Math 251: Matrix Algebra I - Spring 2009

GO TO:

## Important Notes

• (04/27 - 5:10am) The room for the review session (today at 10am) is Aconda Court 113 (at the corner of Volunteer and Cumberland, beside the University Center).

• (04/24 - 10:00am) Here is some info about the final:

• It is this coming Tuesday (04/28), from 10:15am to 12:15pm in our regular classroom.
• We will have a Review Session on Monday at 10am in a room to be announced. (Check this site later.) Keep in mind that I will not prepare a review. I will only answer your questions and solve problems. So, I strongly recommend that you study before you come, so you know what are your difficulties.
• I can take appointments for office hours on Monday, but please try to come to the review and ask your questions there instead! (My schedule might be a bit restricted on Monday.)
• No calculators will be allowed! As usual, also no notes, index cards, books, etc.
• It covers all sections that we covered in the semester. Namely: Sections 1.1-7, 2.1-3, 3.1-3, 4.1-3, 5.1-6, 6.1-3 (see restrictions below), and 7.1-2.
• Note that we have the following restrictions on Chapter 6: the only kind of inner product considered was the Euclidean inner product (i.e., the dot product), so there is not much to Section 6.1 (but you do need to know the properties of the dot product). On Section 6.2, the only thing new (and important) is the orthogonal complement (definition and computation). On Section 6.3, we covered up to pg. 323 (still only considering the Euclidean inner product), but I did show how to get an orthonormal basis for dimensions 2 and 3, and you do have to know that. (You can use your notes, or look at the Gram-Schmidt Process on pg. 324, which is the process for all dimensions.)
• If I ask you to prove some set is a vector space (and it is not a subspace of some vector space), then I will give the definition of a vector space (i.e., the list of properties on pg. 222), so you do not need to memorize it. (You do need to know how to prove something is a subspace, though.)
• The exam has 7 questions, four worth 10 points, one worth 15, one (with four parts) worth 20, and one worth 25 (see below). Most of the questions are computational in nature, and I consider the exam overall straightforward. But, it is a lot of material and you have to know the best ways to compute things, or you might not have enough time.
• The question worth 25 points has 6 parts, but each part should be immediate if you remember the proper results. (Things like the problems in the book that you do ``by inspection''.)
• Of the 100 points, 58 are from new material (Section 5.3 and later). Also, keep in mind that in order to keep the length reasonable, some topics (on which you might have worked very hard when preparing) might not appear in the exam. (Of course, I won't tell you what is there and what is not, as you should learn all the material of the course!)
• To study, I recommend:
• Quickly review your notes and read the book. The most important is to review the definitions, Theorems, and how to compute things (such as, finding bases, orthogonal complements, eigenvectors/eigenvalues, diagonalization, etc.). If necessary, write those in a different sheet of paper so that you don't have to browse your book every time you need to refresh your memory.
• Look at all examples and solved problems (from book and class!).
• Do HW13!!!! There is a lot of material on which you did not have any quiz or HW to turn in. Often this makes students overlook this assignment. But this last part is a very big part of the exam, so you have to be prepared! Look at the Solutions if you need help, but I strongly recommend that you try to work the problems before checking them, so that you realize what are your difficulties and exactly what you were missing. If you have any questions, bring them to the review on Monday.
• Look over the final from my previous M251 course. But be careful, that some topics there are slightly different. More precisely, question 4 deals with a non-Euclidean inner product (but there are some concepts in there that you should know, such as transition matrix, coordinates with respect to an orthonormal basis, etc.). Also question 6(c) deals with orthogonal matrices, which we did not discuss, but look at the orthonormalization of a basis of dimension 2 done in there (since you have to know it).
• Besides the differences mentioned above, note that there were topics left out of the old final that might be in yours!
• Review your old HWs. Redo all problems that you've missed. (You should always learn from your mistakes!) Always look for exactly what went wrong so that you do not repeat the same mistake!
• Look in solutions of solved problems for ideas that can make a solution shorter or simpler.
• Do as many problems as possible from the book.
• When you first start studying, you can look at the book or notes, but by the end, you should be able to do problems without looking.
• Try to choose seven (or so) problems to work on as if you were in the exam: no notes, allow only two hours, etc. This might give you an idea if you are prepared.
• (04/24 - 9:00am) I posted (yesterday) solutions to all the HWs. (Note that the solutions include also problems which have not been assigned.) See the section Solutions to Selected HW Problems below. I also posted the final from the last time I taught 251. It is on the section Final of my Fall 2007 M251 web page.

