Math 54: Honors Linear Algebra

MATH 54: HONORS LINEAR ALGEBRA
Barus-Holley 161, MWF 10:00 - 10:50am

Professor Steven Miller (sjmiller AT math.brown.edu), Kassar House, Room 210, 401-863-1123

OPTIONAL PASS/FAIL EXAM: Monday, May 7th: 8 to 9:30am: MacMilan 115
FINAL EXAM: Wednesday, May 16th 2007: 2pm - 5pm: BH 141

COURSE DESCRIPTION: Linear algebra for students of greater aptitude and motivation. Recommended for prospective mathematics concentrators, and science and engineering students who have a good mathematical preparation. Topics include: matrices, linear equations, determinants, characteristic polynomials, and eigenvalues; vector spaces and linear transformations; inner products; Hermitian, orthogonal, and unitary matrices; bilinear forms; elementary divisors and Jordan normal forms. Provides a deeper and more extensive treatment of the topics in MA 52 and can be substituted for MA 52 in fulfilling requirements. Prerequisite: MA 18, MA 20, or MA 35. NOTE: I will often emphasize the computational aspects of linear algebra, with problems drawn from Linear Programming, Computer Science and Statistics.

The textbook is the fourth edition of Gilbert Strang's Linear Algebra and its Applications (ISBN: 0-03-010567-6), as well as supplemental handouts. The third edition of Strang's book seems to have almost identical material; the only problem you will have if you use the third edition is that the HW problems are numbered differently. You can probably get by with a used third edition, but you are responsible for getting the correct HW problems; I will try and post the HW problems on the webpage. Much emphasis will be placed on efficient algorithms. Please read the relevant sections before class. This means you should be familiar with the definitions as well as what we are going to study; this does not mean you should be able to give the lecture.

GRADING / HW: Homework 20%, Midterm(s) 35%, Final 45%. Exams are black tie optional. Homework is to be handed in on time, stapled and legible. Late, messy or unstapled homework will not be graded. I encourage you to work in small groups, but everyone must submit their own homework assignment. Some potential HW problems for the course are available here.

OFFICE HOURS: Before and after class; if possible email me ahead of time to let me know you are coming, as well as what you want to discuss.

HANDOUTS:

Some potential HW problems for the course are available here.
I've written over the years notes for linear algebra. These are meant to supplement the coursebook, not to replace it.
Notes on Linear Programming.
Notes on The Method of Least Squares.
MIT has a webpage for the book and its linear algebra course: http://web.mit.edu/18.06/www/
Professor Treil's textbook (which is an excellent supplemental text)
Introduction to Random Matrix Theory (from my textbook An Invitation to Modern Number Theory).

HOMEWORK:

