Math 225 Linear Algebra and Matrix Theory

Semester: Spring 2018

Keep Calm and Learn Linear Algebra
Lecture 01 (21468) / Lecture 02 (23453)
Inst : Prof. Asher Auel
asher * auel AT yale * edu
Time : Tue Thu 09:00 - 10:15 am
Tue Thu 01:00 - 02:15 pm
Loct : LOM 206
Office : LOM 210
Phone : (203) 432-4187
hours :

Tue 2:15 pm - 4:15 pm
Text : Linear Algebra, 4th Edition
Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence
Pearson, 2003. ISBN-13: 978-0-13-008451-4.
 Course syllabus and homework schedule.
 Office hours during reading period:
  Mon, Apr 30, 1:00 - 2:00 pm (Charlie)
  Mon, Apr 30, 3:00 - 4:00 pm (Charlie)
  Tue, May 1, 9:00 - 11:00 am LOM 200 (Byungmin)
  Wed, May 2, 3:00 - 5:00 pm Common Room or LOM 201 (Noah)
  Wed, May 2, 6:00 - 8:00 pm Common Room or LOM 201 (Noah)
  Thu, May 3, 9:00 - 10:00 am (Prof. Auel) LOM 205
  Thu, May 3, 12:00 - 2:00 pm (Joon-Hyeok) LOM 206
  Thu, May 3, 4:00 - 5:00 pm (Prof. Auel) LOM 206
 Final Review Sessions :
  Tue, May 1, 3:00 - 5:00 pm LOM 200 (Joon-Hyeok)
  Wed, May 2, 1:00 - 3:00 pm LOM 200 (Byungmin)
  Thu, May 3, 2:00 - 4:00 pm LOM 215 (Prof. Auel)
Discussion Section
Teaching Fellows : Byungmin So and Joon-Hyeok Yim
Section Time : Mon 16:30-18:00 and Wed 13:00-14:15 (Byungmin)
Mon 14:30-15:45 and Tue 16:30-18:00 (Joon-Hyeok)
Office hours : Fri 15:00-16:00 (Joon) and Fri 16:00-17:00 (Byungmin)
Peer Tutors
Peer tutors : Charlie Kenney and Noah Montgomery
Office hours : Tue, Wed 06:00 - 08:00 pm (Charlie)
Mon 08:00 - 10:00 pm, Wed 07:00 - 09:00 pm (Noah)
Loct : Math Dept Common room

Description of course: Linear Algebra can be regarded as the art of solving linear equations. You have seen such equations since middle school: if 2x=4 then find x. In high school, you solved systems of 2 or 3 simultaneous linear equations in 2 or 3 variables. Such systems can be organized into a matrix equation Ax=b, where A is a matrix, x is a variable vector, and b is a constant vector. Linear algebra is a deep investigation into systems of simultaneous linear equations. In the course, we will examine such questions as: How do we know when a system of m linear equations in n variables has a solution? How many solutions can there be? How do we find them? We will mostly be interested in the abstract structure of the space of solutions, which forms an object called a vector space. While such questions might appear to be somewhat isolated, they are actually fundamental to the natural sciences, computer science, economics, and statistics. Furthermore, almost all higher mathematics today (geometry, topology, number theory, analysis, differential equations, etc.) depends on linear algebra in important ways.

This course will provide a rigorous proof-based introduction to linear algebra. The main topics covered will be vector spaces, linear transformations, matrices, systems of linear equations, determinants, eigenvalues, eigenvectors, diagonalization, inner product spaces, spectral theorem, and applications. Time permitting, we will investigate the linear algebra behind Google's PageRank algorithm and Heisenberg's uncertainty principle in quantum mechanics. Math 225 (as opposed to Math 222) is more focused on the abstract aspects of linear algebra and will demand a fair amount of maturity of mathematical thinking, not just rote problem solving. The course strikes a balance between computations, concepts, proofs, and applications. Problem sets will consist of a mix of computational and proof-based problems.

Expected background: Officially, the prerequisite is Math 120 (taken earlier or concurrently). In reality, we will hardly use any calculus or infinite series. However, it is important that you are comfortable with vectors and basic geometry of 3-dimensional space as taught in Math 120 (e.g., vector addition, scalar multiplication, dot product, magnitude, normal vectors, lines and planes in three dimensional space).

Homework 35%
Quizzes   10%
Midterm exam (08 Mar)   20%
Final exam (06 May) 35%
Grades: Your final grade will be based on weekly homework, short quizzes, a midterm exam, and a final exam. Notice that more overall emphasis is placed on exams than on weekly homework assignments in computing your final grade. On the other hand, completing your weekly homework will be crucial to your success on the exams and in the course.
Group work, honestly: Working with other people on mathematics is highly encouraged and fun. You may work with anyone (e.g., other students in the course, not in the course, tutors, ...) on your homework problems. If done right, you'll learn the material better and more efficiently working in groups. The golden rule is:
Work with anyone on solving your homework problems,
but write up your final draft by yourself.
Writing up the final draft is as important a process as figuring out the problems on scratch paper with your friends, see the guidelines below. Mathematical writing is very idiosyncratic - we will be able to tell if papers have been copied - just don't do it! You will not learn by copying solutions from others or from the internet! Also, if you work with people on a particular assignment, you must list your collaborators on the top of the first page. This makes the process fun, transparent, and honest.


(or otherwise the small print)

Homework: Weekly homework will be due in-class on Thursday. Each assignment will be posted on the syllabus page the week before it's due.

Late or improperly submitted homework will not be accepted. If you know in advance that you will be unable to submit your homework at the correct time and place, you must make special arrangements ahead of time. Under extraordinary circumstances, late homework may be accepted with a dean's excuse.

Your homework must be stapled (or otherwise securely fastened) together, with your name clearly written on the top. Consider the pieces of paper you turn in as a final copy: written neatly and straight across the page, on clean paper, with nice margins and lots of space, and well organized.

You might consider taking the opportunity to learn LaTeX.

Do not copy homework solutions from internet resources!

No homework will be due during the week of the midterm exam.

Your lowest homework score from the semester will be dropped.

Exams/quizzes: There will be two in-class 20 minute quizzes, announced a few days beforehand. The midterm exam will take place on Thursday 08 March, 07:30 - 09:00 pm in Davies Auditorium. The final exam will take place 07:00 pm - 10:30 pm on Sunday 06 May.

Your top quiz score will be weighted twice as much as your lowest score.

Make-up quizzes and exams will only be allowed with a dean's excuse.

The use of electronic devices of any kind during quizzes and exams is strictly forbidden.

Homework guidelines: Generally, a homework problem in any math course will consist of two parts: the creative part and the write-up.

  • The creative part: This is when you "solve" the problem. You stare at it, poke at it, and work on it until you understand what's being asked, and then try different ideas until you find something that works. This part is fun to do with your friends; you can do it on the back of a napkin. If you're having trouble, even in understanding what the problem's asking, use the resources available to you: my office hours, teaching assistants' office hours, weekly tutoring sessions, etc. Ask for help as early as you can! This part should all be done on "scratch paper."

  • The write-up: Now that everything about the problem is clear in your mind, you go off by yourself and write up a coherent, succinct, and nicely written solution on clean sheets of paper. Consider this your final draft, just as in any other course. This part you should definitely NOT do with your friends.