2009 March 19 / jj|d|a|v|i|s|@|c|a|r|l|e|t|o|n|.|e|d|u

# Linear Algebra

Carleton College Math 232, Winter 2009, Prof. Joshua R. Davis

## Introduction

In my view, linear algebra is the most important math course. We begin with a simple idea — solving systems of linear equations — but it turns out that this idea pervades mathematics! Just about every higher math course uses linear algebra, and just about every mathematician uses it every day. Others who use math (e.g. physicists, computer scientists) also do a lot of linear algebra. The subject has numerous practical applications — web search engines, error-correcting communications protocols, cryptography, computer graphics, relativity, quantum mechanics, balancing chemical reactions, structural geology, and more. Because so many people use linear algebra, it can be seen from many different viewpoints — matrix/tensor algebra, intersecting hyperplanes, abstract vector spaces, etc. So there is a lot to say about the simple idea of solving systems of linear equations.

The basic course materials are

• Linear Algebra With Applications, 3rd edition, by Bretscher. This is currently the standard Carleton linear algebra textbook. Other editions of the text are not acceptable, because they have different section/exercise numbering.
• Sometimes we'll do exercises using the mathematics software Mathematica. Here are some instructions and notebooks. Remember to right-click (or control-click) on a notebook link to save the notebook to your computer; then open the file in Mathematica.
• Introductory Survey is due at 11:59 PM on the first day of class.
• Here are two pages on splines (see 1.2 #32).
• 232.examples.pdf suggests a bunch of examples for you to work out.
• 232.jordan.pdf describes the Jordan decomposition, which is absent from the textbook.
• Here are our exams.
• Exam 1, Answers (Percentiles: 75th = 70/88, 50th = 59/88, 25th = 44/88)
• Exam 2, Answers (Percentiles: 75th = 35.5/44, 50th = 29.5/44, 25th = 26/44)
• Final Exam (Percentiles: 75th = 58.5/72, 50th = 51.25/72, 25th = 43.75/72)
• Here are the exams from when I taught the course last spring.

Our class meets in CMC 206 during period 2A (MW 9:50AM-11:00AM, F 9:40AM-10:40AM). Here's how you get in contact with me:

Dr. Joshua R. Davis (call me Josh if you like)
E-mail: jj|d|a|v|i|s|@|c|a|r|l|e|t|o|n|.|e|d|u
Office: CMC 327, x4482
Office hours: Sun 12-1, Tue 10-11, Wed 3-4, and Thu 3-4. You can also make an appointment; simply pick a free time from my weekly schedule and e-mail me. You can also talk to me after class.

## Responsibilities

Final grades (A, B, C, etc.) are assigned according to an approximate curving process. By this I mean that there are no predetermined percentages (90%, 80%, 70%, etc.) required for specific grades. The following elements contribute to the final grade.

• Participation: Each class session covers one or two sections of the textbook, which you are expected to read before class. Class will be conducted on the assumption that you have already read and thought about the sections. Attendance is mandatory; furthermore, you are expected to participate actively in group work, discussion, individual exercises, etc. Class participation influences final term grades in borderline cases. You are also required to attend office hours at least once before the first exam.
• Homework: In this course, most of your learning will take place while doing homework. The homework problems are listed on the schedule below. They will be collected once a week. I strongly encourage you to do the homework promptly, rather than waiting for the night before it is due. I may also assign some small writing assignments. Altogether, homework counts for 25% of your grade.
• Exam 1: This is an in-class exam, scheduled for W 1/28, worth 25% of your grade.
• Exam 2: This is a take-home exam, occurring about 2/3 of the way through the course, worth 25% of your grade.
• Final Exam: The final exam takes place Saturday March 14, 3:30-6:00 PM. Self-scheduled final exams are not allowed. The final exam is entirely cumulative and worth 25% of your grade.

## Writing

You are encouraged to work with others on all assignments. Work together to figure out the problems, and then write them up separately, in your own words. You may not copy someone else's work or allow them to copy yours.

Homework is graded for correctness and for presentation. Depending on time constraints, perhaps only a subset of the work is graded; in order to ensure full credit, do all of the assigned problems. Make your paper easy to grade. The problems must be answered in the order they were assigned. Do not write them in multiple columns; just use one column, going down the page. Clearly write each problem's number on the left side of the page. If your paper is messy or disorganized from revisions or erasures, then you may need to recopy it.

How much work should you show? The answer is simple: Write your solutions as if the intended audience is your fellow students. By doing so, you show enough detail that your grader can ascertain whether you yourself understand the material. Your solutions should also be self-explanatory; the grader should not have to refer to the book, to determine whether your solution is correct. In short, if a classmate were to read one of your solutions, then she or he should be able to understand what the problem was and how you solved it.

