MAT 211 (Introduction to Linear Algebra) Section 1

Spring 2006

Department of Mathematics, Stony Brook University

The course meets Mondays and Wednesdays in Lgt Engr Lab 152 from 2:20 pm to 3:40 pm.

Instructor's Office Hours: Math Building 3-111: Monday 1-2:10, Tuesday 12:15 - 1:30 and by appt. You are always welcome to contact me by email (dusa at math.sunysb.edu) either to ask a short question or to set up an appointment to see me.

Grader: Mohamed Hassan

Grader's Office Hours: Monday 5--7pm in Physics C117 and by appt.

Textbook:

Bretscher, Linear Algebra with Applications, 3rd Ed., Pearson/Prentice-Hall

(two copies are available on reserve in the Math/Physics/Astronomy Library)

Link to an on line tutorial on Linear Algebra by Avi Goldman. It is worth looking at this from time to time since it contains useful outside links and notes on the topics of the course.

Click here for a link to the CURRENT HOMEWORK. This page also contains links to solutions.

Course Notes:

(posted May 12) I have now posted the grades. In general you did pretty well on the final, and some of you did extremely well; one student even got 120/120! The average was 84.6/120. Distribution: <50: 7; 60-69: 3; 70-79: 1; 80-89: 5; 90-99: 8; 100-109: 7; 110-120: 3. For most people the final grade was based on the total number of points they had earned, but a few people did better on the final than over the semester and so their grade came from the final. Have a good summer!

(posted May 8) Mohamed asks me to tell you that there is a typo on p 7 of the solutions to the final review sheet. E0 should be (1,0,0) not (0,0,1) and E1 should be (1,1,0) and not (0,1,1). E2 is correct. If you see any other inconsistences please email Mohamed at: mrhassan@ic.sunysb.edu. (You can also email me so that I can post a correction.)

(posted May 5) Here are the SOLUTIONS to the problems on the review sheet.

(posted May 1) The FINAL EXAM is on Wednesday May 10 at 2:00-4:30 pm in our usual classroom. q

(posted May 1) I read over the final review sheet and found a few typos -- in Q 5,7, 10 and 13. Nothing too serious, but you might want to look at the new version now posted.

(posted May 1) My OFFICE HOURS for next week: Monday May 8: 1-2 and Tuesday May 9: 2-4 both times in my office.

(posted April 25) I have just posted the last homework. This is a Half Homework due on MONDAY May 1. It should be graded by Wed May 3. This is the last day of classes. I cannot be there, but Mohamed will take a review class based on a review sheet that is posted HERE. I plan to hold some review sessions just before the final; the times will arranged in class soon, and I will post them here.

(posted April 25) Here are the solutions to the Extra Homework. And here is the second midterm.

(posted April 17) Someone pointed out in class today that there is a typo in question 1 on the Extra Homework (make up exam 2): it should be R^3 and not R^4. I can't change the pdf file at the moment, since its source is on my home computer which is out of commission right now.

(posted April 9) Here are the solutions to Midterm 2. Also I posted Homework 12 today.

(posted April 4) Midterm 2 was harder than I meant it to be, and so I have decided to give those of you who did poorly on it another chance. I have written a simplified version which I am assigning as an extra homework. Everyone is welcome to do it (you can use it as extra practice). However, it will affect the grades only of those who got less than 50 on the midterm. For those people I will add 1/3 of your score on this make up to your midterm grade, to a maximum of 50. i.e. your new Midterm grade will be the MINIMUM of 50 and (Old Midterm grade + 1/3 of make up grade.) This make up is due on April 19, the Wednesday after the break. Click here for the assignment. As always, I expect the solutions you hand in to be your own work, though you can get any help you want, including studying the solutions to Midterm 2 (which I will post very soon.)

(posted April 3) I have slightly rearranged the schedule for the last few weeks of semester. I will give 3 classes on Ch 6 and 4 on Chapter 7. The last class will be review.

(posted April 3) I slightly changed the Homework for this week; one of the problems from Sec 5.5 has been deleted. You certainly don't need to read the whole section to do the remaining problem; look at defs 5.5.1 and 5.5.2 and examples 2 and 4.

(posted April 3) The scores for this Midterm were significantly lower. Average grade: 52.5 I will post solutions very soon. Please come to office hours if you have questions. (I am willing to make special appointments with students who cannot manage the regular times.)

Distribution: < 20: 5; 20-29: 3; 30-39: 4; 40-47: 0; 48-59: 5; 60-69: 4; 70-79: 2; 80-89: 2; 90-99: 4.

Grade cutoffs: < 30: F; 30-47: D; 48-59: C, C+; 60-79: B-,B,B+; 80-89: A-, 90-99: A.

(posted March 31) Are any of you interested in being an undergraduate math TA next semester? Applications forms are in the Math Undergraduate Office; the interviews will be soon. For the first semester you work for 3 credits (taking MAT 475), but in subsequent semesters you get paid. If you are interested, it is much the best to start in the Fall; there are usually no openings in the Spring.