• (01/02 - 1:00pm) Please, check this section often. I will put announcements and important info here.

Back to the TOP.

### Instructor Contact and General Information

Instructor: Luís Finotti

Office: Aconda Court 211-H

Office Hours: M 3:30pm-4:30pm and W 11:00am-12:00pm or by appointment (subject to change!!)

Textbook: H. Anton, ``Elementary Linear Algebra'', 9th Edition. Wiley, 2004. (The ``Applications Version'' of this book is also OK, but more expensive and unnecessary.)

Prerequisite: Math 142 or Math 148.

Class: TTh 9:40-10:55 at HSS 56. (Section 4.)

Exams: Midterms: 02/17 (Tue) and 03/31 (Tue), in regular classroom and time; Final: 04/28 (Tue) from 10:15am to 12:15pm, also in our regular classroom.

Grade: 25% for quizzes + 20% for each Midterm + 35% for the Final. Note the weight of the HWs!

Back to the TOP.

### Exams:

#### Midterm I

The exams were graded and will be returned in class on Thursday (02/19). (You can check your grades in Blackboard.) If you are not in class that day, you have to come by my office to pick up yours. (I will not take them to class anymore.) The average was 69.24, the median was 81 and the highest grade was 98. I am happy with the results, but keep in mind that this was an easy exam.

Here is how I would curve these grades and the number of people with each grade:

 Grade Range # of people A 90-100 10 A- 87-89 2 B+ 83-86 0 B 79-82 1 B- 77-78 2 C+ 73-76 2 C 68-72 4 C- 66-67 0 D+ 63-65 1 D 60-62 1 D- 57-59 0 F < 57 1

Note: This curve is just to give you an idea of how I curve. In the end I will not even look at this curve, I will just curve all the final averages.

Back to the TOP.

#### Midterm II

The exams were graded and will be returned in class on Thursday (04/02). (You can check your grades in Blackboard.) If you are not in class that day, you have to come by my office to pick up yours. (I will not take them to class anymore.)

The results of the exam, as expect, were not as good as the previous exam. The average was 64.6 and the median was 69.5. The highest score was 95. (You can see more statistics in Blackboard.)

Here is how I would curve these grades and the number of people with each grade:

 Grade Range # of people A 90-100 2 A- 87-89 0 B+ 81-86 3 B 78-80 2 B- 72-77 3 C+ 65-71 1 C 60-64 3 C- 57-59 0 D+ 53-56 1 D 50-52 0 D- 47-49 0 F < 47 4

Note: This curve is just to give you an idea of how I curve. In the end I will not even look at this curve, I will just curve all the final averages.

Back to the TOP.

#### Final

Here is the final: Final, Final Solution.

Back to the TOP.

### Course Information

This is a first course on Linear Algebra. We will study matrix algebra, system of linear equations, vector spaces, linear transformations, and eigenvalues/eigenvectors.

The first two of these topics are of computational nature, while the latter topics will be more theoretical, and hence usually considered harder by the students. Therefore, the initial plan is to go rather fast through the computational material, so that we can go over the most challenging topics on a better pace.

#### Course Structure

We will likely cover the following:

• Chapter 1: sections 1-7.
• Chapter 2: sections 1-4. (Section 4 very briefly.)
• Chapter 3: sections 1-3.
• Chapter 4: sections 1-3.
• Chapter 5: sections 1-6.
• Chapter 6: sections 1-3 (no Gram-Schmidt or QR-decomposition), 5.
• Chapter 7: sections 1-3.

The first three chapters will be covered in a faster pace. Chapter 4 might also be covered superficially and briefly. The last three chapters should be the more challenging ones, I hope to have enough time to cover them in a slower pace.

Also, note that this outline is subject to change slightly without prior notice.

#### Homework and Quizzes

Homeworks will be assigned after every class and will be posted at the section Homework of this page. No paper copy of the HW assignments will be distributed in class. It is your responsibility to check this page often! Besides HW assignments, other important information will be posted here. (Check the section Important Notes often!)

The HWs will be collected on Thursdays. Each HW will have problems from the previous week (Tuesday and Thursday lectures). The problems to be turned in, as well as due dates, will be clearly posted here. The homework turned in will NOT be graded (unless it will replace a quiz -- see below) I will look through them and keep comments for myself which will affect only border line cases when deciding the final grade for the course. (Again, unless it will be counted as a quiz, as explained below.) Borderline students who do not turn in HWs consistently will always receive the lower grade.