Please spend at least 1 hour a night reading the material/looking at the proofs/making sure you can do the algebra. Below is a tentative reading list and homework assignments. It is subject to slight changes depending on the amount of material covered each week. I strongly encourage you to skim the reading before class, so you are familiar with the definitions, concepts, and the statements of the material we'll cover that day.
I will try and post the HW on-line so you can use an earlier edition of the book, but you are responsible for getting the HW problems. Some potential HW problems for the course are available here. Extra credit assignments should be handed in separately from your homework.
Week One (1/24 - 1/26):
Read: Chapter 1, Sections 1.1, 1.2, 1.3, 1.4
HW: Work on the problems due on Monday, 2/5; no problems are due the first week because of shopping period. We are not going to do Chapter 1 in order, so you should work on the problems from the sections covered in class. Extra Credit Problem (hand in by March 2nd): Let A be an nxn matrix with integer entries a_ij. Bound the denominators that arise in obtaining an upper triangular matrix in Gaussian elimination as a function of the a_ij.
Week Two (1/29 - 2/2):
Read: Chapter 1 (1.1 through 1.6); Chapter 2 (2.1 through 2.4).
HW: Due Monday 2/5: Sec 1.2: #22; Sec 1.3: #11, #18; Sec 1.4: #10, #20; Sec 1.5: #5, #42, #47; Sec 1.6: #6 (just do the first one, A₁), #11, #40, #52.
Suggested Problems: Sec 1.2: #3; Sec 1.5: #25, #29, #41, #45; Sec 1.6: #19, #43,#65; Review Exercises: Page 65, #1.6 and Page 66, #1.20.
HW: Due Wednesday 2/7: Sec 2,1: # 1, #25, #29; Sec 2.2: #2, #25, #37, #44, #54.
Extra Credit Problems (due 2/7): (1) Show that the inverse of a permutation matrix is the transpose of the matrix. (2) Let S be the set of all infinite sequences (a₀, a₁, a₂, ...) such that sum_{n = 0 to oo} a_n converges. Define addition of two sequences by adding components, and define scalar multiplication by multiply each component by the scalar.Prove or disprove: S is a vector space.
Suggested Problems: Sec 2.1: #4, #7; Sec 2.2: #24, #26, #34, #48, #61, #65.
Week Three (2/5 - 2/9):
Read: Chapter 2 (2.1 through 2.4, 2.6); Chapter 3 (3.1 and 3.2).
HW: Due Monday 2/12: Sec 2.3: #1, #14, #43; Sec 2.4: #3, #16, #37; Sec 2.6: #7, #36, #50; Review Exercises: Page 139: #2.23; Sec 3.1: #11 , #19, #46.
Suggested Problems: Sec 2.3: #39, #40; Sec 2.4: #5, #20, #27; Sec 2.6: #3, #14, #46, #47; Review Exercises: Page 139: #2.10; Sec 3.1 #2, #14, #32, #36,#49, #51.
Extra Credit Problems (due 2/14): Let A be an m by infinity matrix. Give an example of such a matrix where any set of m columns are linearly independent, or prove no such matrix exists.
Week Four (2/12 - 2/16):
Read: Chapter 3 (3.2 and 3.4), Chapter 4 (4.1, 4.2), Professor Treil's textbook (Chapter 3, pages 69 - 83).
HW: Due Wednesday 2/21: Sec 3.2: #12, #13; Sec 3.4: #2, #3, #13, #21; Sec 4.2: #10, #12, #14, #26; Sec 4.3: #3, #18.
Suggested Problems: Sec 3.2: #15, #17; Sec 3.4: #9, #24, #31; Sec 4.2: #1, #4, #19, #30; Sec 4.3: #4, #6, #9, #28, #34.
Extra Credit Problem (due 2/21): Let f(x) = exp(- 1 / x²) if x is not equal to 0 and 0 if x equals zero. Calculate the infinite Taylor series expansion of f(x) about x=0.

