Math 251: Matrix Algebra I - Fall 2010

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## Important Notes

• (12/02 - 11:35am) Posted solution to a couple of problems. See Solutions to Selected Problems below. (I will not post it more here. Just check the section often.)

• (12/01 - 4:40pm) I have just written the final. Here is some info about it:

• It is this coming Monday (12/06), from 2:45 until 4:45.
• I will have office hours from 10:00 to 11:00 on Friday (12/03) and Monday (12/06, day of the exam). If those times don't work, you can write me to make an appointment.
• No calculators will be allowed!
• It will cover sections 1.1-7, 2.1-3, 3.1-4, 4.3-4.9, 5.1-2. Note that sections 4.1 and 4.2 will not be in the exam. Also, very little from Chapter 3 will be in it, and most of the exam is about sections after 4.3.
• The exam has nine questions. One is to put a large matrix in reduced echelon form worth 10 points, one with five items that can be answered quickly, including some direct simple computations, which involves various different sections (including older ones) worth 10 points, one from chapters 1 and 2 worth 10 points, one from chapter 5 worth 10 points, and five from chapter 4, adding up to 60 points.
• As I mentioned in class, I will try to not 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.) But sometimes this is impossible, so be careful with your calculations!
• This exam is very computational and so more straight forward than the previous one, although that one had very familiar problems (from HW and old exams).
• Make sure you know all the easier computational methods: solving systems, matrix operations, inverting matrices, computing determinants, finding coordinates of a vector with respect to a basis, find transition matrices, linear dependence/independence, basis for spaces (like row space and column space, nullspace), finding eigenvalues, eigenvectors, and eigenspaces, diagonalization, etc. (Just to name a few!) Also remember the shortcuts for these (if any)!
• As with the previous exam, to try to avoid simple algebra mistakes, I ask in some (not all) questions for you to set up the corresponding system (without solving it) and to just tell me how the solution would answer the problem.
• 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 (as mentioned above). 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.
• Do HW12, even though you will not turn it in.
• 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 two hours, etc. This might give you an idea if you are prepared.
• You can do most of my old finals (and midterms!). Here are the exceptions (on finals):
• (08/17 - 9:00am) Please, check this section often. I will put announcements and important info here.

• (08/17 - 9:00am) Please, hit the ``Refresh'' or ``Reload'' button on your web browser every time you visit this page, so that you can see the most recent updates.

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### Instructor Contact and General Information

Instructor: Luís Finotti

Office: Aconda Court 211-H

Phone: 974-1321 (please do not ask me to call back -- leave your e-mail)

Office Hours: Tu,Th 10am-11am or by appointment (subject to change!!)

Textbook: H. Anton, ``Elementary Linear Algebra'', 10th Edition. Wiley, 2010. (The ``Applications Version'' of this book is also OK, but more expensive and unnecessary.)

Prerequisite: Math 142 or Math 148.

Class: Tu,Th 2:10pm to 3:25pm at HBB 124. (Section 1.)

Exams: Midterms: 09/28 (Tue) and 11/02 (Tue), in regular classroom and time; Final: 12/06 (Mon) from 2:45pm to 4:45pm also in our regular classroom.

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 Thursday (09/30). (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 71.54, the median was 71.5, and the highest grade was 97. (You can see more statistics at Blackboard.) The grades were reasonably spread out, with a few very good grades. So the result was not as I was expecting, but not too bad.

As far as curving goes, roughly we would have: 90 and above would be the A's range, 78 and above would be B's, 58 and above would be C's. (So, the cut offs for Cs was lowered much more than the ones for A's and B's, to obtain a more even distribution of grades.)

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 Thursday (11/04). (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 79.3, the median was 82.5 and the highest grade was 91. The results of the exam were good, better than the previous exam.

To give a rough idea of how I would curve these grades, an A would be about 90 and above, a B about 80 and above, a C about 67 and above, and a D about 57 and above. (This is pretty close to the usual cut-offs.)

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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### 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 1-3.
• Chapter 3: sections 1-3.
• Chapter 4: sections 1-10.
• Chapter 5: sections 1-2.
• Chapter 6: sections 1-2, maybe 3 with no QR-decomposition.
• Chapter 7: sections 1-2, if time allows.

