MATH 315 D1, Fall 2002.

Linear Transformations and Matrices

You can check the score reports on the web. The average represents your average score for the exams, quizzes and homework graded so far. The ones graded are the ones that appear. "84<90" means that your actual score for the exam was 84 but it was rescaled to 90 because of a diffferent curve used for this particular exam. ** marks the lowest quizzes, homework, and exams that were dropped. The curves for each exam, given in the class, are taken into account: in the calculation of the average, the rescaled grades are used. The lowest ones that are dropped do not contribute to the average. Percentage for exams, quizzes, homework is taken into account.

The lowest average to receive A- is 90, and the lowest average to receive A is 93.33333333333333333333333333333333333333333333333333333333333333333333...

Igor Mineyev, 235 Illini Hall.
mineyev@math.uiuc.edu
Office hours. MWF 3:00-4:00 pm.
Text. Linear Algebra with Applications, sixth edition, by Steven J. Leon.
Class time. MWF 11:00am-11:50am
Class location. 245 Altgeld Hall
Prerequisites. MATH 242 Calculus of Several Variables, or MATH 243 Multivariable Calculus and Vector Analysis, or MATH 245 Calculus II.

The course is designed to introduce you to techniques of linear algebra. The topics covered include matrix operations, determinants, linear equations, vector spaces, linear transformations, eigenvalues, and eigenvectors. You will be required to know some proofs as well.

Homework and exams. The homework will be assigned daily and collected weekly. 10-minute quizzes will be given every week. There will be three 50-minute midterm exams and a comprehensive 3-hour final exam on Wednesday, December 18, 8:00 am - 11:00 am, in the regular classroom. The time of the final exam will not be changed. Please make you travel arrangements accordingly. The quizzes and exams will not be necessarily the same as in the homework, but rather will be based on the homework and the material presented in class. No textbooks, lecture notes, or calculators will be allowed on the exams and quizzes. No late assignments will be accepted. Missed or late assignments, exams, and quizzes count as 0.

Grading policy. Two lowest homework assignments, one lowest quiz and one lowest midterm exam will be dropped. The grading scale will be decided individually for each exam, and all the quizzes and homework, after the scores are obtained, but it is guaranteed that you get at least A- if the score is 90%, at least B- if the score is 80%, at least C- if the score is 70%, and at least D- is the score is 60%. Each of the midterms will count approximately as 17% of the grade, the homework 17%, the quizzes 17%, and the final exam 33%.
The grades will reflect the accomplishment, not the amount of time and effort spent. And you will need to put time and effort in order to do well in this class. Be prepared to work regularly.

Homework and exams policy. It is important to do the homework after each class, even though it will be collected weekly. Please write the problems in the order they are given. Staple the sheets (no folding over corners). You are expected to write complete, grammatical sentences, and to spell correctly. Working together on homework is encouraged. Copying is unacceptable. Make sure you know the material, otherwise you will not be able to do well on the exams.

There will be no "extra credit" or "make-ups". The reason one midterm exam, quiz, and two homework assignments are dropped is to allow for personal emergency situations, when there is a strong reason you must miss a class. Do not miss exams and quizzes (and regular classes) for no reason.

Attendance. It is very important to attend the class. If you miss a class, you are still responsible for learning what was discussed. This includes the homework assigned during that class as well.

Additional help. You are welcome to come to my office during my office hours to discuss homework etc. Additionally, as was announced in class, Professor Irma Reiner has volunteered to tutor students in MATH 315, in B3 Coble Hall, Monday and Thursday from 2:30 to 3:45 pm.

Tentative schedule.

Chapter 1: Matrices and Systems of Equations.
Chapter 2: Determinants.
Chapter 3: Vector spaces.
Chapter 4: Linear transformatioons.
Chapter 5: Orthogonality. (Sections 1-6.)
Chapter 6: Eigenvalues. (Sections 1, 3-6)
Finishing up. Bring your questions for discussion.

The above schedule is tentative. Some adjustments will be made during the course.

The homework below is provided only for convenience for future references. The homework assignments will appear here with a delay. The current homework is given during each class. The best way to know the current homework is to never miss a class. Assignments will be due beginning of each Friday class, at 11 am.

Homework.