Since the HW will not be graded, we will have HW Quizzes every Thursday, at the beginning of the class. You will not have extra time if you are late. The quizzes will consist entirely of HW questions. You will have only ten or fifteen minutes, and so if you hadn't already solved it, you might not have enough time to come up with a solutions. Points will be taken from messy solutions, and you need to show work in all questions (unless stated otherwise)! The same applies to exams and all graded work!

Every once in a while I might grade two or three HW questions instead of giving a quiz. (So, there will be no quiz that day in class.) This will be done whenever I need the extra time or want to better check how well you are doing your HW. The questions will be graded as a quiz (you need to write the solution properly and show work) and will count as a quiz. I will not tell you in advance when that will happen! So, be always prepared.

Calculators will not always be allowed! You will need to know how to perform some computations. Later in the course you might be allowed to use calculators, but you should never depend on it. You can only use your calculator if I explicitly say so! (This includes HW, quizzes, and exams!)

I will do my best to post solutions to the most difficult problems. If I do, they will be posted in this page.

In my opinion, doing the HW is one of the most important parts of the learning process, so the weight for them is greater than the weight of a single midterm, and I will assume that you will work very hard on them.

Also, you should try to come to my office hours if you are having difficulties with the course. I will do my best to help you. Please try to come during my scheduled office hours, but feel free to make an appointment if that would be impossible.

Finally, it is your responsibility to keep all your graded Quizzes and Midterms! It is very important to have them in case there is any problem with your grade. You can check all your scores at Blackboard. (Blackboard will be used only for scores. This is the official site for the course.)

#### Missed Work

There will be no make-up quizzes or exams. If you miss a quiz or exam and have a properly documented reason, your final will be used to make-up your score.

#### E-Mails

You will have to check your e-mail at least once a week, preferably daily. I will use your e-mail (given to me by the registrar's office) to make announcements. (If that is not your preferred address, please make sure to forward your university e-mail to it!) I will assume that any message that I sent via e-mail will be read in a week or less, and it will be considered an official communication.

#### Feedback

I have an On-line Feedback Form where you can anonymously send me your comments and suggestions. I will consider your comments and try to do whatever I can to resolve possible problems before it is too late. So, please, feel free to use it whenever you have any constructive comment or suggestion. (In fact, I would greatly appreciate it.) If you don't want you comments to be anonymous, just send me an e-mail or come by my office and we can discuss the problem.

Back to the TOP.

### Legal Issues

#### Conduct

All students should be familiar and maintain their Academic Integrity: from
Hilltopics 2008/2009, pg. 40:

Study, preparation and presentation should involve at all times the student's own work, unless it has been clearly specified that work is to be a team effort. Academic honesty requires that the student present his or her own work in all academic projects, including tests, papers, homework, and class presentation. When incorporating the work of other scholars and writers into a project, the student must accurately cite the source of that work.

All students should follow the Honor Statement: from Hilltopics 2008/2009, pg. 11:

Honor Statement

``An essential feature of The University of Tennessee is a commitment to maintaining an atmosphere of intellectual integrity and academic honesty. As a student of the University, I pledge that I will neither knowingly give nor receive any inappropriate assistance in academic work, thus affirming my own personal commitment to honor and integrity.''

You should also be familiar with the Classroom Behavior Expectations.

#### Disabilities

Students with disabilities that need special accommodations should contact the Office of Disability Services and bring me the appropriate letter/forms.

#### Sexual Harassment and Discrimination

For Sexual Harassment and Discrimination information, please visit the Office of Equity and Diversity.

Back to the TOP.

### Important dates:

Wednesday, January 7 - Classes begin.

Friday, January 16 - Last day to add, change grade options, or drop a full semester course without a "W".

Monday, January 19 - Martin Luther King, Jr. Day. (No class.)

Tuesday, February 17 - Midterm I.

Tuesday, March 10 - Last day to drop a full term course with a "W".

Monday-Friday, March 16-20 - Spring Break. (No class.)

Tuesday, March 31 - Midterm II, Last day to drop with a WP/WF.

Friday, April 10 - Spring Recess. (No class.)

Friday, April 24 - Last Class Day.

Tuesday, April 28 - Final.

More dates.

Back to the TOP.

Back to the TOP.

Back to the TOP.

### Problems Likely To Be Assigned

This list is subject to change without prior notice. The official assignments will be posted below.

Section 1.1: 1, 2, 3(a), (b), 4(a), (c), 5(a), (c), 6, 11, 12, 13.

