Course Objectives: Walking through Kemeny Hall, you will hear professors and students discussing a myriad of topics. A comment made suprisingly often in an explanation — independent of the subject — is It's just linear algebra. Whatever this subject is, one can be certain that linear algebra is as fundamental to higher mathematics as counting is to arithmetic. But what is it?
In a concrete context, it is the study of matrices and of finding solutions to systems of linear equations. In a more abstract setting it is the study of vector spaces (spaces which have an algebraic structure like $\mathbb R^n$), and of maps between them which preserve linear structure. There are two linear algebra courses at Dartmouth, Math 22 and 24. The first is concerned more with computation and applications and less so with abstraction. This course is concerned with a broader context in which to view linear algebra, and learning to justify assertions by means of rigorous proofs. But both courses strive to reveal the power and beauty of this subject as well as some of its amazing applications.
In the end, we shall split our time. The theorems you learn are your tools, but tools are only useful if you know how to use them; a hammer is a great tool, but not terribly useful until you have learned not to bend too many nails. It is great to be able to determine whether a system of equations has a solution or not, but if it doesn't, you might still be interested in "how close" to a solution you can actually get.
If nothing else, linear algebra is a beautiful and elegant subject, and one that can easily convince you of the power of mathematics.