Description of Course Goals and Curriculum
Companion course to MAT203. Linear systems of equations, linear independence and dimension, linear transforms, determinants, (real and complex) eigenvectors and eigenvalues, orthogonality, spectral theorem, singular value decomposition, Jordan forms, other topics as time permits. More abstract than MAT202 but more concrete than MAT217. Recommended for prospective physics majors and others with a strong interest in applied mathematics. Prerequisite: MAT104 or MAT215 or equivalent.
The MAT 203-204 sequence is an intermediate step between MAT 201-202 (the engineer’s multivariable calculus and linear algebra) and MAT 215-17 or MAT 216-218 (for math majors). The two courses emphasize proof-based mathematics and explore more abstract ideas while also providing concrete applications.
MAT 204 generally follows Penney’s Linear Algebra: Ideas and Applications (3rd edition) in chronological order and covers roughly the same material as MAT 202, but approaches the material from a more theoretical and proof-based perspective. For instance, the course begins introducing the abstract notion of vector spaces as they apply to not only vectors, but also matrices, polynomials and functions. The course repeatedly emphasizes the broader applications of linear algebra to these abstract objects to prepare for applications to Markov processes, the Fourier transform, graph theory, differential equations and quantum mechanics.
MAT 204 begins with vector spaces, Gaussian elimination and various relationships between row space, column space, null space, dimension and linear independence; it then focuses more on matrices and their properties, then more specific vector spaces where structures such as the inner product are defined; in the last few weeks, the course begins to apply the concepts covered to statistical methods such as principal component analysis and various applications named above. In parallel, students are taught MATLAB briefly in precept and through optional problems in problem sets, in preparation for two MATLAB assignments.
Prior knowledge of linear algebra is not required for this course, but the professors move so quickly through the basics of vectors and matrices that it is useful to have prior exposure and experience with these ideas. Students are expected to come to lecture having read relevant textbook readings; problem sets take substantial amounts of time and should not be left to the night before; and exams expect proofs of mathematical statements rather than computation. For a student not accustomed to proof-based mathematics, and even for those who are well-versed with proofs, the course will be challenging and considerably time-consuming. Nevertheless, the course is very rewarding, and students leave the class with very solid foundations for applications of linear algebra in fields beyond vectors and matrices (e.g. quantum mechanics and the Fourier theorem).
Learning From Classroom Instruction
Lectures are designed to supplement textbook readings by covering selected proofs in detail and explaining the motivations behind particular steps, as well as introduce the material. Beside certain topics (such as applications) that are not covered in the textbook and must be learned in class, lectures can be seen as a way of interactively asking for clarification on particular steps in a proof, the approach to a proof, or the place of an idea in the broader context of theory and pure mathematics. Although lectures are not mandatory, it is highly recommended to attend because it will take substantially more time to catch up a missed lecture than to attend it. Precepts are generally used to cover example problems from past exams, but the preceptor will also answer particular questions about the material and introduce some of the material for upcoming lectures. Students should come to precepts having a working knowledge of the lecture material in order to get the most out of this part of the course.
Learning For and From Assignments
There are three forms of graded assessment in this course: weekly problem sets; two quizzes, a midterm and a final; and two MATLAB assignments. In Spring 2015, problem sets accounted for 18%, each MATLAB project 1.5%, each quiz 12%, the midterm 20% and the final 35%.
Problem sets consist of a selection of textbook problems and supplementary problems that highlight the various applications of linear algebra (e.g. image processing, data compression, Lagrange interpolation, spin matrices). Problem sets also usually contain optional MATLAB problems. These are designed to teach students who have little or no prior experience with MATLAB, but even students experienced with MATLAB should attempt or at least read through them because they address aspects that are specific to linear algebra and will help students prepare for the two MATLAB assignments.
Each problem set will take a non-trivial amount of time, particularly because many problems expect the student to think of and write logical proofs – something that cannot be done quickly under time pressure. It is recommended that students collaborate in study groups to approach problem sets, but attempting each problem individually before collaborating will help prepare students for exams for several reasons: firstly, problem sets serve to solidify and consolidate the abstract concepts learned in class (it is important to be able to compute, for instance, eigenvectors, as well as prove relationships between these concepts); secondly, there are tools and techniques introduced in both textbook and supplementary problems that will be very useful in quizzes and exams.