Section 1.2: 1, 3, 5(a), (c), 6(b), (d), 7(b), 8(b), 10(b), 12, 14(a), (b), 17, 21, 22.

Section 1.3: 1, 2, 3(g), (j), (k), 5(a), (b), (d), (h), (j), 7(a), (d), (e), 8, 12(b), 13(a), 14(a), 18(a), 21.

Section 1.4: 1(b), (d), 3(d), 4(b), 5(b), 6(a), 7(c), (d), 9(b), 13, 14, 17, 20(b), 21, 29.

Section 1.5: 1, 3, 5, 7(a), (d), 9, 14, 15.

Section 1.6: 2, 7, 11 (you can use any method here), 15, 16, 17, 23.

Section 1.7: 1, 2, 4, 5, 11, 12, 13 (only matrix A), 15, 18.

Section 2.1: 3(a), (d), 4, 7, 12, 13, 17.

Section 2.2: 1(b), 2, 3, 4, 6, 9, 12, 14(a).

Section 2.3: 2, 4, 5, 6, 7, 12, 14(a), (b), 15(a), (b) (this means use only (a) and (b) from 14 in 15), 16.

Section 3.1: 6, 7, 8, 9, 10, 11.

Section 3.2: 1(a), (e), 2(a), (c), 3, 4, 7.

Section 3.3: 1(a), (d), 2(a), (d), 3, 4(a), (c), 5(a), (c), 9, 12, 16.

Section 4.1: 3, 7, 9(d), 11(c), 14(d), (e), 16, 18(b), 20.

Section 4.2: 1, 3, 5, 7, 9, 12, 16, 20.

Section 4.3: 1, 2(b), (c), 4, 5(b), 6(b), 7, 8, 11, 13, 15, 19(a), (c), 22.

Section 5.1: 1, 5, 6, 7, 9, 11, 12, 15, 18.

Section 5.2: 1, 2(b), (c), 3(c), (d), 4(a), (b), (c), 5(b), (c), 7, 8(a), 10(c), (d), 11(a), (b), 13.

Section 5.3: 1, 3(a), (c), 5, 6(a), (b), 8, 9, 11, 13, 15.

Section 5.4: 1, 3(a), (c), 4(a), (c), 7(b), (c), 8(c), 9(b), 10(b), 14, 16, 18(b), (c), 19, 23.

Section 5.5: 2(c), (d), 3(b), (c), 5(b), 6(c), 7, 8(c), 9(c), 11(b), (c), 12(c), 13.

Section 5.6: 2(b), (c), 3(b), (c), 4, 5, 7, 8, 9, 12, 13.

Section 6.1: 3(a), 4(a), 6, 8(b), 9(a), (d), 10(c), 11, 13(a), 14, 16(c), (f).

Section 6.2: 2, 4, 6(a), 8(a), 11, 15, 18(b), (c), 21.

Section 6.3: 3, 4, 6, 8, 9, 11, 13, 14.

Section 6.5: 2, 4, 5, 6, 9, 11.

Section 7.1: 4(a), (c), (f), 6(a), (c), (f), 7, 9, 10, 11, 20.

Section 7.2: 1, 2, 6, 7, 9, 13, 16, 17, 19.

Section 7.3: 1(b), (c), (e), 3, 5, 7, 9.

Back to the TOP.

### Solutions to Selected HW Problems

Please read: I will try to post here a few solutions. The new solutions will be added to this same file. They might come with no explanation, just the ``answer''. If yours do not match mine, you can try to figure out again. (Also, read the disclaimer below!) You can come to office hours if you want explanations for the answers. Be careful that just because our ``answers'' were the same, it doesn't mean that you solved the problem correctly (it might have been a ``fortunate'' coincidence), and in the exams what matters is the solution itself. I will do my best to post somewhat detailed solutions to the harder problems, though.

Disclaimer: I will have to put these solutions together rather quickly, so they are subject to typos and conceptual mistakes. (I expect you to be a lot more careful when doing your HW than I when preparing these.) You can contact me if you think that there is something wrong and I will fix the file if you are correct.

Solutions to Selected HW Problems (Click on ``Refresh'' or ``Reload'' if you don't see the changes!)

CHANGE LOG:

• (04/23 - 9:20am) Added solutions for HW11, HW12, HW13. (The file also contains some problems not assigned.)
• (03/26 - 1:45pm) Added solutions for HW7, HW8, HW9, and HW10.
• (02/27 - 8:30am) Added solutions for HW6.
• (02/13 - 9:15pm) Added solutions for HW4 and HW5.
• (02/12 - 8:10am) Added solutions for HW2 and HW3.
• (01/28 - 1:00pm) Posted solutions for HW1.

