Math 113: Linear algebra and matrix theory

Stanford University, ``Winter'' 2001


  • (3/23) Here are the grades.
  • (3/17) Exam clarification: in 9a, the matrix A is real. Also, an explanation is expected for every question, including true/false and calculations.
  • (3/16) If anyone has a good set of lecture notes, I would be grateful if I could make a photocopy of them for future reference. Please send me an email.
  • (3/16) Here is the final exam.
  • (3/9) Note that HW#9 is due on Friday 3/16 at 9:00 AM. This way we can go over the problems in class as needed. The final will be posted on Friday evening. I will hold extra office hours on Friday 3/16 from 12:30 - 2:00 in case there are any further questions.
  • (3/1) I will not be able to make it to my office hours on Wednesday 3/7. If you would like to see me, please send me an email, and we can arrange a time to meet on Tuesday or Thursday.
  • (2/26) There is a typo on HW#7: problem 1 should say section 4.2, not 4.3.
  • (2/7) Clarification on HW#4: For 2b, it helps to know that if you row reduce a matrix with linearly independent rows, you cannot get a zero row. (One can prove this by showing that if the rows of a matrix are linearly dependent, then they are still dependent, with slightly different coefficients, after a row operation.) For 4b, the vectors e_i are not the standard basis vectors of R^n; they are just an orthonormal basis for W. (Maybe a different letter would have been better.)
  • (2/7) HW#2 has been graded and can be picked up from the box outside my office, and HW#3 should be done in a couple of days. I'm sorry about the delay. Please don't be discouraged if your score is lower this time; I asked the grader to be more strict. Please note also that the grader will pick up the homework at 5PM SHARP, and will not accept any late homeworks.
  • (2/3) My office hours on Monday 2/5 are cancelled. Instead, I will hold office hours on Friday 2/9 from 1:00-2:30.
  • (2/2) In case you're not already familiar with it, MATLAB is a useful and fun software package for doing all sorts of computations with matrices, such as row reduction, etc. It's available on the main Stanford computers and has online help, and might be useful for one of the problems on HW#4.
  • (1/31) Olivier's office hours on Tuesday 2/13 are cancelled. Instead he will hold office hours on Monday 2/12 from 3-5.
  • (1/30) Problem 5 on HW3 is a little advanced, so we'll count it as extra credit.
  • (1/29) In HW3, problem 6, the vector space is over R.
  • (1/29) As I mentioned in the first class, if you don't know where to start on a homework problem, or if you are not sure if your proof is valid, it will probably help to discuss the matter with a classmate. If you would like help finding people to discuss the homework with, send me your name and contact info, and I will post a list.
  • (1/29) Just to clarify: the material I present in class is what I want you to learn. The book is just an extra resource. Unless you are very quick, you will probably need to go over your lecture notes after class in order to understand the stuff.
  • (1/9) If you don't make it to the first day of class, please be sure to fill out this survey.
  • (1/2) This course will be similar to the course I taught in Winter 2000. However it may be a tiny bit more advanced, and we will be using a different book.

    Homework assignments

    HW#1, due 1/19. Selected solutions.
    HW#2, due 1/26. Selected solutions.
    HW#3, due 2/2. Selected solutions.
    HW#4, due 2/9. Selected solutions.
    HW#5, due 2/16. Selected solutions.
    HW#6, due 2/23.
    HW#7, due 3/2.
    HW#8, due 3/9.
    HW#9, due 3/16 at 9:00 AM.


    Michael Hutchings
    Office hours: Mon, Wed, 1:00 - 2:30, Room 383M.

    Course assistant

    Olivier Daviaud
    Office hours: Tuesday, 3-5; Wednesday, 4-6; Thursday, 3-5, room 380G.

    Course goals

    Our primary goal is (surprise) to teach the fundamental concepts of linear algebra, which are essential in both pure and applied mathematics. Roughly speaking, one uses linear algebra whenever one has a mathematical problem or model involving more than one variable. As a working mathematician, I use linear algebra on a daily basis, perhaps even more than calculus. A detailed list of the topics we will cover is given in the syllabus below.

    A secondary goal is to work on mathematical reasoning and proofs. Proofs are an essential technique in higher mathematics, and I feel that even for someone who does not want to become a mathematician, a facility with proofs is one of the most important things one can learn from the mathematics department. In elementary math courses one has to find answers; in advanced courses one also has to find reasons and explanations. This is a fun game, full of beautiful tricks, but it can be difficult for beginners. In this course we will do some explicit work on proof techniques, to try to provide a transition to more advanced mathematical courses.

    What is the difference between Math 103 and 113?
    As opposed to Math 103, which focuses on computation and applications, the treatment of linear algebra in Math 113 will be more mathematically sophisticated, with a greater emphasis on theoretical understanding and proofs. On the other hand, we will not neglect computation and applications, for as Feynman said, ``If you cannot compute anything, then you know nothing.'' But if you do not understand the theory, you might not know what you are doing in your computations. Anyway, although we will start at the beginning of linear algebra, this course might be a little intense, although hopefully still manageable, if you have never seen a matrix before; and if you have, you shouldn't be too bored. You might want to shop both courses for a while to see which one is more suitable for you.


  • S. Friedberg, A. Insel, L. Spence, Linear algebra, third edition, Prentice Hall.

    This book has the advantage of being comprehensive and sophisticated, with lots of detailed explanations and examples and good problems. We will cover approximately 50% of its contents, and it should be of continued use to you in your future studies. A disadvantage is that some of the basic concepts might be a little hard to pick out amidst all the other stuff that is going on. This is one reason why it is important to come to class, where I will try to explain what I consider to be the most fundamental ideas, in the simplest possible manner. The book should be regarded as a resource for more comprehensive explanations, examples, and further directions.


    Disclaimers: the references below to sections of the book should give a rough idea of what we will be doing, but as explained above we will not always follow the book very closely. The schedule is only approximate.

    Homework policy

  • There will be a weekly homework assignment, consisting of a mixture of computations and proofs, problems from the textbook and problems that I make up. Assignments will be posted on this page, a week before they are due.
  • Homeworks are due on Fridays at 5 PM in the box outside my office door (room 383M).
  • No late homeworks will be accepted.
  • You are encouraged to discuss the homework problems with your classmates. However you must write up your homework on your own.

    Exams and grades

  • There will be no midterm.
  • There will be a take-home final exam, handed out on Friday March 16, and due on Wednesday, March 21 at 11:30 AM. It will be open book, and you can take as much time as you need, although it shouldn't require more than one day.
  • Course grades will be based on 50% homework, 50% final exam. The lowest homework score will be dropped in computing the homework average.
    Up to Stanford Undergraduate Mathematics.
    updated: 2/23/01