## Math 110—Linear Algebra—Fall 2009

### Professor:Mark Haiman

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)

GSI's:

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 helpful.

Required Text: Friedberg, Insel, and Spence, "Linear Algebra," 4th Ed.

Grading policy: Homework 15%, Midterms 15% each, Final exam 40%.
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 web.
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.)

Exams:

• 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 Hearst Gym
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 more than one midterm, see the discussion of Incomplete grades under Grading Policy, above.

### Syllabus

• 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 (Appendices C,D).
• 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 equations (6.3).

• 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 writing them.
• 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 2005, Ribet, 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 room).
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 230 Hearst 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 05, Ribet, Fall 08. Please disregard questions on those exams about unitary, self-adjoint and normal operators, which we did not cover this semester.
• Final Exam Solutions