The first three chapters will be covered in a faster pace. I will slow on chapter 4, as the subject gets more theoretical and difficult.

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 on Tuesdays, 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.

If you like to do your HW early, you can do problems in the list of Problems Likely To Be Assigned below, even before I assigned them. I might change a problem or two for the actual assignment, but that is not very likely. I hope to be able to keep this list ahead of the class, so that you can always start early. It will also be useful if I, for some reason, delay to post problems.

Also, some times I might get too ambitious in posting problems, i.e., I might think we will cover a section during the week, put exercises from it in the next assignment, and then end up not being able to finish it. In this case I might have to take a few problems off the assignment. The bottom line is the following: the assignment is not final until I remove the "More to come" from it. (If you've done problems which were removed, just saved them for the following week.)

Finally, if there is still a "More to come" in an assignment on a Thursday or Friday, please write me right away so that I can update it. If I only realize on Monday (or Tuesday morning!) that an assignment was not complete (with the "More to come" still there) and nobody tried to contact me, I may add new problems just then, giving you little time to finish the assignment. (If I delay in replying, you can proceed with the Problems Likely To Be Assigned.)

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 2010/2011, pg. 41:

Academic Integrity

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 2010/2011, 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:

Wednesday, August 18 - Classes begin.

Friday, August 27 - Last day to add, change grade options, or drop a full semester course without a "W".

Monday, September 06 - Labor Day Holiday. (No class.)

Tuesday, September 28 - Midterm I.

Thursday-Friday, October 07-08 - Fall Break. (No class.)

Tuesday, October 19 - Last day to drop a full term course with a "W".

Tuesday, November 02 - Midterm II.

Tuesday, November 10 - Last day to drop with a WP/WF.

Thursday-Friday, November 25-26 - Thanksgiving. (No class.)

Tuesday, November 30 - Last Class Day.

Monday, December 06 - 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, 8, 9(b), 11(b), 14(b), (c). T/F: (a) to (h).

Section 1.2: 1, 4, 5, 6, 13, 14, 15, 16, 18, 21, 26, 35, 36. T/F: (a) to (i).

Section 1.3: 1, 4(h), 5(h), 6(c), 8(a), (c), 10(a), 11(a), 20. T/F: (a) to (o).

Section 1.4: 6, 8, 10, 12, 15, 18(e), 28. T/F: (a) to (k).

Section 1.5: 2, 3, 5, 8, 10, 14, 21. T/F: all.

Section 1.6: 2, 11, 15, 20. T/F: all.

Section 1.7: 1 to 4, 8, 12, 13, 17, 22, 27. T/F: all.

Section 2.1: 1, 3, 5, 7, 9, 13, 25, 27 to 32. T/F: all.

Section 2.2: 3, 5 to 9, 13, 17, 21, 23, 27, 30. T/F: all.

Section 2.3: 1, 5, 11, 16, 21, 25, 30, 35. T/F: all.

Section 3.1:: 5, 7, 10, 15, 21, 22, 27, 33. T/F: all.

Section 3.2:: 1, 3, 7, 9, 13, 20, 22. T/F: all.

Section 3.3:: 1, 3(c), (d), 5, 6, 8, 9, 15, 17, 19, 22, 25. T/F: all.

Section 4.1: 1, 4, 7, 8, 9, 10, 11. T/F: all.

Section 4.2: 1(a), (b), (c), 2(d), (e), (f), 3(a), (b), 9(a), (c), 10(a), 11(a), (c), 12(a), (c), 20. T/F: all.

Section 4.3: 1, 2(a), 3(a), 4(a), 5(a), 7, 9. T/F: all.

Section 4.4: 1, 2, 3(a), 4(a), 9(a), 11. T/F: all.

Section 4.5: 1, 3, 7, 8(b), 9(b), 13. T/F: all.

Section 4.6: 2(a), 3(a), 4, 5(b), 9, 10, 15, 23. T/F: all.

Section 4.7: 3(a), (b), 5(b), 6(c), 11(a), (b), 12(a), (b). T/F: all.

Section 4.8: 2(c), 4, 5, 7, 13, 17, 18. T/F: all.

Section 4.9: 1, 3, 5(c), (d), 7, 8(a), (b), 12(b), 14, 15, 16, 17, 18, 19, 29, 30. T/F: all.

