Course: MAT202

Instructor: Sprung

S 2015

### Description of Course Goals and Curriculum

Companion course to MAT201. Matrices, linear transformations, linear independence and dimension, bases and coordinates, determinants, orthogonal projection, least squares, eigenvectors and their applications to quadratic forms and dynamical systems. MAT 202 provides an introduction to Linear Algebra for students with little or background in the subject. Most of the students who take this class are engineers, because it is a prerequisite. For the most part, the topics covered in MAT 202 and MAT 204 overlap. However, MAT 202 is more geometric and tangible, while MAT 204 is more theoretical and proof-heavy. Most students take this course in the Spring, having taken MAT 201 in the fall. However, there is minimal overlap between the two courses and they can be taken in any order. The course begins with an introduction to matrices and their utility in solving systems of linear equations, followed by linear transformations in n-dimensional space, using different coordinates systems and bases. These topics are meant to provide students with a good foundation for the rest of the course. The course then focuses on the application aspects of linear algebra. Topics covered include orthogonality and least squares fit (used to plot a line of best fit, given a set of points), determinants (used to calculate areas/volumes of shapes undergoing transformations), eigenvalues and eigenvectors, quadratic forms, singular values, linear dynamical systems (all used to solve systems of linear equations and even differential equations). For someone who is entirely unfamiliar with the course material, the first few weeks might seem too abstract. The course attempts a geometric approach whenever possible, so that students can tangibly see graphs and solutions to problems. We can all visualise 2 and 3 dimensional space, but what about 5 and 6 dimensions? And how can every axis in these dimensions also be mutually perpendicular? It is concepts such as these that make the course challenging. One has to accept that some problems cannot be visualized and must be verified solely on the basis of algebraic laws. The course progresses at a rapid pace, building on the previous weeks of material. For this reason it is extremely important to know the first few weeks of material exceedingly well, since that provides you with a strong conceptual framework. By the end of the course students should be comfortable using matrices to solve an array of different problems involving linear equations in n-dimensional spaces.### Learning From Classroom Instruction

The class is taught in a series of small classroom lectures (at most around 30 students), each taught by a different professor. Perhaps the most important thing to do is to find a professor with whom you both enjoy learning from and teaches in a style that you find helpful. Definitely take the time at the beginning of the year to sit in on a couple of different professors’ lectures. There are a lot of different professors teaching this course; you will understand some of them better than others. As mentioned above, the class can move quickly and builds off of material introduced at the very beginning of the year. Most often, the class follows a structure of introducing a theoretical topic, providing a proof of why some mathematical formula holds and then examples of how this topic and formula are put into practice. After viewing some examples and seeing how the formulas are put into practice, the proof often makes more sense. One effective note-taking strategy is to write down everything said in class, even if it does not make perfect sense at the time. Draw as many diagrams as possible too, because it always helps to be able to visualise a problem, when possible. Later, revisit these notes and see if they make more sense. In the case where things do not make sense, the professors are open to questions and will do their best to explain. If you do not understand something in class, make sure you ask about it as soon as you can. If you have a question, chances are someone else has that question too, so it can actually be helpful to ask the question out loud. If you really do not want to ask the question out loud, then email the Professor or go to his/her office hours. There is also a problem session held with one of the Professors or TAs every week. These session are not mandatory, but attendance is strongly recommended, because they discuss difficult true/false questions. Quizzes and exams always have a true/false section, which is generally the most difficult section to do well in. In addition, these sessions provide an opportunity for students to ask any other questions they might have. One could think of these sessions as a mix of lecture and office hours.### Learning For and From Assignments

Assignments in MAT 202 consist of a problem set due every week, intended to reinforce the material introduced during lecture. The problem-set questions are from the textbook and mimic many of the examples given in both class and on quizzes and exams. Collaboration is allowed and highly beneficial. Assessments for the class come in the form of two quizzes, a midterm and a final exam. The questions for these assessments are very similar to the examples done in class and those found on the problem sets. Additionally, the class provides previous years assessments that contain problems very similar to those on the actual assignments. Working through the previous years assessments and reviewing notes from class on the topics specified for the assessment is a good way to prepare. There is a true/false section that is unlike the homework problems and can be less directly related to the material introduced in class than the other questions posed. However, using both the examples given from past assessments as well as the weekly true/false problem sessions, it is possible to get an idea of what the section will be like. It is worth noting that some of these exams have equation sheets attached to them as the emphasis is on learning why and how to use the equations rather than pure memorization of them. It is worth clarifying with your professor whether an assessment will have a sheet like this in order to optimize your study schedule.### External Resources

The primary resources for MAT 202 are the lectures the textbook. It is highly beneficial to read the chapters being covered in class before lecture, and the examples in the textbook can be very helpful if you ever find yourself stuck during a problem set. Students also have access to lecture notes and a collection of true/false problems(with solutions) that are discussed each week at the problem session. These resources should be highly utilized by students, since these notes are compiled by the same people who set the exams. As good as the textbook and your professor might be, it is always a good idea to get different perspectives and to learn the same topic using different approaches--especially for the more challenging, abstract topics. Some resources you can use are McGraw Study Hall, Khan Academy and even youtube. This practice helps to clarify and concretise the concepts being learnt. Additionally, the mathematical software language Matlab is designed to manipulate and use matrices to perform mathematical computations. It is worth taking an hour of your time to watch youtube videos or even just go to the Matlab site and read through some of the introductory articles in order to familiarize yourself with the basics of Matlab. The software itself can be downloaded for free through Princeton. Not only is this an invaluable programming language to get familiar with (especially if you plan to go into any sort of engineering) but it also can be used to help solve many of the problems that you will see in MAT 202.### What Students Should Know About This Course For Purposes Of Course Selection

This course is more abstract than MAT 103, MAT 104 and MAT 201, which might make it a little more challenging. It is heavier on conceptual knowledge (the true/false section is intended primarily to test concepts), than preceding math courses. The numerical calculations involved are not the focus; rather, the challenge lies in being able to think a little out of the box and really understanding what is occurring conceptually. Knowledge of the subject material has vast applications to solving systems of linear equations that one might encounter in the real world and more specifically in the field of engineering. It is no surprise that this class is a requirement for all BSE students.Linear Algebra with Applications