Wellesley College/ Math Dept/CourseHomePage

 

What's in the Course?

The first part of the course, which comprises the material in Chapters 1-4, is in some ways preliminary to the main part of the course. Chapters 1 and 2 cover systems of linear equations, matrices and operations on matrices, determinants, and the relations between these things. (For instance, how can we find the solution set for a system of linear equations? If the determinant of the coefficient matrix for a system of n linear equations in n variables is nonzero, what does this tell us about the system of equations?). Chapter 3 offers a review of Math 205 material concerning vectors in R2 and R3. Chapter 4 previews vector spaces and linear transformations in this same context of Euclidean n-spaces.


The remainder of the course, which begins in Chapter 5, covers the theory of abstract vector spaces and linear transformations, a theory that both is beautiful in its own right and also has applications in virtually all the natural sciences and social sciences and to many problems in industry, finance, and management. You can get an idea of what some of these applications are by browsing the glitzy photos and paragraphs that appear between the index (p xvi) and the text (p1) or, perhaps better yet, by leafing through Chapter 11.

To give you an idea of what abstract vector spaces and linear transformations are, let me quote from the description on the Math Department's web page:

"Whenever one encounters a bunch of gizmos (functions, for instance) that can be added to form new gizmos and multiplied by scalars to form new gizmos, and for which a few familiar rules of algebra hold, one has what's called an "abstract vector space". The gizmos, whatever they may be, are called vectors, or abstract vectors. A linear transformation is a special kind of function from one abstract vector space to another. If you've ever wished that sin(A+B) could just be sin A + sin B and sin(kA) could be k sin A, you were wanting the sine function to be linear.

Math 206 defines "abstract vector space" and "linear transformation" axiomatically and then deduces a very pretty story. Geometry, perhaps unexpected, reappears--and the abstract story that unfolds has some very concrete, computational aspects as well. "

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As the use of the terms "axiomatically" and "deduced" suggests, there is considerable emphasis in this course on proofs. Quoting again from the web-page description:
"Proof is the gold standard of mathematical discourse, and linear algebra is especially well suited for learning to do proofs. Many aspects of the course material are abstract, yet the logic involved is quite direct and the proof paths are often guided by very concrete ideas. In the course of learning some linear algebra, a student who takes this course can learn to read and understand proofs, to judge for herself whether a proof is valid or not, to write proofs clearly and carefully, and to create basic proofs on her own. This emphasis on reasoning skills is valuable for virtually any student, whether or not she intends to major/minor in mathematics."

If you had a proof-oriented Euclidean geometry course in high school, Math 206 may remind you of it, although the subject matter is different. Math 206 satisfies the "proof-course" prerequisite for all 300-level mathematics courses. Since writing proofs and finding your own proofs is something you may not have hand much experience with, emphasis will be given to learning what constitutes a proof, how to properly write a proof, and how to find or create a proof of your own.

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What's required and how much does it count??

Daily work, including homework, classwork, and any quizzes: approximately 15% of course grade
Graded Problems (one or two generally due each Monday, Thursday): about 30% of course grade
Tests (2): about 30% of course grade
Final Examination: about 30% of course grade

Homework will be due most Mondays and Thursdays, as will the graded problems. The graded problems should be submitted separately from the check-off HWK. Although these assignments will usually be due one week after they are assigned, you should plan to do each assignment for the very next Monday or Thursday class meeting, so that you have an opportunity to ask questions on any problems that cause you difficulty. Expect tests at about the 6th week of the semester and the 12th week of the semester. Quizzes, if any, will occur at somewhat random intervals (but will be announced in advance).

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What text? Which sections of the text?

Elementary Linear Algebra Applications Version, 8th edition, by Howard Anton and Chris Rorres, John Wiley & Sons, Inc., New York, 2000.

We will cover most of the (new) material in the first eight chapters and some of the material in Chapters 9 and 11, perhaps some in Chapter 10.

When and where does the course meet?

Class meetings are Mondays, Alt2 Wednesdays , and Thursdays, in SC 268 from 11:10 - 12:20.
The Wednesday meetings are on the following dates: February 6, 20; March 6, 27; April 10, 24.


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How do I contact the instructor?

Office: SC 370

Ext: 3131

E-mail:
For e-mail within First Class, use: Alexia Sontag. Otherwise use asontag@wellesley.edu.

