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
(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
(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
(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
(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
HW#1, due 1/19. Selected
HW#2, due 1/26. Selected
HW#3, due 2/2. Selected
HW#4, due 2/9. Selected
HW#5, due 2/16. Selected
HW#6, due 2/23.
HW#7, due 3/2.
HW#8, due 3/9.
HW#9, due 3/16 at 9:00 AM.
Office hours: Mon, Wed, 1:00 - 2:30, Room 383M.
Office hours: Tuesday, 3-5; Wednesday, 4-6; Thursday, 3-5, room 380G.
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
- (1/10 - 1/12) Solving systems of linear equations. (Section 1.4
and Chapter 3)
- Elimination, reduced row-echelon form.
- Basic theorems on the existence and uniqueness of solutions.
- Matrix notation.
- (1/17 - 1/31) The geometry of vectors.
- Linear combinations and span. (Section 1.4)
- Subspaces of R^n. (Section 1.3)
- Linear independence, basis, and dimension. (Sections 1.5 and 1.6)
- The null space and column space of a matrix. (Section 2.1)
- Affine subspaces and the set of solutions to Ax=b. (Section 3.3)
- Inner products, orthogonal projection, Cauchy-Schwarz and
triangle inequalities. (Section 6.1)
- Application: least-squares approximation. (Section 6.3)
- Orthonormal bases and Gram-Schmidt orthogonalization. (Section 6.2)
- Vector spaces in general. (Section 1.2)
- (2/2 - 2/14) The algebra and geometry of linear transformations.
- Representing linear transformations from R^n to
R^m by m by n matrices. (Section 2.2)
- Examples: rotations, reflections, projections.
- Composition of linear transformations and matrix
multiplication. (Section 2.3)
- Injections, surjections, isomorphisms. (Appendix B, Section 2.4)
- Inverses. (Section 3.2)
- Representing linear transformations in different coordinate
systems. (Sections 2.2 and 2.5)
- (2/16 - 2/23) Determinants. (Chapter 4)
- Axioms for determinants.
- Further properties and computation.
- Cramer's rule.
- Geometric meaning of determinants: volume distortion, preservation
of orientation, invertibility.
- Existence of determinants.
- (2/26 - 3/12) Deeper understanding of linear transformations.
- Eigenvectors and eigenvalues. (Section 5.1)
- Diagonalizability. (Section 5.2)
- Symmetric matrices are diagonalizable.
- Applications: systems of linear ODE's (Section 2.7), Markov
chains (Section 5.3), the inertia tensor (if time permits - not in the book).
- (3/14 - 3/16) Summary and review.
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
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.
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