Section 4.10: 3, 5, 8, 9, 11, 13(b), 14(b), 15, 17, 18, 21, 23, 24. T/F: all.

Section 5.1: 3, 4, 5, 6(a), (c), (e), 7(a), (c), (e), 8(a), (c), (e), 9(a), 10(a), 11(a), 12, 14, 18. T/F: all.

Section 5.2: 1, 2, 4, 5, 6, 8, 9, 15, 17, 19, 23. T/F: all.

More to come!

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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:

• (12/02 -- 11:35am) Posted solutions to 4.10.18 and 4.10.24 from HW11.
• (10/29 -- 12:45pm) Posted solutions to HW6.

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### Homework

HW1 - Due on Tuesday 08/31:

Section 1.1: 1, 8, 9(b), 11(b), 14(b), (c). T/F: (a) to (h).

Section 1.2: 1, 4, 5, 6, 13, 14, 15, 16, 18, 21, 26, 35, 36. T/F: (a) to (i).

Section 1.3: 1, 4(h), 5(h), 6(c), 8(a), (c), 10(a), 11(a), 20. T/F: (a) to (o).

HW2 - Due on Tuesday 09/07:

Section 1.4: 6, 8, 10, 12, 15, 18(e), 28. T/F: (a) to (k).

Section 1.5: 2, 3, 5, 8, 10, 14, 21. T/F: all.

HW3 - Due on Tuesday 09/14:

Section 1.6: 2, 11, 15, 20. T/F: all.

Section 1.7: 1 to 4, 8, 12, 13, 17, 22, 27. T/F: all.

HW4 - Due on Tuesday 09/21:

Section 2.1: 1, 3, 5, 7, 9, 13, 25, 27 to 32. T/F: all.

Section 2.2: 3, 5 to 9, 13, 17, 21, 23, 27, 30. T/F: all.

Section 2.3: 1, 5, 11, 16, 21, 25, 30, 35. T/F: all.

HW5 - Not to turn in (due to the exam):

Section 3.1: 5, 7, 10, 15, 21, 22, 27, 33. T/F: all.

Section 3.2: 1, 3, 7, 9, 13, 20, 22. T/F: all.

Section 3.3: 1, 3(c), (d), 5, 6, 8, 9, 15, 17, 19, 22, 25. T/F: all.

HW6 - Due on Tuesday 10/12:

Section 4.1: 1, 4, 7, 8, 9, 10, 11. T/F: all.

HW7 - Due on Thursday 10/28 (note the date!! -- it will be HW, not quiz!):

Section 4.2: 1(a), (b), (c), 2(d), (e), (f), 3(a), (b), 9(a), (c), 10(a), 11(a), (c), 12(a), (c), 20. T/F: all.

Section 4.3: 1, 2(a), 3(a), 4(a), 5(a), 7, 9. T/F: all.

HW8 - Not to turn in (due to the exam):

Section 4.4: 1, 2, 3(a), 4(a), 9(a), 11. T/F: all.

Section 4.5: 1, 3, 7, 8(b), 9(b), 13. T/F: all.

Section 4.6: 2(a), 3(a), 4, 5(b), 9, 10, 15, 23. T/F: all.

HW9 - Due on Tuesday 11/09:

Section 4.7: 3(a), (b), 5(b), 6(c), 11(a), (b), 12(a), (b). T/F: all.

HW10 - Due on Tuesday 11/16:

Section 4.8: 2(c), 4, 5, 7, 13, 17, 18. T/F: all.

HW11 - Due on Tuesday 11/23:

Section 4.9: 1, 3, 5(c), (d), 7, 8(a), (b), 12(b), 14, 15, 16, 17, 18, 19, 29, 30. T/F: all.

Section 4.10: 3, 5, 8, 9, 11, 13(b), 14(b), 15, 17, 18, 21, 23, 24. T/F: all.

Section 5.1: 3, 4, 5, 6(a), (c), (e), 7(a), (c), (e), 8(a), (c), (e), 9(a), 10(a), 11(a), 12, 14, 18. T/F: all.

Section 5.2: 1, 2, 4, 5, 6, 8, 9, 15, 17, 19, 23. T/F: all.

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

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