Math 204: Linear Algebra


Think-it-through
Think-it-through: Michael Cary
Offered:     	Spring 2016
Instructor:  	Kevin J. Mitchell
Office: 	Lansing 305 
Phone:  	(315) 781-3619
Fax:    	(315) 781-3860
E-mail: 	mitchell@hws.edu


Class:       	M-W-F 11:15 to 12:10 in Napier 201.
             	Final Exam: Tuesday, May 10, 2016 1:30 PM
Text:           Linear Algebra & Its Applications, 4th ed, by David C. Lay

Course Website: http://math.hws.edu/~mitchell/Math204S16/index.php         

WeBWorK:     WeBWorK Home Page for Math 204
             WeBWorK Instructions and FAQs
             WeBWorK Syntax and List of Functions

About Math 204

Math 204 serves as an introduction to the core of the mathematics curriculum. Unlike in a calculus course, students in this course are presumed to have a serious and enduring interest in mathematics. Most students who take this course continue on to major or minor in mathematics or a related field. Correspondingly, there is a seriousness of purpose that I expect in your approach to this course. Generally, students who take Math 204 have done well in their previous mathematics courses. Consequently, the pace is quicker and there are no labs. You will need to be more independent in your study to be successful in this course, for example, you will need to do more problems and create more examples on your own.

The content of Math 204 is used throughout most upper level mathematics courses, whether they are applied or theoretical. The main object of study, vector spaces, is sufficiently general so that many "systems" fall under this category. From physics, you may be familiar with vectors as "arrows" in 2- or 3-dimensional space that represent forces. But there are other real 2- and 3-dimensional spaces, for example, the set of complex numbers or the set of quadratic polynomials, respectively. Further there are also infinite-dimensional vector spaces such as the set of all differentiable functions.

What allows us to call each of these objects a vector space is a shared underlying general structure and it is precisely this common structure that is the focus of Math 204. But what good is it to recognize that a system has the structure of a vector space? The point, of course, is that we (you) will prove theorems about all vector spaces in general that can then be applied to any particular vector space encountered. Once you know a system satisfies the properties of a vector space, lots of other structure automatically follows.

The other key notion in the course is idea of a linear transformation, which is a special type of function. Linear transformations tell us how we can associate the elements or "vectors" of one vector space with those of another. Examples include geometrical transformations such as rotating a plane about the origin or reflecting the plane in a line. Other examples include differentiation, which takes one set of functions and "maps" them to another set, or multiplication by \(\small i\), which takes one complex number and produces another.

Vector spaces and their transformations may be used to model a wide variety of phenomena from how a lumber company should harvest trees in a forest, to how 3-dimensional objects should be drawn on a 2-dimensional surface such as a computer screen, to predicting which team will win the World Series and how many games it is likely to take, to how Google ranks pages for its search results. As time allows, we will investigate various applications.


Goals and Outcomes

The first and foremost goal is to develop a familiarity with certain basic mathematical structures and operations that pervade higher-level mathematics. Specifically, by the end of the course you should be comfortable with the following:

A second goal is to begin to develop the tools and the precise language that is used throughout all advanced mathematics. This language will help you to carry out sophisticated and rigorous analyses and arguments (proofs) in both written and oral work.

This course material is integral to almost all upper-division courses in our curriculum. As such, this course should be your main priority this semester. Consequently, you should expect to spend 10 to 12 hours per week outside of class on work associatied with the course.


Maple Software

During the term I will expect you to do some of your work using Maple. Maple is a computer algebra system that can handle a wide variety of mathematical calculations, not just linear algebra. You will be able to use Maple to check some of your homework answers, though you will still need to show me intermediate hand calculations. Most importantly, Maple will make it possible for you to do more realistic problems that involve many variables and more complex calculations. I expect that you will be able to pick up much of the syntax on your own from the examples that I post online.

The Colleges have a site license for Maple, but the software can only be accessed from a computer on the Colleges' network (e.g., in Gulick, the Library, Rosenberg, Stern).


Prerequisites, Expectations, and Assessment

The minimal prerequisite for this course is Math 131. However, the Department strongly recommends that students take Math 135 (or, perhaps, CPSC 229) before taking this course as preparation for reading and writing proofs. This recommendation is based on the experience of previous students.