Week Five (2/21 - 2/23):
Read: Chapter 4 (4.1 through 4.4), Professor Treil's textbook (Chapter 3, pages 69 - 83), Notes on The Method of Least Squares.
HW: Due Monday 2/26: Sec 4.4: #16; Review Exercises: Page 230: #4.5, #4.9, #4.16.
Suggested Problems: Sec 4.4: #30.
Week Six (2/26 - 3/2):
Read: Chapter 5 (5.1, 5.2, 5.5 -- in this order).
HW: Due Monday 3/12: Sec 5.1: #1, #6, #18, #39, Sec 5.2: #9, #25; Sec 5.5: #12, #41, #44;
Suggested Problems: Sec 5.1: #4 (this problem will be easier after we do Sec 5.4), #14, #26, #40; Sec 5.2: #7, #19, #25; Sec 5.5: #21.
Week Seven (3/5 - 3/9): MIDTERM INFORMATION: The midterm will be a closed-book, in-class exam on Wednesday, March 7th. The test will cover all material through determinants (chapter 4).
Read: Chapter 5: (5.5, 5.3, 5.4 -- in this order)
HW: Due Wednesday 3/21: Sec 5.3: #15, #19, #29; Sec 5.4: #38, #41.
Suggested Problems: Sec 5.3: #8; Sec 5.4: #1, #4, #6; Sec 5.5: #21; Sec 5.3: #8; Sec 5.4: #1, #4, #6.
Week Eight (3/12 - 3/16): Optional 2nd Midterm in class on Monday, March 12th.
Read: Chapter 5: 5.4, 5.3. Read Appendix B on Jordan Canonical Form. You will NOT be responsible for the proofs of Jordan Canonical Form, but you WILL be responsible for the statement (ie, there may be a problem on the final where you need to know the statement of JCF). You can also read my notes on JCF (MSWord file, PS file)
HW: Due Wednesday 3/21: Sec 5.3: #15, #19, #29; Sec 5.4: #38, #41; Sec 5.3: #15, #19, #29; Sec 5.4: #38, #41. Page 427: #1: Find the Jordan form (in three steps!) of the 2x2 matrix that is all 1s, and the 3x3 matrix that is all zero except for the first row (which is 0 1 2), #7: Suppose that A² = A. Show that its Jordan form J = M^-1JM satisfies J² = J. Since the diagonal blocks stay separate, this means J_i² = J_i for each block; show by direct computation that J_i can only be a 1 by 1 block, J_i = [0] or [1]. Thus, A is similar to a diagonal matrix of 0s and 1s.
Week Nine (3/19 - 3/23):
Read: Chapter 6: 6.1, 6.2, 6.4; Chapter 3: 3.3 and handout on The Method of Least Squares (we will start with 6.1 on Monday; however, you should be prepared for the lecture on least squares at any time); Professor Treil's textbook (pages 189 - 191)
HW: Due Wednesday 4/4: Page 170: #6, #13; Page 316: #1, #4, #8; Page 326, #1, #26, #37, #43; Page 345: #5, #7, #8; Page 304: #24a. Prove that if λ₁, ..., λ_n are the eigenvalues of A then λ₁^k, ..., λ_n^k are the eigenvalues of A^k (if the eigenvalues are distinct the proof is easy; however, you need to do some work if there are repeated eigenvalues. The problem is what happens if there are not enough eigenvectors: you could go from having eigenvalues 2, 2, 2, 4, 4 to 2, 2, 4, 4, 4! I can think of at least four different proofs using techniques from the class so far; three of them work regardless of whether or not there are repeated eigenvalues). Also skim the notes on Linear Programming over the break.
Suggested Problems: Page 346: #16.
Week Ten (3/26 - 3/30): BREAK: NO CLASS: Continue reading (especially the notes on Linear Programming) and working on HW.
Week Eleven (4/2 - 4/6):
Read: Chapter 7: 7.1, 7.2, 7.3, the handout on fast multiplication, and the Notes on Linear Programming.
HW: Due Monday 4/9: Page 357: #3, #4, #9, #18, #19, #20; Page 366: #1, #2, #4. From the fast multiplication handout: page 80: #1, #2.
Suggested Problems: From the fast multiplication handout: verify equation (2.27) on page 77, and on page 80: #3, #4.
Week Twelve (4/9 - 4/13):
Read: Notes on Linear Programming.
HW: Due Monday 4/16: From the Notes on Linear Programming handout: Page 5, Exercise 1.2; Page 7, Exercises 1.5; Page 10, Exercise 1.10 (here A' has M rows and k columns, and the k columns are linearly independent); Page 17, Exercise 1.14.
Suggested Problems: From the Notes on Linear Programming handout: Page 7, Exercises 1.3, 1.4; Page 9, Exercise 1.7; Page 10, Exercise 1.11.
Week Thirteen (4/16 - 4/20):
Read: Professor Treil's textbook (pages 109 - 118), handout on Introduction to Random Matrix Theory (read the first 20 pages; I realize that there is a lot of material there, some of it assuming material you haven't seen. We will cover a subset in class, but you should skim the material so as to have some sense of the topic).
HW: Due date TBD: From the Notes on Linear Programming handout: Page 20: Exercises 1.15, 1.16. Professor Treil's textbook: Page 116: #1.3, #1.4, #1.5, #1.8, #1.9. From my handout on Introduction to Random Matrix Theory: Page 8, #1.1.4; Page 10: #1.2.3 (if you cannot solve the problem in general, if you prove it for 2x2 and 3x3 matrices you can receive half credit).
Suggested Problems: From Professor Treil's textbook: Page 116: #1.6, #1.7. From my handout on Introduction to Random Matrix Theory: Page 10: #2.2.2.
Week Fourteen (4/23 - 4/27):
Read: Notes on Introduction to Random Matrix Theory (read the first 20 pages; I realize that there is a lot of material there, some of it assuming material you haven't seen. We will cover a subset in class, but you should skim the material so as to have some sense of the topic).
HW: TBD
Please spend at least 1 hour a night reading the material/looking at the proofs/making sure you can do the algebra. Below is a tentative reading list and homework assignments. It is subject to slight changes depending on the amount of material covered each week. I strongly encourage you to skim the reading before class, so you are familiar with the definitions, concepts, and the statements of the material we'll cover that day.