Back to the TOP.

### Homework

HW1 - Due on Thursday 01/22:

Section 1.1: 1, 2, 3(a), (b), 4(a), (c), 5(a), (c), 6, 11, 12, 13.

Section 1.2: 1, 3, 5(a), (c), 6(b), (d), 7(b), 8(b), 10(b), 12, 14(a), (b), 17, 21, 22.

Section 1.3: 1, 2, 3(g), 5(a), (b), (d).

HW2 - Due on Thursday 01/29:

Section 1.3: 3(j), (k), 5(h), (j), 7(a), (d), (e), 12, 13(a), 14(a), 18(a), 21.

Section 1.4: 1(b), (d), 3(d).

HW3 - Due on Thursday 02/05:

Section 1.4: 4(b), 5(b), 6(a), 7(c), (d), 9(b), 13, 14, 17, 20(b), 21, 29.

Section 1.5: 1, 3, 5, 7(a), (d), 9, 14, 15.

Section 1.6: 2, 6, 11, 15.

HW4 - Due on Thursday 02/12:

Section 1.6: 16, 17, 23.

Section 1.7: 1, 2, 4, 5, 11, 12, 13 (only matrix A), 15, 18.

Section 2.1: 3(a), (d), 4, 7, 9, 12, 13, 17.

HW5 - Not to be turned in! Just work on them to prepare for the exam:

Section 2.2: 1(b), 2, 3, 4, 6, 9, 12, 14(a).

Section 2.3: 2, 3, 4, 5, 6, 7, 12, 16.

HW6 - Due on Thursday 02/26:

Section 3.1 6, 7, 8, 9, 10, 11, 13.

Section 3.2: 1(a), (e), 2(a), (c), 3, 4, 5(a), 7.

HW7 - Due on Thursday 03/05:

Section 3.3: 1(a), (d), 2(a), (d), 3, 4(a), (c), 5(a), (c), 9, 12, 16.

Section 4.1: 3, 7, 9(d), 11(c), 14(d), (e), 16, 18(b).

HW8 - Due on Thursday 03/12:

Section 4.2: 1, 3, 5, 7, 9, 12, 16, 19, 20.

Section 4.3: 1, 2(b), (c), 8, 11, 13, 15, 22.

HW9 - Due on Thursday 03/26:

Section 4.3: 4, 5(b), 6(b), 7, 19(a), (c).

Section 5.1: 1, 5, 6, 7, 9, 11, 12, 15, 18.

HW10 - Do not turn in!:

Section 5.2: 1, 2(b), (c), 3(c), (d), 4(a), (b), (c), 5(b), (c).

HW11 - Due on Thursday 04/09:

Section 5.2: 7, 8(a), 10(c), (d), 11(a), (b), 13.

Section 5.3: 1, 3(a), (c), 5, 6(a), (b), 8, 9, 11, 13, 15.

HW12 - Due on Thursday 04/16:

Section 5.4: 1, 3(a), (c), 4(a), (c), 7(b), (c), 8(c), 9(b), 10(b), 14, 16, 18(b), (c), 19, 22(a), 23.

Section 5.5: 2(c), (d), 3(b), (c), 5(b), 6(c), 7.

Section 5.5: 8(c), 9(c), 10(c), 11(b), (c), 12(c), 13.

Section 5.6 2(b), (c), 3(b), (c), 4, 5, 7, 8, 9, 12, 13.

Section 6.1: 16(c), (f).

Section 6.2: 2, 4, 11, 15, 18(b), (c), 21.

Section 6.3: 3, 4, 9, 11, 13, 14.

Section 6.5: 2, 3, 5, 6, 9, 11.

Section 7.1: 4(a), (c), (f), 6(a), (c), (f), 7, 9, 10, 11, 20.

Section 7.2: 1, 2, 6, 7, 9, 13, 16, 17, 19.

And that's all!!

PLEASE, HIT ``REFRESH'' (OR ``RELOAD'') IN YOUR BROWSER WHEN VISITING THIS PAGE!!!!!!! I usually get messages asking for the update in the HW when it has already been updated. Since I change this page often, some times the browser don't see the changes. But, if you hit refresh and there is still problems missing, feel free to write me.

If it is already Friday afternoon and there still is a ``More to come'' after the HW assignment due on the coming Monday, write me an e-mail at lfinotti@utk.edu, and I'll update it and let you know.

Back to the TOP.