Staple your week's worth of solutions into a single packet, in the order they were assigned. Packets that are not stapled are unacceptable. I will not accept packets that are not stapled. Staple your packet. Your packet? Staple it.

## Special Accommodations

During the term, you have one free pass to hand in an assignment late. Here is how you activate it. Instead of handing in your assignment, send me e-mail declaring that you are using your late pass and proposing a new due date. If the due date is extended only a couple of days, then no explanation is necessary; if you need longer, then convince me. Use your free pass wisely. Otherwise, no late assignments are accepted, except in extreme circumstances that are truly beyond the student's control.

If some medical condition affects your participation in class or your taking of exams, let me know in the first week of class, so that we can make arrangements. You may want to visit the dean's Disability Services/Resources page first.

## Schedule

Problems marked in bold must be handed in. Do the other problems, but do not hand them in. Remember that this quantity of problems is the minimum that I recommend to pass the course.

DayTopicReadingProblemsDue Day
01 M 1/05solving linear systems1.11.1 #02, 04, 10, 15, 18, 19, 22, 28, 29, 37, 4004
02 W 1/07row reduction1.21.2 #09, 10, 18, 24, 29, 32, 34, 36, 4204
03 F 1/09matrix algebra I1.31.3 #04, 05, 07, 14, 18, 24, 28, 29, 30, 47, 48, 51, 54, 5807
04 M 1/12linear transformations2.12.1 #02, 03, 05, 07, 18, 20, 32, 35, 37, 44, 4807
05 W 1/14geometry of transformations2.22.2 #03, 05, 14, 16, 24, 27, 29, 33, 40, 4207
06 F 1/16inverse transformations2.32.3 #04, 19, 28, 34, 40, 41, 45, 5410
07 M 1/19matrix algebra II2.42.4 #07, 11, 12, 16, 23, 28, 35ab, 39, 44, 57, 72, 84, 8610
08 W 1/21image, kernel3.13.1 #05, 06, 18, 20, 33, 35, 36, 43, 44, 45, 4810
09 F 1/23span, independence3.23.2 #06, 15, 19, 24, 30, 36, 37, 39, 42, 46, 53, 5413
10 M 1/26basis, dimension3.33.3 #16, 23, 32, 35, 43, 44, 47, 52, 61, 62, 6313
11 W 1/28EXAM 1 (DAYS 01-08, SECTIONS 1.1-3.1)
12 F 1/30coordinates3.43.4 #08, 17, 25, 29, 40, 49, 52, 57, 61, 66, 73, 74, 7616
13 M 2/02vector spaces4.14.1 #03, 06, 19, 22, 26, 39, 44, 45, 47, 5416
14 W 2/04linear transformations4.24.2 #09, 17, 26, 33, 48, 55, 62, 72, 74, 77, 78, 8116
15 F 2/06matrices for transformations4.34.3 #02, 05, 15, 16, 19, 54, 59, 7018
-- M 2/09MIDTERM BREAK
16 W 2/11orthonormality5.15.1 #06, 12, 15, 18, 27, 37, 42, 4518
17 F 2/13Gram-Schmidt5.2 (omit QR)5.2 #14, 29, 3321
18 M 2/16orthogonal transformations5.35.3 #06, 09, 10, 15, 16, 27, 30, 31, 38, 42, 52, 5321
19 W 2/18least squares5.45.4 #04, 07, 09, 20, 30, 36, 3921
20 F 2/20inner products5.55.5 #01, 06, 09, 18, 20, 2624
21 M 2/23determinants6.1EXAM 2 IN PROGRESS
properties of determinants6.2EXAM 2 IN PROGRESS
22 W 2/25geometry of determinants6.3 (until Cramer)6.1 #07, 28, 38, 43, 44, 49
6.2 #05, 23, 37, 41
6.3 #04, 14, 18, 19, 20, 46
24
intro to eigenstuff7.17.1 #01, 02, 03, 04, 05, 06, 07, 08, 18, 19, 3424
23 F 2/27computing eigenvalues7.27.2 #12, 22, 29, 34, 4327
24 M 3/02computing eigenvectors7.37.3 #09, 20, 28, 31, 37, 38, 44, 46ab27
25 W 3/04diagonalization7.47.4 #16, 26, 59
232.jordan.pdf #01, 02, 03
27
26 F 3/06complex eigenvalues7.57.5 #19, 25, 32, 3628
27 M 3/09spectral theorem8.18.1 #11, 12, 23, 24, 36never
28 W 3/11Hamming code
-- Sa 3/14FINAL EXAM (3:30-6:00)