MATH 54: HONORS LINEAR ALGEBRA
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:
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.
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, A_{1}), #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 (a_{0}, a_{1}, a_{2}, ...) 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 / x^{2}) 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 A^{2} = A. Show that its Jordan
form J = M^{-1}JM satisfies J^{2} = J. Since the diagonal
blocks stay separate, this means J_{i}^{2} = 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 λ_{1},
..., λ_{n} are the eigenvalues of A then λ_{1}^{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.