Math 110—Linear Algebra—Spring 2012
Professor Haiman's Office Hours: W 10:30am-12:30pm, 855 Evans
Hall. RRR Week: W-Th 11-12.
Daub Office hours Fri 1-3, 737 Evans
Degany Office hours Tues 11:35-2:00, 1060 Evans
- Thomas DeIonno Office hours Wed 12-2, 1093 Evans
McIvor Office hours Thu 11:30-1:30, Fri 11-12, 1070 Evans
Lectures: TuTh 2-3:30pm, 10 Evans Hall
Discussion Sections: Wednesdays,
Course Control Number: 54242
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 helpful.
Sheldon Axler, Linear Algebra Done Right, Springer, 2nd edition
Grades based on homework (20%), final exam (40%) and two midterm exams
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.
For assignments and due dates, see Reading and Homework Assignments,
Each assignment will consist of a number of Exercises,
relatively straightforward problems which will not be graded,
and a smaller number (from 1 to 3) of Problems,
which will be graded. You do not have to hand in the
Exercises, but you should nevertheless do them carefully, both to
check your understanding of the material and as preparation for the
Problems and for exams. Generally speaking, exam questions will be
similar to the Exercises or some of the less difficult Problems.
You are free to discuss ideas on how to solve the 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 web.
No late or make-up homework will be accepted. When calculating
grades, we will drop two scores and use only your best scores
Two midterm exams during lecture hour:
- Midterm 1, Thursday Feb. 16, in two rooms:
- Last names A–M in 105 Stanley (opposite Evans across the
- Last names N–Z in 10 Evans (the regular lecture room)
- Midterm 2, Thursday Mar. 22, same rooms as Midterm 1
Final Exam Monday, May 7, 11:30-2:30pm (Exam Group 2), Room
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.
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 both midterms, see the discussion of
Incomplete grades under Grading Policy, above.
- Vector spaces.
- Subspaces. Intersections and sums; direct sums.
- Span, linear independence and bases. Dimension of a
finite-dimensional vector space.
- Linear maps.
- Nullspace, range and rank of a linear map.
- Matrix of a linear map.
- Invertible linear maps.
- Eigenvalues and eigenvectors.
- Inner product spaces.
- Orthonormal bases and the Gram-Schmidt procedure.
- Orthogonal projections; applications.
- Self-adjoint and normal operators; spectral theorem.
- Positive operators and isometries; polar and singular-value
- Operators on complex vector spaces.
- Characteristic polynomials and minimal polynomial.
- Jordan form.
Reading and Homework Assignments
Homework problems will be due on Fridays by 3pm at your GSI's office
or mailbox: please follow your individual GSI's instructions as to
where to turn it in. Note that in the mail room, the mailbox is
below the name label. This has caused some confusion since Daub,
Degany and DeIonno are all in the same column of mailboxes.
- Reading for Lectures 1-3: Axler, Chapters 1 and
2. Homework Assignment 1, due Friday,
Jan. 27. Solutions
- Reading for Lectures 4-6: Axler, Chapters 2 and
3. Homework Assignment 2, due
Friday, Feb. 3. Solutions
- Reading for Lectures 7-8: Axler, Chapter
3. Homework Assignment 3, due
Friday, Feb. 10. Solutions
- Midterm 1 is Thursday, Feb 16, in two rooms (see above). The
subject matter is the contents of Homeworks 1 to 3, that is, Axler
chapters 1-3 plus the material from the lectures
on F2 and on span, independence and bases for
infinite-dimensional spaces which was covered in the homework
exercises. You can find the first midterm (with solutions) from my
Fall 2009 Math 110 class along with links to some older exams
Math 110 Fall 09.
Midterm 1 Solutions
- Reading for Lectures 9-11: Axler, Chapter
3, continued. Homework Assignment 4, due
Friday, Feb. 24. Solutions
(updated Mar. 5).
- Reading for Lectures 12-13: Axler, Chapter 10 (Change of Basis
only), Chapter 4 (review of polynomial algebra), Chapter
5. Homework Assignment 5, due
Friday, Mar. 2. Solutions
(updated Mar. 5).
- Reading for Lectures 14-15: Axler, Chapter
5. Homework Assignment 6, due
Friday, Mar. 9. Solutions
(this link erroneously gave HW 5 before, now fixed).
- Reading for Lectures 15-16: Axler, Chapter
6. Homework Assignment 7, due
Friday, Mar. 16 (updated Mar. 14: the assigned problem was too hard,
so I changed it). Solutions
- Midterm 2 is Thursday, Mar. 22 in the same two rooms as before:
last names A–M in 105 Stanley, last names N–Z in 10 Evans. The
subject matter is the contents of Homeworks 4-7.
Midterm 2 Solutions
- Reading for Lectures 18-20: Axler, Chapter 6: Orthogonal
Projections and Minimization; Linear Functionals and
Adjoints. Homework Assignment 8, due
Friday, Apr. 6. Hint for Problem 1: the hardest part is
figuring out how to use the hypothesis ||Pv||≤||v||. To
get an idea about this, try working out an example of a
projection P in R2 which is not orthogonal,
and see for which vectors the hypothesis
- Announcement: Prof. Haiman's office hours on Wed., Apr. 4 will be
1-2:30pm (back to normal next week).
- Reading for Lectures 21-22: Axler, Chapter 7: Self-Adjoint and
Normal Operators, Isometries. Don't miss Lecture 21! It will cover
some topics not in the book. Homework
Assignment 9, due Friday,
Apr. 13. Solutions
- Reading for Lectures 23-24: Rest of Axler, Chapter 7, but
omitting Normal Operators on Real Inner Product
Spaces. Homework Assignment 10, due
Friday, Apr. 20. Solutions
- Reading for Lectures 25-26: Axler, Chapter 7: Singular Value
Decomposition. Axler, Chapter 8, omitting Square Roots and The
Minimal Polynomial. Homework Assignment
11, due Friday,
Apr. 27. Solutions
- Professor Haiman's RRR week office hours are W-Th 11-12.
- Final Exam Monday, May 7 (details under Exams, above). You can
bring two note sheets to the final. Please also bring (and use!)
scratch paper. The final will cover material from the whole course,
with extra emphasis on subject matter from Homeworks 8-11.
- 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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