HW1 begins
1.1 # 1ad, 2bd, 6afh, 9.
1.2 # 2bc, 3, 5aeg, 6, 8, 9, 12.
1.3 # 1, 2, 3, 4; learn rules on pp.46-47, prove rules 1-4, 7.
HW2 begins
1.3 Learn rules 1-4 on p.56, and prove them. Learn statements of Theorem 1.4.3 and Corollary 1.4.4. Prove Theorem 1.4.1, and I and II on p.71.
(You do not need to write the "statements" in the homework. Only learn them. You do need to write the proofs in the homework when asked to "prove".)
1.4 # 1, 2, 4, 8, 9fg, 12.
2.1 # 1-3.
2.2 # 1, 3.
HW3 begins.
2.2 # 6, 7.
2.3 # 1, 2bce, 5.
3.1 # 1, 4, 5, 7-9; 11. Learn the definition of a vector space on p.129 (not to write in homework); write a proof of Theorem 3.1.1 on p.131.
3.2 # 1-4. (For numbers 1-3, as usual: write the proof if the statement is correct, provide a counterexample if the statement is false.)
HW4 begins
3.2 # 5-7, 10, 11.
3.3 # 1, 2, 5-7, 10, 11, 13, 14, 16. Prove Theorem 3.3.2, p.150. Then assume V=Rⁿ and prove the theorem in a different way. (Hint: You can use theorems from Chapter 1.)
HW5 begins.
3.4 Prove the "better" versions of Theorem 3.4.1 and of Theorem 3.4.3, given in class. Prove the Claim (a "better" part of Theorem 3.4.4):
Let V be a vector space of dimension n>0.
a) In any spanning set of V, there is a subset that is a basis for V.
b) Any set of independent vectors in V can be extended to a basis.
# 3, 4, 5abc, 7, 8, 9b, 10.
3.5 #1-6, 9.
HW6 begins
3.3 # 12
3.6 Prove Theorem 3.6.5, p.177 (explain why the dimension of N(A) equals n-r); prove Theorem 3.6.6, p.178; # 1-3, 6-10, 16, 20.
4.1 Prove (i), (ii), (iii), p.193; # 1, 3, 4. Prove Theorem 4.1.1, p.194; # 5-8.
HW7 begins
4.1 # 16ac-17ac, 20.
4.2 # 2-8, 18.
4.3 # 1-4, 7, 11, 12.
HW8 begins
5.1 # 1ab-2ab, 3ab, 4, 5, 8a, 10, 11.
5.2 # 1, 2a, 3, 4.
HW9 begins (due Friday, Nov. 15)
5.2 # 6, 8, 12, 14, 15.
5.3 # 1-3.
HW10 begins
5.3 # 5a, 6, 9ab (in a only prove the equality Pb=b).
5.4 # 1, 3-11; prove the Cauchy-Schwarz inequality (with details).
HW11 begins
5.5 # 1-3, 5, 6, 12, 15, 29.
5.6 # 1, 3-4, 7.
6.1 # 1abcfgh, 2,3.
HW12 (not to turn)
6.3 # 1-6.

Also, any of the following problems might appear at any of the exams or quizzes:

Chapter 1 test # 1-8. (In # 3, you are allowed to "cheat" and use determinants, even though they vere not discussed in Chapter 1. The completely correct way to do this problem, without cheating, would be to write a proof along the lines of the proof of Theorem 1.4.3. In any case, you will need to use elementary matrices as well.)
Chapter 2 test # 2-10.
Chapter 3 test # 1-2, 4-9.
Chapter 4 test # 1-3, 7, 9.

In addition to the numbers above, you are expected to know the material (in patricular, the definitions) that was discussed in class.

Notes on how to organize homework, quizzes, and exams.

Always write complete solutions. Correct anwers with no solution do not score points.
Explicitly specify the answer to each problem.
Use the equality sign, "=". Quite often, when you need to show that one thing is equal to another, you write a sequence of equalities connecting the two things. This is the best way to present a solution, whenever possible.
Use the equality sign correctly. The thing on the left side of "=" must actually be equal to the thing on the right side of "=".
Do not write statements that are obviously wrong, for example "2=5" or "A=a_i,j". (Why wrong?)
(Or "our teacher is tough". :)
When you explain things by words, write complete grammatical sentences.
Write the solutions to the problems in the order they were assigned. Write the homework number, section numbers, then problem numbers.
Staple the weekly homework together, to make sure no part of it is lost. (No folding over corners.)
PRINT your "Last Name, First Name" legibly on the top of the homework. Also write "MATH 315 D1".
Turn the homework at the beginning of the class it is due.

Additional notes.

The first hour (that is, 50 minute) exam was on Monday, October 7. The mean was 71.7 and the median 74.

The second hour exam was on Friday, November 8 (sections 3.5-5.2). The mean was 85.4 and the median 89.

The third hour exam was on Monday, December 9. The mean was 65.7 and the median 66. There was homework due Friday, December 6, but no quiz.