Class conference on FirstClass: named MATH206-S02 . Within the class conference there will be a subconference named "HearYe" that I will use for official announcements. You should check that subconference regularly. There will also be a subconference named Q&A that you can use for any and all questions about the course. I encourage you to check that subconference on a regular basis, as well, and to answer each others' questions when you can. I will also check Q&A frequently and respond to questions there.

 

Open Office Hours (subject to occasional change)

Monday 5:15 - 6:15 p.m.
Tuesday 3-4 p.m.
Wednesday 2:15 p.m. - 3:15 p.m.
Thursday 4:30 - 5:30 p.m.

Office Hours also by appointment and/or chance. If you wish to see me and can't make my office hours, let me know and we'll arrange an appointment.

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What other resources are available?

I anticipate that many or all of you will wish to see me in office hours with questions about the course material, and I enjoy working with individual students, so don't hesitate to come see me. If you find that you wish to see me and can't come during office hours or wish to have an individual appointment, just let me know and we'll arrange something. Here are some additional resources you may wish/need to use.

FirstClass conference: Asking questions on the FirstClass conference for this course (Math206-S02) is a good way to ask quick questions or share ideas, as well as to carry on a more extended discussion about a confusing problem or topic. The main conference will have a subconference titled "Q&A", which is where you should post questions. I will check Q&A fairly regularly and respond to any questions that haven't already been answered by others. There will also be a subconference titled "HearYe!" where I will post announcements for the class. You should be sure to read all announcements posted in the HearYe! subconference.

In Math Dept corridor (outside SC 362): Lots of old linear algebra books. These can be a good source for extra problems or alternate explanations. To sign a book out, just write your name on the posted list.

Other students in the class. Most students find it helpful to form study groups. I encourage you to do this and may require that you work with others on certain assignments.

LTC Tutors: Contact the Learning and Teaching Center to get a tutor assignment form, then see instructor.

Note: Students with disabilities who are taking this course and who need disability-related accommodations should speak with me as soon as possible. Barbara Boger, Director of Programs for the Learning and Teaching Center, and James Wice, Director of Disability Services are available to assist students in arranging these accommodations.

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What can I learn/accomplish in this course?

Mathematics per se. You can learn the basic concepts, facts, and techniques of linear algebra. See What's in the Course? or browse the text for more specific information.

Reasoning. In all of your work in this course, you can improve your reasoning skills, your analytical skills, your proof-writing skills, and your problem-solving skills, all of these both in a specifically mathematical sense but also in a very general sense. You can strengthen your ability to reason logically in mathematics and to write correct proofs.

Working with others on harder problems. I hope that you will work with classmates on some of the assignments and supporting work in this course. In this way, you can acquire some experience working with colleagues or peers and you can work on more difficult problems than might be feasible for you to solve alone.

Working on open-ended problems.By now you will have had considerable experience with what we might call algorithmic mathematics, the sort where you're applying a particular strategy or procedure to a recognizable kind of problem. You have probably had less experience with solving more open-ended kinds of problems, problems for which it may not be at all clear, at first, how to solve the problem. With some of the problems in this course, especially those where you find or create proofs of your own, you can acquire additional experience with the more open-ended situation and thereby broaden your understanding of what it is to do mathematics.

Reading mathematics. In reading assignments from the text, in your work on graded problems, and perhaps in other readings as well, you can practice the skill of, and enjoy the pleasure of, reading mathematics. Edward Rothstein writes (on page 16 of his book Emblems of Mind: The Inner Life of Music and Mathematics) that "reading even ordinary language is complicated, but reading mathematics presents a different order of complexity; it involves a return to the thinking that went into the writing". To return to the thinking that went into the writing, one must read actively, pondering as one reads, solving problems as one reads. Reading mathematics, however, as opposed to listening to mathematics or watching someone else do it or explain it, does have some advantages. One can travel at one's own pace, take a side trip, backtrack, explore on one's own.

Writing mathematics. In your work on the graded problems, especially, you can practice the skill and the art of writing mathematics (and writing about mathematics). Communicating your ideas and your reasoning clearly and accurately/correctly is an important aspect of doing mathematics, as important as "getting a correct answer" or figuring out what makes a proof work. Writing can also be an important tool for learning and for solving problems.

Explaining mathematics. In group work on graded problems, in study groups you form on your own, in class discussions, and when I ask you to present HWK solutions in class, you can practice explaining mathematics to your classmates and orally communicating your ideas and your questions. You can practice using language carefully and structuring oral arguments.

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