathematics and theoretical mathematics. It is utterly important in many subsequent mathematics courses in the mathematics major, and it is also useful in the extreme for many applications not only in higher mathematics, but also in the sciences, including the physical sciences as well as economics and the social sciences. It is even indispensible in some parts of business and the managerial sciences. It may very well be the most important course in mathematics you ever take, regardless of your future academic goals. Yes, it even rivals the Calculus in importance!
What is the source of this central role it plays?
From the applications side of things, in many, many situations, the object of study is governed by a system of linear equations. At a lower level, linear algebra can be thought of as a collection of techniques for solving such systems, using a variety of methods and tools, especially matrices. Furthermore, even in situations where the phenomena under study is non-linear, one often makes a linear approximation as a first step in studying and understanding the phenomena. Think about it - this is exactly what you do in differential calculus!
From the theoretical side of things, many geometrical phenomena are best described using the notion of vectors. This is also true of sciences where geometry is important, such as Physics. Furthermore, it turns out that the theory of linear equations (mentioned in the last paragraph) can also be best understood using the language of vectors and certain sets of vectors called vector spaces. In this way, linear algebra (the theory of matrices and vector spaces) can be viewed as a sort of universal language with which to state (and solve) problems in physics, geometry and algebra.
At a higher level, these vector spaces also turn up in other advanced mathematics courses, so this language can be used to describe (and solve) problems in Graph theory (MC302), Abstract algebra (MA319 & MA320 and beyond), and at the graduate level, even of problems in Analysis and Topology.
Finally, at more of a meta-mathematical level, vector spaces are defined axiomatically, and theorems about them are proved from the axioms using deductive and inductive reasoning. Many mathematical subjects are treated the same way (Graph theory, Group theory, Ring theory, Lie Algebra theory, Topology, etc.), so the way the knowledge is organized and developed for vector spaces and the linear maps between them provide a sort of blueprint for a large portion of modern mathematical theories to emulate.
The way the course is organized, we follow some of the rough divisions outlined above. The first part of the course, Chapters I - II, treats linear algebra in a very concrete way. Here, we focus on systems of linear equations, and learn techniques to solve them. Here is where we will be able to exploit the software Mathematica. We also start to learn the language of linear algebra by covering vectors in Euclidean space and matrices.
In the second part of the course, Chapters III & IV, we slowly shift away from the concrete to a more abstract approach and learn about vector spaces. Here, we begin to look at things from a more theoretical point of view, and begin to think more like a mathematican than like someone who is a user of mathematics.
The last part of the course (Chapters V - VII) goes into more detail with the theory and covers some more advanced material. Even though this is the most abstract part of the course, it is never very far from the applications to both mathematics and to science. Applications appear throughout the book, and we will cover them as time permits.
In summary, we intend to cover the first four chapters of the text in detail, and most of Chapter V. Following this, we will choose the topics from the remaining chapters as time and interest permits. Occasionally, I may bring in examples and topics from outside the text to supplement the material.
