Course MAT202

Introduction to Linear Algebra

Introduction to linear algebra, mostly in real n-space. Companion course to MAT201. Introduces more algebraic methods needed to understand real world questions. Whereas MAT201 develops calculus in a multivariable setting, this course develops fundamental algebraic tools involving matrices and vectors to study linear systems of equations and Gaussian elimination, linear transformations, orthogonal projection, least squares, determinants, eigenvalues and eigenvectors and their applications.  Emphasizes concrete computations over more theoretical considerations.   Because this is an algebra course, it is necessarily more abstract than MAT201, and requires some general arguments and consideration of exceptional cases on exams, mostly in the form of true/false questions, an intermediate between formal proofs and concrete computational questions.

Topics

Matrices, linear transformations on real n-space, linear independence and dimension, bases and coordinates, determinants, orthogonal projections, least squares, eigenvalues and eigenvectors and their applications to quadratic forms, dynamical systems and differential equations. Complex eigenvalues and eigenvectors are also covered in the 2 by 2 and 3 by 3 cases.

Description of classes

Classes meet 3 times per week, for 50 minutes.  Sections are generally offered MWF at 9, 10, 11 and 12:30 in both semesters.

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 70%.  These exams are graded by all the instructors and graduate student AI's together to ensure uniformity across all sections. Typically there are two in-class quizzes, the same for all sections. Homework and quizzes together usually account for about 30% 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, 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.

Notes
  • MAT201 and MAT202 can be taken in either order, but we recommend you take MAT201 first. The course does not use much calculus, and although it treats many of the same topics as MAT201, it does so more algebraically. Either treatment can reasonably come first.
  • The least abstract and most computational of our three introductory linear algebra courses (MAT202, MAT204, and MAT217); provides a very solid introduction to the subject sufficient for most future engineers and scientists.
  • MAT175 is intended for students who will not take futher mathematics courses at Princeton but in rare cases it may be possible for a highly motivated student who received a grade of at least B+ to attempt MAT202 afterwards.  Such a student should expect to work extremely hard in order to succeed as this course demands much greater mathematical maturity than does MAT175.  (Students who have already taken MAT175 should not take MAT201.)
  • Although MAT104 is listed as a prerequisite, the course does not require much calculus knowledge, although prior experience with vectors is very useful. This course does however require strong motivation and some mathematical maturity; students who are not following the standard science/engineering tracks should use caution.
Who Takes This Course
  • The typical student is an incoming freshman or sophomore with plans to major in engineering or one of the sciences; however, many other students with quantitative interests (e.g. economics or finance) take this course, especially those with possible graduate work in mind. It gives a solid introduction to linear algebra suitable for most students who want to use mathematics as an analytic tool in later studies in other fields.  Most students in MAT202 in the fall semester are sophomores who took MAT104  and MAT201 in their freshman year. Most students in the spring are freshmen continuing from MAT201
  • Students who consider a major in physics should take MAT204 or MAT217 instead, as do many of the more mathematically inclined future scientists and engineers.  These are better linear algebra choices for students who plan to take 300-level math courses here at Princeton. 
  • Prospective math majors should probably take MAT217 instead (after MAT214 or MAT215). Some students who are more interested in applied math opt instead for MAT204.
  • Students interested in econometrics should take this course instead of (or in addition to) MAT175.
  • AB COS majors are not required to take MAT201. They need only MAT202 (or MAT204 or MAT217).
Placement and Prerequisites

 While MAT201 is normally taken first, it is not a prerequisite. The main requirement is the maturity and self-reliance that students usually learn there. Strong interest in thinking rigorously about problems, rather than learning cook-book type algorithms is required.

Sample Material

Once you have tried the problems, you can check your answers: Quiz One Solutions, Midterm Exam Solutions, Quiz Two Solutions, Final Exam Solutions.

Additional Sample Finals (without solutions):

FAQ
  1. I already took linear algebra in high school, do I have to take this course?
    • Many students in MAT202 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 MAT202 even though some material will be review.  And if you really love math, you might be able to consider taking MAT204 or MAT217 instead.
    • Take the sample final. 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?
    • In rare cases, the placement officer will decide that your prior work is indeed equivalent to MAT202 at Princeton. It will be helpful if you can bring your graded exams from the course you took to show him/her. You may also need to take an exam to demonstrate your knowledge.
  2. Can I take MAT201 and MAT202 in the same semester?
    • It is not impossible, but we do not recommend it. It makes midterm week particularly unpleasant, but if you have a very good reason for it and you are a very strong student, it can be done. It will likely mean that you will get a lower grade in one of them that you would otherwise have done.
  3. 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 3 hours to complete, although this can vary quite a lot depending on your background and goals.  To do well on exams, you need to work through a lot of extra problems. All in all you should be ready to spend up to then hours per week working outside of class.  Because the material is more abstract than a calculus course, it often takes more time to digest the new material. Last minute cramming is especially unwise for this course.
  4. 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 202.  See our detailed guidelines for summer courses.
  5. I have more questions that are not answered here. What should I do?
    • First, check the undergraduate home page or the general FAQ 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.  In addition, representatives from the math department will be available at freshman registration.