Math 110—Linear Algebra—Fall 2009
Professor Haiman's Office: 855 Evans Hall, hours M
10:30-11:30, W 2:30-3:30 (Reading/Review Week hours MW 11:30-1:30)
- Charley Crissman,
743 Evans, hours Th 10-12, F 12-1
- Amit Gupta, 845 Evans Hall, hours W 2-3, Th 10-12
Mitchell, 745 Evans, hours W 10-11, Th 11:30-12:30, F 11-12
- Ivan Ventura,
1039 Evans, hours Tu 2:30-3:30, Th 1:30-2:30, F 2-3
Lectures: MWF 1-2pm, 100 Lewis
Discussion Sections: Wednesdays, see schedule.
No section meeting on the first day of classes, Wednesday, August 26.
Course Control Number: 54554
Prerequisites: Math 54 or equivalent preparation in linear
algebra at the lower division level. Some prior experience with
mathematical reasoning and proofs, for example Math 55, is also
Required Text: Friedberg, Insel, and Spence, "Linear
Algebra," 4th Ed.
Grading policy: Homework 15%, Midterms 15% each, Final exam
Incomplete grades are rarely given, and only in documented cases of
illness or serious personal or family emergency. A grade of
incomplete for missing work can only be given if you having passing
grades on your other work.
Homework: Due at the beginning of lecture on Mondays
(except HW 1 due Fri 9/4, because Mon 9/7 is a holiday). Returned in
sections on Wednesday. Homework will not be accepted late. Due to
limited resources, only some of the problems from each homework set
will be selected for grading.
You are free to discuss ideas on how to solve the homework problems
with other students. However, you must write your solutions
independently. It is not permissible to copy solutions worked out in
a group, or from students who took the class before, or found on the
When calculating grades, we will drop your two lowest homework scores.
(Since not all homeworks have the same weight, the actual algorithm we
will use is a little more complicated: we will drop whichever two
homework scores leave you with the highest percentage of the total
possible score on the remaining homeworks, then use that percentage to
figure your grade.)
At midterm exams you may bring one (ordinary size) sheet of notes,
written on both sides. Two sheets are allowed for the final exam. No
other books, calculators, computers or other aids may be used. Space
to write answers will be provided on the exam paper, but you should
bring your own scratch paper.
- Midterm 1 - Friday, Sept 25 during lecture hour. Last names
A-K in 10 Evans, L-Z in 100 Lewis.
- Midterm 2 - Friday, Oct 23 during lecture hour. Last names A-J
in 50 Birge,
K-Z in 100 Lewis.
- Midterm 3 - Friday, Nov 20 during lecture hour. Last names
A-J in 50 Birge, K-Z in 100 Lewis.
- Final Exam - Wednesday, Dec 16, 5-8 pm (Exam group 12), Room 230
No make-up exams will be given. If you miss one midterm, your score
on the following exam will count for that midterm. However, you
cannot "miss" a midterm retroactively after turning in your exam. If
you miss the final or more than one midterm, see the discussion of
Incomplete grades under Grading Policy, above.
- Axiomatic definition of vector space; vector geometry in
R2 and R3 as examples. Properties of
vector arithmetic (FI&S sections 1.1-1.2).
- Fields; examples Q, R, C, F2
- More vector spaces: Fn, Mm×n,
spaces of functions, polynomials, and sequences (1.2).
- Subspaces (1.3).
- Linear combinations. Solving systems of linear equations (1.4).
- Linear span (1.4).
- Linear dependence and independence (1.5).
- Bases and dimension (1.6).
- Linear transformations. Nullspace and image ("range"), rank (2.1).
- Matrix representation of linear transformations (2.2).
- Matrix multiplication and composition of linear transformations (2.3).
- Invertible linear transformations and invertible matrices (2.4).
- Isomorphic vector spaces; coordinates with respect to a basis (2.4).
- Change of coordinates (2.5).
- Elementary matrices and row operations (3.1).
- Finding the inverse of an invertible matrix (3.2).
- Rank of a matrix (3.2).
- Matrix form of a system of linear equations; description of
solution set, criteria for existence and uniqueness of solutions (3.3)
- Row echelon form and Gaussian elimination (3.4).
- Determinants (4.1-4.5).
- Eigenvalues and eigenvectors (5.1).
- Characteristic polynomial. Trace, determinant and eigenvalues (5.2).
- Diagonalization. Criteria for diagonalizability (5.2).
- Applications: Powers of a matrix. Fibonacci numbers. Systems of
first order constant coefficient linear differential equations (5.2).
- Matrix exponential (5.3).
- Cayley-Hamilton Theorem (5.4).
- Inner product spaces; norm. Cauchy-Schwartz inequality (6.1).
- Orthonormal bases. Gram-Schmidt process. QR decomposition (6.2).
- Orthogonal complement; orthogonal projection (6.2).
- Adjoint of a linear operator (6.3).
- Least-squares approximation; minimum-norm solution of a system of
- Additional topics if time.
Reading and Homework Assignments
- Reading for Lectures 1-4: Sections 1.1–1.4 and Appendices
C,D. Problem Set 1, due Friday Sept. 4.
PS 1 Solutions—Problems 1, 5 and 7
will be graded (10 points each). Please ignore which problems are
marked for grading on the solutions! I changed my mind after
- Reading for Lectures 5-7: Sections 1.5–1.6.
Problem Set 2, due Monday Sept. 14. PS 2 Solutions—Problems 2 and 4 will be
graded (10 points each).
- Reading for Lectures 8-10: 1.6 on Lagrange Interpolation; 2.1. Problem Set 3, due Monday Sept. 21. PS 3 Solutions—Problems 4, 5 and 7 will be
graded (10 points each).
