Course MAT204

Advanced Linear Algebra with Applications

Companion course to MAT203, a more abstract treatment of linear algebra than MAT202, but more concrete than MAT217.  Introduces basic algebraic tools such as matrices, vector spaces and linear transformations, bases and coordinates, eigenvalues and eigenvectors and their applications.  Exams test for thorough conceptual understanding as well as computational fluency. In this course we assume that students will not need much help to master standard, straightforward calculations, allowing the instructor to emphasize more subtle aspects of the definitions and important exceptional cases that come up in applications.  Offered Spring semester only.


Covers all the major topics in the first seven chapters of the textbook, Linear Algebra: Ideas and Applications by Richard C. Penney (3rd edition).  These topics include: linear systems of equations, linear independence and dimension, linear transforms, determinants, real and complex eigenvalues and eigenvectors, orthogonality, spectral theorem, singular value decomposition and Jordan forms.

Description of classes

Classes meet 3 times per week, for 50 minutes.  Sections are generally offered MWF at 10, 11 and 12:30 in the Spring semester only.

The course is organized into small sections of 20 to 30 students. There is one course head who coordinates with all the instructors to write the exams. All students have the same homework assignments and take the same midterm and final exam. The midterm and final count for the bulk of the course grade, typically about 65%.  These exams are graded by all the instructors and graduate student assistants together to ensure uniformity across all sections. Typically there are several take-home quizzes, the same for all sections. Homework and quizzes together usually account for about 35% of the course grade.  The homework may include a very small matlab component (about 3% of the course grade).

In order to do well in the course,  we anticipate that most students will need to spend approximately ten hours per week reading the text (before the corresponding lectures), reviewing class notes, solving homework problems and working through lots of extra practice problems to prepare for quizzes and exams.  The course will be quite fast-paced and it is essential to work steadily throughout the semester.  Frequent feedback will be given to help students keep up and monitor progress.

  • Most students will be continuing from MAT203 (Vector Calculus), which is offered in the Fall semester only. Although MAT203 and MAT204 can be taken in either order, we recommend you take MAT203 first.
  • The homework problems and computer projects (if any) are meant to be complementary to the exams, rather than preparatory.
  • Late homework submissions will not in general be accepted.  Collaboration on homework is allowed, even encouraged, but copying the work of another student is not permitted.  Each student in a collaborating group should write up the solutions in his or her own words and record the names of all collaborators.  Graders may subtract points for illegible or unstapled homework submissions.
  • All exams or quizzes will be timed and closed-book.  No technology may be used for these exams.  Collaboration is not allowed.
Who Takes This Course
  • The typical student in this course is strongly interested in physics as a major, but this course is suitable for anyone with a strong background in high school math (minimally, a 5 on the BC calculus exam and 750 or more on math SAT) and a serious interest in how mathematics can be applied to other fields such as economics, science or engineering. 
  • Naturally, since math and physics have much in common, many of the students in MAT204 are also considering math as a major. If you think you are more likely to major in math rather than physics, we recommend that you take our most rigorous linear algebra course (MAT217) instead (after completing MAT215 or some equivalent introduction to rigorous proof and formal mathematical argument). Don't worry if you cannot decide for sure, or if you change your mind. You can still decide to be a math major if you take MAT204.
  • If you cannot imagine yourself as a math or physics major, then this course is probably not for you.
  • Students who took AB calculus only and who may consider majoring in science or engineering should start in MAT104. (In rare cases, for students with an AB score of 5, a math SAT above 760 and a very strong interest in math as a major, we recommend MAT214 or MAT215. If that describes you, please talk to the placement officer at freshman registration.)
  • Many of the more mathematically inclined economics majors (or others interested in a finance certificate) take MAT201/202 instead.  Future economics majors can minimally fulfill their math prerequisites with MAT175 instead, but this does not give adequate preparation for taking more advanced math courses at Princeton.
  • AB COS majors are not required to take multivariable calculus. They need only linear algebra (MAT202 or MAT204 or MAT217) and can safely sign up for linear algebra without taking multivariable calculus first.  However MAT217 requires some prior background with formal mathematical proofs.
Placement and Prerequisites

We don't use much calculus in this course, but you will need to have the mathematical maturity and independence that most students acquire by studying calculus seriously in courses like MAT203. This is an algebra course, necessarily more abstract than calculus courses at the same level. You absolutely must have a serious interest in thinking rigorously about problems involving space and time, even though we will not spend much time on applications. It will be assumed that routine calculations like solving a linear system of equations are already familiar or are very easily mastered. Your instructor will assume that you can read the textbook and learn nuts-and-bolts computations without much help. Instead the lectures will focus mostly on developing a deeper, more conceptually sophisticated grasp of the main ideas of linear algebra as you generalize simple techniques for working with matrices and vectors and solving linear systems to more interesting situations in higher dimensions.