(posted March 30) I decided to make the next HW a half HW. It will be posted today. I haven't graded the exam yet, but it's obvious you found it harder than the first one. Hang in there! The grade cut offs will be adjusted to make the exam fair. And those of you who know you did badly, please COME to CLASS and DO the Homework -- it really makes a difference. It's never too late to start. We will begin the determinant on Monday. This is completely new material, not related to Ch 4 or Ch 5.

(posted March 28) ANOTHER TYPO on Review sheet. In 5(iv) I should ask for the matrix of T, not its coordinates. Here are the Solutions to Review sheet.

(posted March 22) Here is the REVIEW SHEET for the second midterm. We will discuss it in class on MONDAY. Note that it has a typo: Definition 5.1.3 should be Def 5.3.1. Note also that there is a half Homework due on Monday (5 short questions on Sec 5.3).

(posted Mar 6) IMPORTANT NOTICE ABOUT FINAL EXAM The final is on Wed May 10 from 2-4:30 in the usual classroom. The time posted on the departmental syllabus was incorrect.

(posted Mar 6) The class did Midterm 1 very well. Average 70, distribution: 0-39: 6; 40-49: 1; 50-59: 3; 60-69: 3; 70-79: 9; 80-89: 7; 90-100: 10. Congratulations! Very rough grade equivalencies: < 50 = F; 50-59 = D,D+; 60-69 = C; 70-89 divided equally among C+ to A-; 90-100: A.

(posted Feb 27) Mohamed pointed out that my solution to sec 3.1 number 22 is wrong (page 2 of sols to HW5). The matrix $A$ has rank 2: the second row in the reduced form should be 0 5/2 -5/2 which is a multiple of the last row. In fact the second row is 5/4 times the last row. Hence there is a solution iff (b_2-3/2 b_1) = 5(b_3 - 3 b_1)/4; ie iff 9 b_1 + 4 b_2 - 5 b_3 = 0. I hope this is correct now.

(posted Feb 2) Here are some notes on vectors that should help with the homework this week. I will probably add to these notes over the weekend.

(posted Jan 25) The most important thing for you to do right now is to review/learn the geometric properties of vectors in 3-space. You need to know the parametric equations for a line, a little about the dot product (i.e. two vectors are perpendicular if their dot product vanishes), and how to determine a plane in 3-space by a point on it and the normal vector. (Don't worry at this stage about how to use the dot product to calculate other angles. That'll come up later.) This is all explained in the on-line text; Ch 1, section 2. I recommend you do all exercises in Sec 2.1 and Ex 2:10, 2:11, 2:13, 2:14 from sec 2.2.) You might also do ex 34 and 35 from p 22 (sec 1.2) of Bretscher. Almost any first year calculus text that does more than differentiation and integration (so is not a "Brief calculus") has a chapter on vectors. Multivariable calculus texts also usually have a chapter on vectors. Wikipedia also has useful information: look up the words: vector, plane.

(posted Jan 28) The homework due 2/1 is now posted. You can find find more notes about vectors, lines and planes on the website of the other section.

Fred Girao, the TA for section 2, has office hours in MLC (Math Learning Center) on Tuesdays 1-2pm. The MLC itself is open M-W 10-9; Th 10-6; F 10-2.

Prerequisites/Corequisites:

You are assumed to have had at least one semester of calculus. If you have not yet studied integration, you should be taking the relevant calculus course (e.g. MAT 126) concurrently with this one, as some important problems and examples in this course require a knowledge of integration.

The Nature Of The Course:

This course is an introduction to the theory which has developed around the solution of systems of linear equations. The importance of this theory as a tool in the social, natural, and mathematical sciences cannot be overestimated. (To get some idea of why this is the case, click here, check out the links of interest below, or take a look through your textbook.) You should keep this in mind throughout the semester, especially if the course material ever seems "too weird" or "too abstract" to be useful.

You should also keep in mind that this course is quite distinct in nature from others you have taken. Regardless of your performance in previous math courses, do not be discouraged if you find yourself wrestling with a problem or a concept for hours. Doing many computations is essential to understanding the material, but mindlessly applying memorized techniques while ignoring their theoretical framework will not get you very far. It helps to be proactive in analyzing and even creating examples (not necessarily complicated ones!) that illuminate the theory.

Course Format:

You will get most out of the classes if you prepare beforehand by reading the relevant section in the textbook before class. I am always glad to answer questions during class. Since this class has no recitations, I will aim to set aside some class time each week for doing examples and discussing the homework. If you have more questions, please talk to me after class or come to my office hours (or go to the Math Learning Center in Math Building S-240A.) There will be a review session before each exam, to be scheduled later.

Some links of interest

A nice expository paper on the use of linear algebra in search engines.

A useful online linear algebra text with many worked examples and exercises with solutions.

Homework:

As always in a mathematics class, doing the homework is an essential part of learning the course material. It is often good to study together with other students. However the work you hand in must be written in your own words, not copied from someone else. The homework assignments are listed here. Due dates for homework assignments are listed in the schedule below. They are due at the beginning of class on their due date, typically Wednesday unless it just before a test. (The first one (a half homework) is due on Wednesday January 25!) Submissions consisting of multiple pages must be stapled together. If you cannot get to class, hand them in to my office (Math 3-111) before class. Late homework will be penalized severely (by at least 20%), and will not be accepted if it is too much overdue. The instructor will hand the homeworks back at the beginning of class. The grader has the final say on all homework grades.

