Math 251: Matrix Algebra I - Summer 2009

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## Important Notes

• (07/01 - 8:30am) I have just written the final. Here is some info about it:

• It is this coming Thursday (07/02), during regular class time. (So you will only have one hour and a half.)
• I will not answer questions before the exam. (Unless you ask them before 1:30pm, of course.)
• See the notes below on office hours.
• No calculators will be allowed!
• It officially covers all sections, but it will be concentraded on sections 5.3-6, 6.3, 6.5, 7.1-2. (Section 7.3 is not part of the exam!) Of course, you do have to know how to compute determinants, inverses, find reduced echelon form, etc.
• The exam has only five questions. One is to put a large matrix in reduced echelon form worth 10 points, one with thirteen items that can be answered quickly (if you remember the important concepts) which involves various different sections (including older ones) worth 40 points, one computation (from older chapters) worth 15 points, one on chapter 7 worth 15 points, and one long one on chapter 5 (mostly) worth 20 points.
• As I mentioned in class, I will try no have you computing reduced echelon forms over and over again, and give you some matrices in echelon form already to use, so that I can test you on concepts independently of computational mistakes. (That's why there is a question dedicated to computing reduced echelon form.)
• 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 (finding bases, determining linear dependency, finding eigenvalues and eigenvectors, diagonalizing, complete/reduce to bases, transition matrices, orthonormalization, 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!).
• 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 calculations shorter or solutions simpler. Even if you can solve it correctly, there might be a faster way.
• Do as many problems as possible from the book.
• Look the my final from the course I taught in the Fall of 2007 and the final from the course I taught in the Spring of 2009, which is closer to how our exam will be. You can try to do them as a ``practice exams''. But remember: those might be a bit longer than ours, and might involve more from earlier sections. Also, question 4 (which involves inner products different from the Euclidean) and 6(c) (which involves 7.3) from 2007 are not on material that will be in our exam. (On the other hand, in 6(c) there is an orthonormalization, if you want to take a look at it.)
• 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 a few problems (similar to the outline above) to work on as if you were in the exam: no notes, allow only 90 minutes, etc. This might give you an idea if you are prepared.
• (06/29 - 9:40am) I will leave town shortly after our final. My plan is to grade your exams on Thursday afternoon and submit the grades on the same day. If you need or want to see me, it must be on Friday! (Likely around 11am, but we can try to figure out a time, as long as I can meet everyone who wants to see me in a small interval.)

• (06/29 - 9:35am) Unfortunately, I might not be available for office hours on the day of our final (07/02). I will have to leave town the day before and I might not be in town until noon or so.

I will try to get here as soon as possible, but more likely it will be around 12:30pm. If you want to come, between 12:30pm and 1:30pm, write me an e-mail! If I don't get an e-mail until 9am, I will just not hurry and take my time to come back to town. (Keep in mind that even if we make an appointment I might be late. It's a doctor's appointment, and you know how this things can run late...)

I strongly recommend that you start studying early and come to office hours on Wednesday instead of Thursday, though. The exam will be one hour and a half long, so I will not take questions in the first half hour as with the midterms.

I apologize for the inconvenience, but there was no alternative.

• (06/10 - 11:30am) My office hours are now MWTh from 11 to 12. This way you can come the day of the exams to ask questions.

• (05/20 - 12:00pm) Please, check this section often. I will put announcements and important info here.

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### Instructor Contact and General Information

Instructor: Luís Finotti

Office: Aconda Court 211-H

Office Hours: MWTh 11am-12pm 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: MTWThF 1:30pm to 3pm at HBB 103. (Section 1.)

Exams: Midterms: 06/11 (Thu) and 06/25 (Thu), in regular classroom and time; Final: 07/02, also in our regular classroom and time.

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

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### Exams:

#### Midterm I

The exams were graded and will be returned in class on Friday (06/12). (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 81.5, the median was 86 and the highest grade was 100. The grades were divided among very good grades (above 80) and pretty bad (below 60). This exam should be considered an easy one.

I usually post a curve, but I will not this time. People below 60 would like get D's or less, while people above 80 would likely get B's and above.

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#### Midterm II

The exams were graded and will be returned in class on Friday (06/26). (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 72.7, the median was 78.5 and the highest grade was 92. The results of the exam, as expected, were not as good as the previous exam.

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#### Final

Here is the final: Final, Final Solution.

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### Course Information

Firstly, this is a summer course, in which 16 weeks are squeezed into 5. So, as you can imagine, the pace is quite fast. You cannot just ``catch up on the weekends'' in a course like this, as by then we will have covered way too much material. You should catch up immediately if you fall behind, as you will not be able to follow classes and things just start to accumulate in a faster pace than you will likely be able to catch up. I strongly recommend that you review, do problems, and study every day.

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 the selected problems and due dates 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 due Mondays and Thursdays, when you will either turn in your HW or take a quiz on problems taken directly from the HW set due on that day. In the former case, two or three problems will be graded and count the same as a quiz. In the the latter, the HW will not be collected at all. You will not known in advance if the HW will be collected or if there will be a quiz, and hence you should be prepared for either one.

The quizzes will take place at the beginning of the class. You will not have extra time if you are late. You will have only ten or fifteen minutes to take a quiz, and so if you hadn't already solved the HW problems, you might not have enough time to come up with a solutions.

Points will be taken from messy solutions in all assignments, and you need to show work in all questions (unless stated otherwise)! (This same applies to quizzes, HWs, exams and all graded work!)

Calculators will not always be allowed! I've given in the past the students the option of using calculators in second part of the course, but due to how this would affect the exams, they have always chosen to not use them. So, I decided to just take the ``no calculators'' policy from the beginning. (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, HW, 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.

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### 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.

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### Important dates:

Monday, June 1 - Classes begin.

Wednesday, June 3 - Last day to add, change grade options, or drop a full semester course without a "W".

Thursday, June 11 - Midterm I.

Friday, June 19 - Last day to drop a full term course with a "W".

Thursday, June 25 - Midterm II.

Thursday, July 2 - Last Class Day and Final.

More dates.

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### 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, 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, 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, 9, 12, 13, 17.

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

Section 2.3: 2, 3, 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, 13.

Section 3.2: 1(a), (e), 2(a), (c), 3, 4, 5(a), 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, 19, 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, 22(a), 23.

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

UPDATED! Sections 6.1 and after have been updated now (06/26 -- 3:50pm)

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.

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

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### 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:

• (06/30 - 11:00am) Posted solutions for HW7 and HW8. Note that some the file has solutions to some problems which were not assigned.
• (06/24 - 8:35am) Posted solutions for HW4, HW5, and HW6.
• (06/09 - 9:25am) Posted solutions for HW2 and HW3.
• (06/05 - 11:40am) Posted solutions for HW1.

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### Homework

HW1 - Due on Thursday 06/04:

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, 13(a), 14(a), 18(a), 21.

HW2 - Due on Monday 06/08:

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, 15, 16, 17, 23.

HW3 - Not to turn in (Practice for Midterm):

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.

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

Section 2.3: 2, 3, 4, 5, 6, 7, 12, 14(a), (b), 15(a), (b) (this means use only (a) and (b) from 14 in 15), 16.

HW4 - Due on Monday 06/15:

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

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

Section 4.2: 19, 20.

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

HW6 - Due on Monday 06/22:

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, 22(a), 23.

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

HW8 - Not to turn in (practice for the final):

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

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