[Note: lectures will be held regularly every week, except
on the official university
holidays (i.e., F 11/24). Any other change of the schedule must be agreed with
the instructor. Students' requests to reschedule academic obligations in order
to
observe their religious holidays will be definitely honored.]
2. Policies
Attendance: Attendance of the lectures is not mandatory, nevertheless it
is
strongly recommended. In fact, I will try to pursue a quite different approach
to the subject, in comparison with the one in the textbook and although I will
regularly post several handouts, as a complementary reference for what covered
in
class, they will not be meant to (and cannot) substitute the lectures.
Moreover, in-class participation, discussions and requests of clarification will
be
highly encouraged, as a peerless opportunity for you to acquire the mathematical
way of approaching problems through abstraction (in particular for freshmen,
taking
their first course in MAT).
Homework: Each week, a set of homework exercises will be assigned. They
will
consist of both theoretical and computational problems. Completing the homework
is the most important thing you can do to master the material of this course!
Your homework will be posted on the Blackboard and due (generally) on Mondays
by 2 pm; it should be put in the mailbox out of my office (1210 Fine Hall). Late
submissions will not be accepted, unless prior arrangements have been agreed.
People usually find it helpful to work on homework together with other students;
I highly recommend forming study groups to work on assignments. However, I
require that you write them up individually!
Please staple your homework, number your pages and be as much neat and
understandable
as you can.
Exams: There will be a midterm and a final (both
in-class, unless different speci-
ed). The dates will be decided later on.
Quizzes: There will be several in-class quizzes from time to time (about
one every
two weeks ). The dates, the topics and the form of the test will be set later and
announced in advance.
Other: It will be possible, for willing students, to get extra credit
working out
individually (under my supervision) a topic or an application, not covered in
class,
and presenting it to the rest of the class. Please, contact me if you are
interested.
Grades: The relatives weights will be:
|
Homework: |
15 % |
|
Quizzes: |
15 % |
|
Midterm: |
30 % |
|
Final: |
40 %. |
3. Syllabus
This course is an introduction to the foundations of linear algebra and its
applications
to different fields. Our orientation (as opposed to that of MAT 202) will be
mainly theoretical; nevertheless, we will discuss various applications to other
fields.
I will prove theorems in class and you will be expected to prove theorems as a
part
of your coursework. My hope is that you will leave this course with both
competency
in the techniques of linear algebra and familiarity with the mathematical
way of approaching problems through abstraction.
This is a (very) tentative list of topics that, time permitting, will be
covered:
• Vector spaces and subspaces:
definition of a vector space and a subspace, linear dependence and independence
of vectors, spanning sets, bases and dimension.
• Algebra of matrices:
definition of matrices, addition and matrix multiplication, special matrices,
triangular factorization , determinant and rank of a matrix and their
properties.
• Systems of linear equations:
definition of systems of linear equations (LS) and homogeneous ones (HLS),
matrix representation, set of solutions (its representation and properties ),
Gauss-Jordan elimination method , Kroenecker theorem, Rouché - Capelli
theorem and Cramer theorem.
• Representation of vector spaces and subspaces:
coordinate (matrix) representation w.r.t. a basis, change of basis, cartesian
equations of a vector subspace, associated HLS and parametric equations.
• Linear transformations:
definition of a linear application, matrix representation, cartesian equations,
change of bases and matrix of a linear transformation, kernel and
rank, equations of the image and counter-image of vector subspaces, eigenvalues
and eigenvectors of a linear operator , eigenspaces, multiplicity of an
eigenvalue , basis of eigenvectors, diagonalizable operators and their
properties,
characteristic polynomial and algebraic multiplicity of an eigenvalue,
diagonalization of symmetric and hermitian matrices, Jordan Form.
• Bilinear and quadratic forms , inner products and
orthogonality:
definition of bilinear and quadratic forms and their matrix representations,
diagonalization of symmetric bilinear forms, Lagrange algorithm and
Sylvester's theorem; inner products , angles, orthogonality, orthogonal vectors
and subspaces, orthogonal bases, Gram-Schmidt process.
• Applications and examples:
time permitting and according to the interests of the class, a selection
of applications and examples, taken from differential equations, Fourier
analysis, game theory etc... will be presented. Further applications might
be investigated (and lectured to the class) by individual students, as part
of an independent work for extra credit.