Homework reading and practice exercises will be assigned at the beginning of each class. It is extremely important to do the practice problems and I encourage working in small groups on them. This can be very helpful in understanding the material. Come by individually or in small groups for help when you need it. Once or twice a week, there will be an assignment consisting of selected problems to hand in for grading. Unless otherwise stated graded assignments are to be your own work without collaboration. Assignments will be due at the beginning of class. No late assignments, please; they will be marked down. Once in awhile we may have a five-minute quiz on definitions. Any such quiz would be announced in advance.

There will be frequent WebWork online exercises that review the material and concepts we are currently covering. You will get immediate feedback from these exercises that will allow you to assess your progress. Further, if you submit an incorrect answer, you may return to the problem and work on it again until you get it right (assuming it is not a true-false or multiple-choice problem). Students typically earn 90-100% on this part of the course. You may find these problems frustrating at first because you will have to be quite careful in entering your answers. Stick with it!

In addition to the graded homework, there will be three hour tests (during the evening) and a final exam on the dates listed below. The final exam may include a project component using Maple that would be assigned late in the term and would be due at the final exam. The hour tests will be cumulative but will concentrate on more recent material. It is impossible to construct fair makeup exams in mathematics. For your own protection, my policy is that there are no makeup examinations. If for some extraordinary reason you find you are unable to take an exam, let me know as soon as possible, certainly well before the exam is administered.

As prospective mathematicians it is important to participate in a wide variety of mathematical activities. Hence, an additional requirement is to attend two Math/CS Department Colloquia. There will be many talks this semester, as we will be hiring two new faculty. If you attend such a talk, please be sure to fill out the questionnaire concerning the candidate.

Your course grade will be determined as follows. Note the exam days and times that extend beyond the usual class time:

Exam 1. Monday, February 15 (6:30-8:30pm): 16% 
Exam 2. Thursday, March 10 (6:30-8:30pm):  16%
Exam 3. Monday, April 18 (6:30-8:30pm):    16%
Homework (and Quizzes, if any):            24%
Math Colloquium Attendance:                 4%
Final Exam Tuesday, May 10:                24%

Because of the nature of this course, its assignments, and its assessment, your attendance and participation are crucial. Mathematics is learned by regular, sustained, attentive effort over an extended period. Only when such effort has been invested will the concentrated study for an exam have any benefit. Therefore, attendance at class is required. Unexcused absences may adversely affect your grade; certainly more than three absences will lower your grade. If you must miss a class for some reason beyond your control, talk to me about it in advance. Beyond six absences will likely result in your expulsion from the course.

Finally, common courtesy demands that you be on time for class and that you do not leave the room during class (unless you are ill). This will help you, your classmates, and me to give our full attention to the course.


Tips for Success

My best advice is to take good, complete notes during class. Even if you don't understand every detail during a lecture, with some patience you should be able to review each day's lecture and understand everything we did. If you don't, then you should come to see me. Here are a few simple things that you can do to be more successful in the course:

Outline of Weekly Readings

The outline below is fairly ambitious and will be adjusted as necessary during the term.

Dates Reading Topic
Weeks 1-4 Sections 1.1-1.10 Linear Systems and Matrix Transformations
Monday, Feb 15 Test 1 Chapter 1
Weeks 5-6 Sections 2.1-2.3, 2.6? Matrix Algebra
Weeks 7-8 Sections 3.1-3.2 Determinants
Thursday, March 10 Test 2 Material from Chapters 2 and 3
Weeks 9--12 Sections 4.1-4.7 Vector Spaces
Monday, April 18 Test 3 Chapter 4
Weeks 13-14 Sections 4.9?, 5.1--5.3 Markov Chains?, Eigenvalues and Eigenvenctors, Diagonalization
Tuesday, May 10, 1:30-4:30 Final Exam Cumulative

Additional Resources

There is a course website where I will be posting assignments, some "notes" that supplement what we do in class, and the answers to homework assignments (after they are due). Please bookmark this site: http://math.hws.edu/~mitchell/Math204S16/index.php

As noted earlier, I hope that you will learn to use Maple this term to be able to do more complicated applications or projects using this software. Here are some resources which can be accessed easily from links in the online version of this syllabus or at the course website. To use Maple you will need to be logged on to one of the Colleges' networked computers (e.g., in Gulick, the Library, Stern, Lansing 310).

Hobart and William Smith Colleges: Department of Mathematics and Computer Science