(This is the course web page for Section 101 of MATH 223; Section 102 also has a course web page.)

Lectures: Mondays, Wednesdays, and Fridays, 10:00-10:50 AM, room MATH 104 (Mathematics Building)
Office hours: Tuesdays 3-4 PM and Thursdays 10:30-11:30 AM
Office: MATH 212 (Mathematics Building)
Course web page: www.math.ubc.ca/~gerg/index.shtml?223
Email address:
Phone number: (604) 822-4371

Course textbook: Friedberg, Insel, and Spence, Linear Algebra, fourth edition, Prentice Hall, 2003. Please let me know if you encounter problems buying the textbook from the UBC bookstore. This textbook is "required", in that we will refer to theorem numbers from the book and some homework problems will be given as problem numbers in the book; however, nobody will ever be checking whether you own a copy. The book Linear Algebra and its Applications (4rd edition) by Lay is another possible reference for the material, although some of the notation will be different.

Course description: Linear algebra is the study of vector spaces and of linear functions from one vector space to another - although this description isn't very helpful if you don't already know what linear algebra is! Linear functions are exemplified by polynomials of degree one in several variables, like 3x-7y+z (as opposed to polynomials of higher degree, such as x2-5y7+z or xyz); matrices are a tool for writing down these functions. Vector spaces are sets where two elements can be added together, or an element multiplied by a constant, to obtain another element, like the ordinary Euclidean plane R2. Linear algebra deals, roughly speaking, with "flat" objects as opposed to "curved" objects; however, the subject can also address more abstract objects with no particular geometric meaning. In this wider sense, linear algebra is both incredibly useful in mathematics and also incredibly well-understood: we know how to answer almost any question that arises in linear algebra.

MATH 223 is aimed at excellent students (typically honours students, although anyone may enroll) who can go through the material at a faster pace and at a higher level of abstraction than in MATH 152 or MATH 221.

Prerequisites: To enroll in MATH 223, students must have either passed MATH 121 or else earned a score of 68% or higher in one of MATH 101, MATH 103, MATH 105, or SCIE 001.

Evaluation: The course mark will be computed from the following components in the proportions indicated.

  • Homework assignments: 10%
  • Two in-class midterms: 20% each
  • Final exam: 50%

To pass the course, students must receive a passing score (at least 50%) for their homework average and also earn a passing score on the final exam, as well as having an overall course mark of at least 50%. Note that the course marks might, at the end of the semester, be scaled upwards in order to make the grades comparable to previous years.

The homework part of the course will consist of 10 assignments, due roughly weekly except for the weeks of the midterm exams (the due dates are posted here). Each student's lowest homework score will be dropped, and the other nine scores will be averaged to determine the term homework score. Some homework problems will be computational, but others will require students to justify general statements (that is, students will be expected to write some proofs, in addition to understanding proofs). Students are allowed to consult one another concerning the homework problems, but your submitted solutions must be written by you in your own words. Students can be found guilty of plagiarism if they submit virtually identical answers to a question, or if they do not understand what they have submitted.

There will be two midterm exams, in the usual classroom at the usual time, on Friday, October 7 and Wednesday, November 9.

There will be a three-hour-long final exam sometime between December 6 and December 20. The precise date and time will not be announced until sometime in October. Students are forewarned not to make travel plans that might conflict with their final exams; it is a UBC policy that travel plans are not a valid excuse for missing or rescheduling a final exam.

Every now and then, a student might be unable, due to extraordinary circumstances, to finish assignments or attend midterms or the final exam. Students who miss the final exam should contact the Faculty of Science directly: they have a formal mechanism for dealing with that situation. Students who must miss a homework assignment should contact the instructor before the assignment is due; students who must miss a midterm exam should contact the instructor before the date of the midterm. Assuming the absence is for legitimate reasons, the course grade will be calculated from the remaining work (there will not be any makeup assignments or exams).

Advice for success:

  • Stay caught up! Mathematics is a very cumulative subject: what we learn one week depends crucially on understanding what we learned the week before. Students who fall behind early struggle to catch up for the rest of the course.
  • Put in the hours! Remember the 2-to-1 rule for university courses: expect to spend an average of 2 hours outside of class for every 1 hour spent in class. In our course, that means 6 hours per week, in addition to coming to lectures, is quite reasonable (and some students will spend more than that). Jump right in and start spending that time; don't wait until later in the course.
  • Work on the homework problems! It's tempting to try to find some short cut to obtaining the answers, such as taking dictation from a fellow student or searching the internet. Besides the fact that cheating in this way violates UBC's academic misconduct policies, it's important to realize that working on the homework is the primary way for you to learn the course material. Learning to do mathematics is like learning to do anything else: you can't learn how just by watching someone else do it.
  • Don't give up! In earlier math courses, everything we needed to be able to do might have been conveniently written in boxed formulas that we can instantly apply. In more abstract mathematics courses, however, we don't always immediately know the correct way to proceed; sometimes trial and error is necessary, and there's nothing at all wrong with this. Trying, struggling, going back to another idea, making mistakes, fixing them - these are all part of the learning process.
  • Come to office hours! If you are stuck in the middle of a homework problem or a concept from the course, you are on the cusp of a great learning moment. I am very happy to help you see the way past that obstacle. During my office hours, my sole responsibility is to talk to students about MATH 223 - so don't be shy about coming. (An additional resource for students in MATH 223 is the math department's drop-in tutoring. TAs who are signed up to provide help for MATH 221 are reasonably likely to be able to help with MATH 223 questions as well.)

