MAT217 Honors Linear Algebra

Schedule: (usually) Tuesdays and Thursdays, 11AM-12:20PM and/or 1:30PM-2:50PM, with a Friday precept, in Spring semester only. Includes optional evening review sessions run by undergraduate course assistants on a schedule that is set up after classes begin.

Brief Course Description: A rigorous introduction to linear algebra with an emphasis on proof rather than applications. Topics include vector spaces, linear transformations, inner product spaces, determinants, eigenvalues, the Cayley-Hamilton theorem, Jordan form, the spectral theorem for normal transformations, bilinear and quadratic forms. This course is a continuation of 215 and gives a more thorough and theoretical treatment of linear algebra than courses like 202 and 204. Continues the development of proof-construction skills begun in 215.

Who takes this course? This course is intended for prospective mathematics majors, but it is also for suitable for non-majors who like to think abstractly. It covers the basics of linear algebra from a highly theoretical point of view while at the same time gives more practice in understanding and constructing formal mathematical arguments. It is ideal for students with a strong mathematical background who want to learn more about proofs in preparation for upper-division course work in the mathematics department. It is an especially good choice for students with a serious interest in applied mathematics, theoretical physics or computer science.

Prerequisites and Placement Information:  Most students are continuing from 215 in the fall. Sometimes students are continuing from 203 or 216 instead. Like 215 and 216, a very strong aptitude for mathematics and genuine mathematical curiosity are essential. Do you like mathematics, especially the part where you try to understand why a method works or why a definition encapsulates a particularly useful point of view? Do you want to make your own conjectures and figure out for yourself whether a mathematical statement is true or false? Would you like to learn how to construct a clear and convincing, even iron-clad, argument to justify a mathematical claim? Would you like to develop an appreciation for the intrinsic value and power of mathematical thinking, separate from considerations of real-world applications or utility?

Dealing with algebraic topics, this course is in some ways more abstract and unfamiliar than 215, which provides the theoretical underpinnings to already familiar ideas from high school calculus.  It is quite difficult to judge how much time you will need to find your footing in this class, and students should be prepared to invest a lot of time early on, especially if this is the first proof-based class they are taking.

FAQ: 

• This is your first course with rigorous proofs. Will it be too hard?  The first few classes will give you a lot of information and the system is flexible for the most part. Different people have very different learning styles and interests, and it is hard to predict how you will feel in your first hard-core algebra class. Some people think learning to construct proofs is actually easier in an algebra course, but it can be hard to get your bearings in an algebra class if everything is new. It may be especially difficult to see the point of what you are doing, and you may be one of those people who is more successful learning proofs in 215 because it develops your proof skills through more familiar topics from calculus. If linear algebra and proof-based mathematics are both unfamiliar, you may find it useful to sit in on both 204 and 217 for the first few weeks, doing problem sets for both classes, before you decide which is better for you. Go to office hours and take advantage of the help available in the evening problem sessions where enthusiastic math majors will help you learn how to think like a mathematician. If the course seems interesting but hard, you should persevere. The grading curve is generous to encourage interested students who want to give this kind of thinking a serious try without undue academic risk. Consult your instructor for advice after the first few weeks if you are still worried about how things are going. Placement adjustments can be made even after the drop/add deadline if necessary.

• How hard should I expect to work in this class?  It can vary quite a bit for individual students, but we expect that you will need to spend substantial time outside of class, reading the textbook and reviewing your notes in order to understand the proofs and the definitions from the textbook well enough to be able to adapt them to create new proofs and construct counterexamples. Following up in office hours or at the undergraduate-led problem sessions or with your study group will likely be important for understanding all the new ideas. The problems require both creativity and knowledge, and so you will need to spend unpredictable amounts of time in active contemplation while you wait for inspiration or the right insight. All in all, you should be ready to spend at least 10 hours per week working outside of class. 

• You are probably interested in physics or engineering, but proof-based math like that in MAT215-217 seems kind of pointlessly abstract and hard or uninteresting.  Both 202 and 204 are very solid introductions to linear algebra that introduce the main ideas more concretely. Either is sufficient to fulfill the linear algebra prerequisites for those majors. Even if you are eventually planning to do more theoretical work, you may find it easier to learn the theory later, when you need it, and in a context where it feels more relevant. Different people learn different ways and you will do better in a course that feels like a better fit for your current interests.

• You have quite a bit of experience with proofs from independent reading or university-level analysis and algebra courses you took in high school or from extracurricular activities like math camp. Many of the topics for the course are already familiar to you.  Try the Sample Problems. If you know how to do these problems already, then taking 216-218 instead might be the right plan. You can discuss this with the math department representatives when you are on campus, or consult one of the other contacts for this course.