  • Because this course is more algebraic and abstract compared to calculus, many students find it difficult in a new way. Generally speaking, in this course, once you understand the ideas well, the actual calculations are relatively simple. It is much less possible to power through vaguely understood problems using your well-honed computational skills. You may well find that this course takes much more time than MAT203 if you have trouble with these more abstract topics. Others, of course, find algebra more to their taste than intricate calculations with curves and surfaces in space were, and thus enjoy MAT204 more than MAT203.
  • Because this course is more abstract, it will probably give you a better sense of what it is like to be a math major than you can get from MAT203.
  • If you were happy with the way calculus was taught in high school, you may prefer the style of MAT202 instead. For most engineering majors, MAT202 is an equally good choice of linear algebra course and will give you the background you need for your upper division courses in other departments.
  • If you plan to be a physics major, MAT204 is the recommended course; so start here, but be realistic! You will learn a lot of linear algebra in MAT202 if you are not ready for MAT204.
  • Because MAT204 covers basically the same mathematics as MAT202, but more deeply and with more abstraction, most students find MAT204 to be more challenging. For most students who are not planning to major in math or physics, it is mostly a matter of taste whether this extra challenge is worthwhile. It may take a few days of going to both courses to figure out which version appeals to you more.  We grade MAT204 on a somewhat more generous curve than MAT202 with the aim of eliminating grades as a serious consideration in the choice between these courses.

General information about our courses, placement and important contact information can be found on the undergraduate home page.

Sample Material

Working problems from these sample quizzes from Spring 2011 can give you a good general idea of the expectations and content in this course as you think about which course is right for you. Just reading the questions or the solutions can be very misleading however. Try the problems yourself!

Solutions to these exams are not guaranteed. There may be typos, but the more important information is in the questions!

  1. How much work is this course?
    • Most math courses require a steady time commitment.  We expect that the weekly problem sets will take at least three hours to complete, although this can vary quite a lot depending on your background and goals.  To do well on math exams, you need to work through a lot of extra problems.  All in all, you should be ready to spend up to ten hours per week working outside of class.  Instructors will assume that students in this course have a strong commitment to becoming real experts in the long run, and so are willing to invest more time and energy in the course.
    • It is quite difficult to judge how much time you will need to master the more abstract parts of the course. For many students this is a big adjustment, unlike the math courses you have taken before, so be prepared to invest quite a lot of time early on, learning how to think about proofs and counterexamples and adapting old techniques to complicated new situations, rather than chugging through computational problems just like the textbook and lecture examples.
  2. I already took linear algebra in high school. Do I really have to take this course?
    • Many students in MAT201/202 and MAT203/204 have had some multivariable calculus and/or linear algebra before, but rarely with the same depth and thoroughness. If you need the course for upper division courses in your major, then you are probably better off to take linear algebra even though some material will be review.  The sample quizzes and exams posted for this course should give you a pretty clear idea of whether you already know the material taught in this class. 
    • If you can't solve the problems on the sample exams for this course, try the sample final for MAT202. Can you do any of the problems? For most students, the answer will be no. Review your old notes and try again. Can you do at least 60% of the exam?  If not, you really should take a linear algebra course here at Princeton.  It is then a matter of taste and interest which linear algebra course to choose.
    • In rare cases, the placement officer will decide that your prior work is indeed equivalent to MAT202 (or maybe even MAT204) at Princeton. It will be helpful if you can bring your graded exams from the course you took to show the placement officer. He/she may also require you to take an exam to demonstrate your knowledge.
  3. Can I take MAT203 and MAT204 in the same semester?
    • No, they are never offered in the same semester. You can however take them in either order, and you can substitute MAT201 for MAT203 or MAT202 for MAT204 to satisfy most programs or departments that require these courses. Both MAT201 and MAT202 are serious courses and, for most students, it really doesn't matter which version you choose.
  4. If I think MAT204 is too hard, what should I do?
    Of course you can consider switching to MAT202, especially if you know that you don't want to be a physics major.
    If you are a physics major and you did OK in MAT203, try to stick it out, but take advantage of all the help available to you. Talk to your instructor about help options for your particular course. Although MAT202 and MAT204 cover basically the same topics, they may cover them in a quite different order. If you wait too long it may be rather difficult to make the switch.
  5. If I think MAT204 is too easy, what should I do?
    • Have you had a quiz yet? You may be in for a surprise. The general math FAQ page has a whole section on how Princeton's lower division courses work which may give you a more realistic view of the situation.
    • If you are also taking PHY104 or PHY106, you might consider just enjoying the fact that MAT204 is too easy. The combination of a demanding physics class and a demanding math class has been responsible for quite a few academic distress stories at Princeton.
    • Consider taking MAT217 instead, but first check out MAT215. You should have some solid experience writing proofs to do well in MAT217. If you are interested in math or physics as a major, taking MAT215 might be a good choice for you in your first year at Princeton. Talk to your instructor or the placement officer for advice.
  6. If I want to switch courses, what should I do?
    • Talk with your MAT204 teacher. You may also want to talk with the teacher in charge of the other course.  Or consult Jennifer Johnson, the person who supervises most of the lower division courses for the math department.  During the drop/add period it is quite simple to switch.  It can also be arranged after the official drop/add period ends, but the sooner you can decide the better!
  7. I would like to switch sections within MAT204.  What should I do?
    • If you have a time conflict, there is usually no problem. If you want to switch to another section at the same time, be prepared for a possible refusal --- the instructor may not have room for another student.  Consult the department's drop/add procedures.
  8. I can't fit this course into my schedule. Can I take this course for Princeton credit at another university?
    • Yes, but it may be difficult to find an equivalent course. Many linear algebra courses at other universities cover only about half of MAT202 and the equivalent of MAT204 will likely be an upper division course for math majors at many universities.  Upper division courses are rarely offered in summer sessions.   If you find a possible candidate course, you need to apply for summer course approval from the math department.
  9. I have more questions that are not answered here. What should I do?
    • First, check the undergraduate home page for more information about how our courses work in general and about who to contact if you need to discuss your situation with someone from the math department.  Also: representatives from the math department will be available at freshman registration and at the departmental open houses during registration week.