Problem sets are generally based on material covered in class during the week and serve as good consistent preparation for exams. To get the most out of a problem set, students should make note of original ideas or clever tools that proved valuable in approaching a problem. Throughout the week, students could also compile and maintain a list or mind map of concepts and links between them (e.g. “the dimension of the row space = the dimension of the column space = rank”) for a high-level mental map of how various concepts relate to each other. Students should, however, be careful not to cut corners with proofs of theorems in the textbook or in lecture. The initial reaction may be to discard proofs in favor of the main idea of the theorem itself, but as proofs of similar theorems or lemmas are a substantial portion of quizzes and exams, students should consider stepping through proofs carefully and writing them out themselves in their own time. In general, students should aim to gain an intuitive understanding of how concepts relate to each other, and also why, fundamentally, they relate in that way.
Quizzes and exams have a mixture of calculation-based questions (e.g. “diagonalize the following matrix”) and proof-based questions (e.g. “Show that any n x n complex matrix A satisfying A3 = A is diagonalizable”), usually involving one question with several true/false sub-questions (prove if true, find a counterexample if false). Thus there is a mixture of direct application and computation, and synthesis and application to unfamiliar problems. To prepare for these assessments, students should first ensure they conceptually understand processes that will be used in the computational part of the quiz/exam, and be able to quickly and reliably apply them to compute the desired answer. Second, students should also aim to be comfortable with manipulating relationships between concepts for the proof part of the quiz/exam by doing practice problems (in addition to the problem set if this can be managed) and practice exams (a number of past exams are usually uploaded a few weeks before each quiz/exam). Many students in the past have benefited from maintaining a list of important mathematical definitions, theorems and properties, along with lay explanations or analogies of what a definition or theorem is saying in essence, and brief outlines of two or three steps that are critical to the proof of each theorem. This serves both as revision after class to consolidate understanding and as pre-exam study material (exams are closed-book, closed-notes, no calculators). For the T/F questions, students should aim to become familiar enough with the material that they can quickly decide whether or not a statement may be true. They should also be careful not to spend too much time explaining a proof for these questions because they may be pressed for time for the remainder of the exam.
MATLAB assignments are worth a very small portion of the overall grade (although each assignment is essentially as important as one problem set – both about 1.5% of the overall grade) and are not assessed in any quizzes or finals. However, MATLAB is a very useful and important tool in numerous fields from pure and applied mathematics to physics, biology and engineering. Thus, although the assignments are together worth only 3%, time should be taken to understand how MATLAB works and how to use it in linear algebraic (and other) applications.
External Resources
As with any other course at Princeton, office hours are an incredibly valuable resource. A few McGraw study halls are also available for MAT 204, although support is limited. It may be helpful to reach out to friends who have previously taken MAT 204 or friends studying to be math majors to ask specific questions. There are also great online resources (e.g. Khan Academy, although it focuses more on application than proof, lecture notes from other universities’ equivalent courses, the Nipissing University linear algebra help site – http://algebra.nipissingu.ca/tutorials/vectors.html ) that may help clarify anything students are not certain about.
For checking computation on problem sets and practice exams, MATLAB is a very useful resource, although care should be taken to not become too reliant on it, since it is not available during quizzes or exams.
What Students Should Know About This Course For Purposes Of Course Selection
MAT 204 teaches linear algebra with both practical and theoretical outlooks. Students leave the course with a far deeper understanding of the nature of linear algebra and its applications in signal analysis and theoretical physics than students who take MAT 202, but also with the practical linear algebra background for physics and engineering. Thus MAT 204 is particularly good for physics and engineering-track students who appreciate and want to learn more about the theoretical foundations of what they are learning. While not as mathematically rigorous as MAT 215-218, this course (together with MAT 203) prepares students well for higher level math courses such as analysis, algebra and topology. It also serves as a good foundation for the mathematics of quantum mechanics and signal analysis. Students who were concurrently studying this, e.g. through ISC 233-234, ELE 201 or PHY 207, found that this provided them with a deep appreciation for the linear algebraic foundations of this material.
Nonetheless, MAT 204 is a particularly time-consuming class, much more so than MAT 202. The weekly problem sets, if combined with a lot of other coursework, can become relentless and can lead to weekly all-nighters. It is important to remember, however, that staying up all night to work on 1.5% of the overall grade is not as valuable as sleeping and studying more effectively for an exam or quiz worth 12% or 20%. Many students begin linear algebra in MAT 204 and drop to MAT 202 without much difficulty because similar material is covered in the first few weeks of the two courses.
In general, most engineers take MAT 202, some engineers and physicists take MAT 204 (which is nominally recommended for physics and math majors, although only a few math majors take MAT 204), and most math majors take MAT 217. A lot of additional information about MAT 204 and similar linear algebra courses can be found here: http://www.math.princeton.edu/undergraduate/course/mat204