If all goes as planned, 11 full homeworks and 3 half homeworks will be assigned during the semester. Your total homework grade will be the sum of the grades of your best 10 homeworks. (One of these might consist of two halves.) Each full homework will be graded on a 20-point system as follows. Five problems (not necessarily the five most difficult ones) will be graded completely, and they each account for 3 points. In order to receive full credit for your solutions to these, each step must be stated clearly and in the correct order, and each statement in English must be a complete and correct sentence. The remaining 5 points are accounted for by a substantial attempt at solving all the remaining problems; a penalty of at least 1 point will be incurred for each one which has not been adequately attempted. A sequence of relevant calculations and/or a list of relevant ideas counts as an adequate attempt, whereas leaving a blank or writing "I don't know" does not.

Examinations:

Please note that examinations are not graded on a curve, nor are the final grades curved. Thus you are not competing against your fellow students for a limited number of top grades. There will be two methods of computing the final grade, and you will get the better of the two. One is a weighted average of your work throughout the semester; the other is based entirely on the final, but your grade here will be somewhat discounted. Thus in order to get a C on the basis of the final alone you would have to get at least a C+ on the exam. If at any point in the semester you are seriously concerned with your standing in the class, you are invited to discuss your concerns in detail with me.

Any use of cellphones, calculators, books, or notes while an exam is underway will be considered cheating. If you miss an exam for an acceptable reason and provide me with an acceptable written excuse, the relevant exam will be dropped in computing your course grade. A letter stating that you were seen by a doctor or other medical personnel is not an acceptable document. An acceptable document should state that it was reasonable/proper for you to seek medical attention and medically necessary for you to miss the exam (for privacy reasons the note/letter need not state anything beyond this point). Incomplete grades will be granted only if documented circumstances beyond your control prevent you from completing 50% or more of all class assignments.

Grading:

Your raw grade will be based on your examination performance and homework, weighted as follows:

Exam I	25%
Exam II	25%
Final Exam	30%
Homework	20%

The grade you receive in the course will be the maximum of your raw grade and 90% of your final exam grade.

DSS advisory:

If you have a physical, psychological, medical, or learning disability that may affect your course work, please contact Disability Support Services (DSS) office: ECC (Educational Communications Center) Building, room 128, telephone (631) 632-6748/TDD. DSS will determine with you what accommodations are necessary and appropriate. Arrangements should be made early in the semester (before the first exam) so that your needs can be accommodated. All information and documentation of disability is confidential. Students requiring emergency evacuation are encouraged to discuss their needs with their professors and DSS. For procedures and information, go to the following web site http://www.ehs.sunysb.edu and search Fire safety and Evacuation and Disabilities.

Schedule (tentative):

The following is the basic syllabus. Please read the relevant parts of the book before class.

Day of	Homework due	Sections Covered
January 23		1.1 (Introduction to Linear Systems)
January 25	Half Homework 1	1.2 (Matrices, Vectors, and Gauss-Jordan Elimination)
January 30		1.3 (On the Solutions of Linear Systems; Matrix Algebra)
February 1	Homework 2	2.1 (Introduction to Linear Transformations And Their Inverses)
February 6		2.2 (Linear Transformations in Geometry)
February 8	Homework 3	2.3 (The Inverse of a Linear Transformation)
February 13		2.4 (Matrix Products)
February 15	Homework 4	3.1 (Image and Kernel of a Linear Transformation)
February 20		3.2 (Subspaces of R^n; Bases and Linear Independence)
February 22	Homework 5	3.3 (The Dimension of a Subspace of R^n)
February 27	Half Homework 6	3.4 (Coordinates); Review
March 1		Exam I (on everything from 1.1 up to and including 3.3)
March 6		4.1 (Introduction to Linear Spaces)
March 8	Homework 7	4.2 (Linear Transformations and Isomorphisms)
March 13		4.3 (The Matrix of a Linear Transformation)
March15	Homework 8	5.1 (Orthogonal Projections and Orthonormal Bases)
March 20		5.2 (Gram-Schmidt Process and QR Factorization)
March 22	Homework 9	5.3 (Orthogonal Transformations and Orthogonal Matrices)
March 27	Half Homework 10	5.5 (Inner Product Spaces)
March 29		Exam II (on everything from 3.4 up to and including 5.3)
April 3		6.1 (Introduction to Determinants)
April 5	Half Homework 11	6.2 (Properties of the Determinant)
April 17		6.3 (Geometrical Interpretations of the Determinant; Cramer's Rule)
April 19	Homework 12	Ch 7.1: Dynamical systems and eigenvectors
April 24		7.2 (Finding the Eigenvalues of a Matrix)
April 26	Homework 13	7.3 (Finding the Eigenvectors of a Matrix)
May 1		7.4 (Diagonalization)
May 3	Homework 14	Review
May 10		Final Exam (Cumulative)