Prerequisites: Before beginning this class you are expected to have had the equivalent of a second semester course in calculus. It would be helpful for you to have taken multivariable calculus as well, although you should be able to take this course without having had the equivalent of Math 21a.

Sectioning: If you have an email account, log on to the Harvard computer system, then type "ssh section@ulam.fas.harvard.edu" instead of "pine," and follow the instructions. Or if you'd prefer, you can try sectioning by going to the Math Department's home page, http://www.math.harvard.edu and clicking on the sectioning link in the upper right corner of the page. If you are having problems sectioning then please contact Susan Milano via email at milano@math.harvard.edu by noon on Thursday.

Classes, Problem Sessions and Course Assistants: Math 21b is taught in sections which meet three hours per week. The philosophy behind the sections is that it is far better to work on math in smaller groups than in one large, impersonal lecture setting. This gives you a better opportunity to ask questions in class, and interact with your teacher. Make sure you take advantage of this arrangement and try to get the most out of being in these smaller groups. Any questions you ask in class will likely be ones that other students will want answered as well, so get over any hesitation you might have and ask questions as the material is presented. There is no class participation grade, so go ahead and ask away - you won't be penalized for doing this, no matter how trivial or simple you think your questions might seem. Remember, the class is being held for you to learn the material, not just to give you a time to copy notes off of a blackboard, so be sure to get help when you need it and stay involved in your class.

You will also be attending a problem session led once per week by a Course Assistant (CA). Course Assistants grade homework and hold weekly problem sessions. They also attend the 21b classes with you, so you will get to know them well during the semester. The problem sessions are an important part of the course and will be devoted mainly to working problems and reviewing material. Even if you find you're not having difficulty doing the homework problems, you should still make a habit of going to these sessions. A schedule of all of the problem sessions will be posted in the Science Center and on our course web page, so that if you have a scheduling conflict with your particular class section's problem session, then you should still be able to attend another problem session.

Question Center: In addition to class, problem sessions and office hours, the math department runs a Math Question Center in Loker on Sunday through Thursday evenings from 8pm to 10pm. The Question Center is staffed by Course Assistants as well as by graduate students and other teaching staff. This is a good place to meet with other students in your class to do homework. You should feel free to drop in any time you want a bit of help, or if you just want to solidify your basic math understanding by doing some review problems.

Homework: There is no question that the best way to learn math is by doing math. Homework exercises are an essential part of any math course. I know from personal experience that if you just go to a math class and watch the teacher work problems, but don't actually try doing any problems on your own, then there is very little chance you will really learn what is going on. It is also very unlikely that you'll do well on math exams without working through homework problems ahead of time! While doing homework, don’t just crank through computations and write down answers - think about the problems posed, your strategies, the meaning of your computations, and the answers you get. The main point is not to come up with specific answers to the specific problems you're working on, but to develop an understanding of what you're doing so that you can apply your reasoning to a wide range of similar situations. It is very unlikely that later on in life you will see exactly the same math problems you're working on now - so learn the material in such a way that you are prepared to use your general math knowledge in the future, not just how to apply particular formulas for very specific problems.

You're encouraged to form study groups with other students in the class so that you can discuss your work with each other. Your Section Leader or CA can provide names and contact information for everyone in your section to help you out with this. Although you can work together with your classmates, all work submitted must be written up individually. Make sure that even if you do work in groups, that you come away with the ability to explain everything you end up writing up in your homework. You're cheating yourself of learning if you just copy down someone else's answers.

There will generally be two problem sets each week. Assignments will be graded by your Course Assistant and will typically be returned to you at the following class meeting. We will then post solutions to the homework on the course website. Check the solutions so that you can learn from your work. To make it possible to post solutions as soon as possible, and in light of the fact that getting behind in a math class is one of the most uncomfortable things you can do to yourself, homework must be turned in on time. We will make a general policy of dropping your 3 lowest homework grades. Because of this, please do not try to harass your CA into accepting a late homework - the homework policy is a coursewide policy, and it would be unfair if certain CAs were lenient while others weren't. If you want to gripe about homework, please come see me instead! Although it may seem as though you're doing homework all the time at first, most students have been thankful in the end that the course was designed to keep you working throughout the semester - it means less last minute catching-up for you when you're getting ready for the tests.

There will be times when problems for homework will look different from what is discussed in class. For some classes we might ask you to read through a section we will cover in class ahead of time so that when you then see it covered in class, you will be able to follow along much more easily (as opposed to seeing it for the very first time in class). As an incentive to do this prereading, we might ask you to do one or two very straightforward questions from that section for homework, even though it hasn't been covered yet in class.

Also, test and exam questions will be similar to the problems which you have worked on in class and on homework, but only up to a point - they will not simply be copies of problems you have already seen. Be prepared to spend some time thinking during tests, not just spending time busily write down formulas.

Textbook: Our main course book this semester will be Bretscher's Linear Algebra with Applications (second edition), available at the Harvard Coop. This textbook was written by a math teacher who used to teach this very course here at Harvard. Note that the earlier first edition has a number of sections in a different order from the second edition, so it would be a good idea to make sure that you get the more recent second edition version for this semester's class.

Technology: In general, technology is a good thing, but as with everything, sometimes too much of a good thing can lead to problems. With the advent of graphing calculators and mathematical software programs, such as Matlab and Mathematica it is now possible to do an amazing number of things almost instantaneously that would otherwise take hours or days to do by hand. For instance, try multiplying two 12 by 12 matrices together by hand. Computers can help you with your math skills and instincts by reducing the time you spend doing burdensome computations, however they should not be relied on to such an extent that they keep you from developing your own skills. They should be used as an aid, but note that without a good understanding of the underlying mathematical concepts, the computer will quite happily mislead you without your even knowing it.

While we are on the topic of technology, as a general policy, you should feel free to use graphing calculators or computer software during the semester as long as they are used as tools to help you learn and explore math, and not as crutches that keep you from developing your own understanding. To the extent that the main point of the course is for you to develop confidence in your mathematical abilities independently of such tools, we will design the course so that for the most part you won't need to use a graphing calculator to do homework problems. Also, we will not allow the use of calculators on the exams as this puts people with different models of calculators at a possible disadvantage to one another, as well as ends up testing how well you can use a calculator instead of how well you've learned the basic mathematical concepts. We will make sure that the problems on the tests require minimal calculation, to allow you to spend your time demonstrating your mathematical knowledge, not your calculating ability.

Tests: There will be several opportunities during the semester for you to show off your math knowledge. Because of the need to have everyone take tests at a common time, something which is practically impossible to do early during the day, the midterms are both scheduled in the evening. It is your responsibility to let your section leader know as soon as possible of any potential conflicts. It is also generally the case that it is your responsibility to resolve any scheduling conflicts - there are only two of these evening tests during the semester, and they should take precedence over any other obligations that you might have.

First Test: Thursday, October 24^th, from 7:30 to 9:30 pm in Science Center auditorium A

Second Test: Tuesday, November 19^th, from 7:30 to 9:30 pm in Science Center auditorium A

Final Exam: tentatively scheduled for Monday, January 13th, rooms TBA