Here are some tips for writing up your solutions:

  1. Justify all of your answers, even numerical answers. Simply stating the answer isn't enough; in other words, include general facts that allow you to draw your specific conclusions. Write enough to distinguish your solution from someone else who doesn't understand the material but is a very good guesser.
  2. Conversely, stating general facts isn't enough if you don't connect them to the problem at hand. For example, someone could write "v is in the span of S because there exists a linear combination of vectors in S that equals v, since the corresponding system of linear equations is consistent and therefore has at least one solution." This sentence correctly describes four different ways of saying the same thing, but never actually indicates why any one of those four things is true for the specific data in the problem.
  3. One rule of thumb is the telephone test: if you read your solution out loud over the phone to your friends (who are in the class but haven't thought about that problem themselves), would they be completely convinced, or would they have to ask you to clarify parts of the solution? If you'd need to clarify it to your friends on the phone, you should clarify your solution in writing.
  4. Suppose a statement makes an assertion about every vector, every matrix, every function, or the like, and you have to determine whether it's true or false. If it's false, then you can simply give one example where the statement is false. If it's true, though, then examples won't suffice: you need to write a general proof.
  5. It's very easy to make calculation mistakes (or even copying mistakes) when doing Gaussian elimination. Take advantage of the fact that you can always check your answer!
  6. Make sure you write all the relevant details, rather than expecting the reader to deduce them from the context (the grader can only grade what's written, not what you were thinking when you wrote it). For example, when you introduce a real number a or a free parameter t, does it represent any possible real number, or a specific real number from somewhere else in the problem?
  7. While it is possible to write all the proofs in this course without using mathematical induction, it becomes much easier to write (and read) solutions written using induction, once you get used to it. Lots of textbooks have sections about induction (not ours, sadly); I also found a quick introduction and a deeper treatment on the web.

Topics to be covered in this course:

Sep 7-9 Sections 1.1, 1.2, 1.3 Introduction to course, vector spaces, subspaces
Sep 12-16 Sections 1.4, 3.4 Linear combinations, systems of linear equations, Gaussian elimination
Sep 19-23 Sections 3.4, 1.5, 1.6 Gaussian elimination continued, linear independence, bases
Sep 26-30 Sections 1.6, 2.1 Dimension, linear transformations, null spaces, ranges
Oct 3-7 Sections 2.2, 2.3; Midterm #1 Matrices, composition, matrix multiplication
Oct 12-14 Sections 2.4, 2.5 Invertibility, isomorphisms, changes of coordinates
Oct 17-21 Sections 2.5, 3.1, 3.2 Elementary matrices and operations, rank, inverses
Oct 24-28 Sections 3.3, 4.4 Solutions of systems, determinants
Oct 31-Nov 4 Appendix D, Sections 5.1, 5.2 Complex numbers, eigenvalues, eigenvectors
Nov 7-9 Section 5.2; Midterm #2 Diagonalizability
Nov 14-18 Sections 5.2, 5.4 Diagonalizability continued, Cayley-Hamilton theorem
Nov 21-25 Sections 6.1, 6.2 Inner products, norms, Gram-Schmidt orthogonalization algorithm
Nov 28-Dec 2 Sections 6.2, 6.5 Orthogonal complements, orthogonal diagonalizability of symmetric matrices

Use of the web: After the first day, no handouts will be distributed in class. All homework assignments and other course materials will be posted on this course web page in PDF format. You may access the course web page on any public terminal at UBC or via your own internet connection.

Why do some people get better quickly when they work hard, while others don't seem to progress as fast? One answer is that deliberate practice is much more effective than going through the work just for the sake of finishing it. From a Freakonomics blog post (boldface is my emphasis): “For example, in school and college, to develop mathematics and science expertise, we must somehow think deeply about the problems and reflect on what did and did not work. One method comes from the physicist John Wheeler (the PhD advisor of Richard Feynman). Wheeler recommended that, after we solve any problem, we think of one sentence that we could tell our earlier self that would have ‘cracked’ the problem. This kind of thinking turns each problem and its solution into an opportunity for reflection and for developing transferable reasoning tools.”

Tim Gowers, a Fields Medalist and world-class mathematical expositor as well, is writing a series of essays on logic, mathematical foundations, and constructing proofs. Anyone who takes the trouble to thoughtfully read all these essays will definitely become better able to write and speak the language of mathematics.

An additional resource for students in MATH 223 is the math department's drop-in tutoring. TAs who are signed up to provide help for Calculus/Linear Algebra are always present and are reasonably likely to be able to help with MATH 223 questions as well; you can also see TAs who are signed up to provide help for Proofs.

For the homework assignments:

  • Don't turn in solutions to the Practice Problems. Most of these problems are easy, and most of them have answers in the back of the book, so they will be useful to you to check your understanding of the material. They will also be a great resource for reviewing for midterms and the final exam.
  • Write up and hand in solutions to the Homework Problems. Please have your solutions to these problems in the correct order on your pages. Please staple the pages together (unstapled solutions will not be accepted).
  • Write clearly and legibly, in complete sentences; the solutions you hand in will be better if you think about how to best phrase your answer before you begin to write it.
  • The Bonus Question is exactly that: you can skip it entirely if you want, but you can also solve it and earn extra marks on your assignment. (Note that your final homework average for the course can't be above 100%—but individual homework scores can!)

All homeworks are due at the beginning of class (10:00 AM) on the indicated days. The links below lead to the most up-to-date versions of the homework (reflecting any needed corrections, for example).


Solutions to homework

Other downloads