First, you will learn the fundamental algorithm for solving systems of linear equations, which is called “Gauss-Jordan elimination” or “row-reduction”. Most calculations in the subject boil down to this algorithm. Second, you will learn an abstract framework (“vector spaces and linear maps”) that captures the essential features of systems of equations. This framework is sufficiently general to model many other situations in mathematics. Third, you will learn a method (“bases and coordinates”) for translating any question in the abstract setting into a concrete question about systems of linear equations that can be answered with row reduction. Along the way, we will discuss a number of applications of these ideas to problems in science and engineering. At the end of the day, you will master powerful tools with a very broad range of applicability.
This course will also give you an introduction to abstraction and higher mathematics. We will be careful about definitions and will give careful arguments, i.e., proofs, justifying everything we do. You will increase your ability to write clear and precise explanations of mathematical facts.
First, because it is really useful. Mathematicians sometimes say that all of mathematics is calculus and linear algebra. This isn't completely true, but it's close. In science and engineering, linear algebra is a tool used every day, for example: linear approximations of functions (think interpolation and extrapolation), interpreting data (least squares regressions and lines of best fit), signal processing, computer graphics, internet search, random processes, even the foundations of quantum mechanics.
Second, you are a top student in Math, and this course is designed to challenge students like you.
Third and most importantly, because it is really interesting! You will see the power of the axiomatic method in isolating and analyzing what is essential.
Yes! You did well in calculus, proving that you know how to work hard and master complex material. Although the material in this class is at times somewhat abstract, the underlying ideas are pretty simple (a lot simpler than things like continuity and limits).
The TAs and I are commited to helping you learn this material and succeed in the course. Keep at it and you will do well.
See the last section below for more hints on how to succeed in this class and advanced math classes in general.
This is a first course in linear algebra for mathematically ambitious and well-prepared students. You will learn the fundamental algorithms and computational recipes as well as the theoretical underpinnings necessary for more advanced mathematics. Enrollment in the course is by invitation only and typically requires a score of 5 on the BC Calculus AP exam or a grade of A in Math 1552. Success in this course will prepare you for 2000-level courses in multivariable calculus, differential equations, discrete mathematics, and foundations of mathematical proof.
There will be two 80-minute lectures and two 50 minute recitations each week. Lecture time will also include clicker quizes and some discussion. Recitations will be devoted to active learning, i.e., working problems and discussing difficulties. There will be weekly homework, two mid-term exams, and a final exam.