Math 251: Matrix Algebra I - Summer 2010

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## Important Notes

• (07/03 - 5:20pm) Section 6.3 will not be in the exam. I will remove it from the HW assignment as well.

• (07/03 - 5:10pm) I have just written the final. Here is some info about it:

• It is this coming Wednesday (07/07), 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.)
• I will have office hours also Wednesday from 11:00 to 12:00, besides the usual ones on Tuesday. I can also talk to you after class on Tuesday if you need.
• No calculators will be allowed!
• It officially covers all sections, but almost half will be on material after 5.3. From Chapter 6 the most important part is how to compute orthogonal complements (Section 6.2). From Chapter 7, the most important part is how to compute eigenvalues and eigenspaces (Section 7.1).
• The exam has only six questions. One is to put a large matrix in reduced echelon form worth 10 points, one with ten items that can be answered quickly, including some direct simple computations, which involves various different sections (including older ones) worth 40 points, one computation (from older chapters) worth 10 points, one on chapter 7 worth 10 points, one about subspaces worth 10 points, and one long one on chapter 5 (mostly) worth 20 points.
• As I mentioned in class, I will try not to 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.)
• This exam is very computational and so more straight forward. The question on subspaces is simple and similar to your HW.
• 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 of row/column/null spaces, determining linear dependency, finding eigenvalues and eigenvectors, completing/reducing sets to bases, finding coordinates with respect to bases, 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 at my old finals and midterms, as I observed below.
• 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.
• (07/02 - 7:35pm) You can check my previous finals to study: final from Fall 2007, final from Spring 2009, final from Summer 2009. Here are the problems you should not look at. From Fall 2007, skip 4, 5, 6(c), (d). From Spring 2009 skip 2(e), (f), 5, and the part about "decide if A is diagonalizable". From Summer 2009 skip 3(f), (g), (k), and 4(c). (You can also look at the past midterms!)

• (07/02 - 5:00pm) I posted the HW (see HW section), posted all solution (see Solutions section) already. The (comprehensive) final will cover only the material of the HWs assigned. I will post more information about it soon, but you should start studying right away!

• (06/29 - 5:00pm) Information about the results of the second midterm was posted below. Check the section Midterm II below.

• (06/01 - 8:00am) 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: MTuTh 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: MTuWThF 1:30pm to 3pm at HSS 56. (Section 1.)

Exams: Midterms: 06/15 (Tue) and 06/29 (Tue), in regular classroom and time; Final: 07/07 (Wed), 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 Wednesday (06/16). (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 76.9, the median was 75.5 and the highest grade was 92. (You can see more statistics at Blackboard.) The grades were concentraded in the 70 to 80 range. So the result was good, but not great, as I was expecting.

As far as curving goes, I would not change much from the standard values. Roughly, 85 and above will be the As range, 75 and above would be Bs, 70 and above would be Cs.

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.

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#### Midterm II

The exams were graded and will be returned in class on Wednesday (06/30). (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 63.7, the median was 63 and the highest grade was 100. The results of the exam, as expected, were not as good as the previous exam.

To give a rough idea of how I would curve these grades, an A would be about 83 and above, a B about 69 and above, a C about 60 and above, and a D about 50 and above.

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.

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#### Final

Here is the final: Final, Final Solution. The average was 78.42 and the highest grade was 100. (You can check your grades in Blackboard.)

Note that the finals will not be returned, since they have to be kept on file. If you want to see your final or check your average, make an appointment.

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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 know 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 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 2009/2010, pg. 40:

The university expects that all academic work will provide an honest reflection of the knowledge and abilities of both students and faculty. Cheating, plagiarism, fabrication of data, providing unauthorized help, and other acts of academic dishonesty are abhorrent to the purposes for which the university exists. In support of its commitment to academic integrity, the university has adopted an Honor Statement.

All students should follow the Honor Statement: from Hilltopics 2009/2010, 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:

Thursday, June 3 - Classes begin.

Monday, June 7 - Last day to add, change grade options, or drop without a "W".

Tuesday, June 15 - Midterm I.

Monday, June 21 - Last day to drop with a "W".

Tuesday, June 29 - Midterm II.

Monday, July 5 - Independence Day (no class).

Wednesday, July 7 - 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.

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:

• (07/02 - 4:45pm) Posted solutions for HW8 and HW9. (Note that there are more problems in the solution than assigned.)
• (06/25 - 5:25pm) Posted solutions for HW7.
• (06/25 - 9:10am) Posted solutions for HW6.
• (06/21 - 4:45pm) Posted solutions for HW5.
• (06/18 - 6:05pm) Posted solutions for HW4.
• (06/11 - 4:50pm) Posted solutions for HW3.
• (06/10 - 3:55pm) Posted solutions for HW2.
• (06/07 - 4:20pm) Posted solutions for HW1.

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### Homework

HW1 - Due on Monday 06/07:

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).

HW2 - Due on Thursday 06/10:

Section 1.2: 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.

HW3 - Not to be turned in (due to exam):

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.

HW4 - Due on Thursday 06/17:

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.

HW5 - Due on Monday 06/21:

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).

HW6 - Due on Thursday 06/24:

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.

HW7 - Not to be turned in (due to exam):

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).

HW8 - Due on Friday 07/02:

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.

HW9 - Not to be turned in:

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.

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).

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

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

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

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

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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