- Reminder: Midterm 1 is Friday, Sept. 25, covering material from
problem sets 1-3. The exam will be held in two rooms: last names
A-K in 10 Evans, L-Z in 100 Lewis (the regular lecture room).
Here are some first midterm exams, with solutions, from this
course in previous years: Holtz,
Fall 2006, Ribet, Fall
2002, Ribet, Spring
Fall 2008. These exams were given later in the term than our
first midterm, so for now please disregard questions on material that
we have not covered yet. I think you will find the remaining
questions helpful for review.
- Midterm 1 Solutions
- Reading for Lectures 11 and 12: 2.1 continued, especially Theorem
2.6; 2.2. Problem Set 4, due Monday
Sept. 28. PS 4 Solutions—Problem 3 will be
graded (10 points).
- Reading for Lectures 13-15: 2.2 (continued), 2.3. Problem Set 5, due Monday Oct. 5. PS 5 Solutions—Problems 3 and 5 will be
graded (10 points each).
- Reading for Lectures 16-18: 2.4, 2.5. Problem Set 6, due Monday Oct. 12.
Correction: the reference to "Problem 2" in Problem 5
should have been to Problem 4. PS 6
Solutions—Problems 3 and 5 will be graded (10 points each).
- Reading for Lectures 19-21: 3.1, 3.2, and optionally 2.6 for more
information about Problem 5 on this week's problem set. Problem Set 7, due Monday Oct. 19. PS 7 Solutions—Problems 1 and 2 will be
graded (10 points each).
- Reminder: Midterm 2 is Friday, Oct. 23, covering Problem Sets 4-7,
but excluding problem 5 from Problem Set 7. Of course, you are also
responsible for knowing earlier material. The rooms are different this
time: A-J will be in 50 Birge, K-Z in 100 Lewis (the regular lecture
For review, I advise going over exercises from Sections 2.1-2.5 and
3.1-3.2 in the book. Many of the book's exercises are similar to what
you may expect on the exam. You may also find it useful to have
another look at the mock midterm from Holtz and the first midterms
from Ribet, above. You might also look at Ribet, Fall
02 Midterm 2 (problems 1,2), Ribet, Spring
05 Midterm 2 (1 a,c), and Ribet,
Fall 08 Midterm 2 (problem 1). The other problems on these
previous exams are on material we have not covered yet.
- Reading for Lectures 22-23: 3.3, 3.4. Problem Set 8, due Monday Oct. 26. PS 8 Solutions—Problem 3 will be
graded (10 points).
- Midterm 2 Solutions
- Reading for Lectures 24-26: 4.1-4.5. Problem
Set 9, due Monday Nov. 2. Problem 1 corrected 10/28.
Problem 2: clarification/hint—det(A)
divisible by the product p in part (b) means that there
exists a polynomial
f(x1,...,xn) such that
det(A)=f·p. You may assume without proof that the
vanishing det(A)=0 for
xi=xj implies that det(A) is
divisible by p. Now compare degrees of the terms in p
with those in det(A) to prove that f must be a constant.
PS 9 Solutions—Problems 1 and 3 will
be graded (10 points each).
- Reading for Lectures 27-29: 4.4-4.5 continued, 5.1-5.2. Problem Set 10, due Monday Nov. 16. No
homework is due on Monday Nov. 9, since Nov. 11 is a holiday, so your
next section meeting is Nov. 18. Problem Set 10 covers Lectures
27-29. Problem Set 11, covering Lectures 30-31, will be posted later
but will also be due on Monday Nov. 16 along with PS 10.
- Reading for Lectures 30-31: 5.2, 5.3 through Theorem 5.12. In
lecture we will also discuss the exponential of a matrix, as presented
in Section 5.3, Exercises 20-24. Problem Set
11, due Monday Nov. 16 with PS 10. PS
10 and 11 Solutions—PS 10 Problems 1 and 5 and PS 11 Problem
3 will be graded (10 points each).
- Reminder: Midterm 3 is Friday, Nov. 20, in the same rooms as
Midterm 2. This exam covers material through PS 11, with the emphasis
on PS 8-11, although you are responsible for knowing the earlier
material too. For review, you may wish to look again at the three old
Midterm 2's from Ribet, linked to above. I also recommend going over
exercises from Sections 3.3-3.4, 4.1-4.5 and 5.1-5.3 in the book.
- Reading for Lectures 32-33: 5.4. Problem Set
12, due Monday Nov. 23. PS 12
Solutions—Problem 3 will be graded (10 points).
- Midterm 3 Solutions
- Reading for Lectures 34-38: 6.1-6.3. Problem
Set 13, due Friday, Dec. 4. The GSIs will hold review
sessions during Reading/Review Week. You can get your graded PS 13
returned there. PS 13
Solutions—Problems 1 and 5 will be graded (10 points each).
- Prof. Haiman's office hours during Reading/Review week will be
Monday and Wednesday, 11:30am-1:30pm.
- The Final Exam is Wednesday, Dec 16, 5-8 pm (Exam group 12), Room
Gym. You may bring two note sheets to the final exam.
The final covers all course material, with some extra emphasis on
sections 5.4 and 6.1-6.3 of the text, which have not been previously
covered on the midterms. Because time was limited, I did not pose
many questions requiring proofs on the midterms. Since you have 3
hours for the final exam, you should expect a few more proofs.
For review, I suggest going over problems from the book, and these
final exams from previous years: Holtz,
Fall 06, Ribet,
Fall 02, Ribet, Spring
Fall 08. Please disregard questions on those exams about unitary,
self-adjoint and normal operators, which we did not cover this
- Final Exam Solutions
Prof. George Bergman has written some Notes on sets, logic, and mathematical
language, which you may find helpful in reading, writing and
understanding some of the definitions and proofs in this course.
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