SYLLABUS MATH 416 SECTION E13:
ABSTRACT LINEAR ALGEBRA
SPRING 2016
1. General information
• Instructor: Ivan Contreras
– E-mail: icontrer at illinois dot edu
– Website: http://math.illinois.edu/ icontrer/
– Office: 107 Altgeld Hall
– Office hours: Wed. and Fri.: 2 PM-3 PM or by Appointment.
• Course information:
– Lecture and Discussion Time: Mo. Wed. Fri. 1:00 PM-1:50 PM
– Room: 140 HRY BLD
– Course website: http://math.illinois.edu/ icontrer/MATH16B.html, see also Moodle.
– Textbook: Linear Algebra, by Friedberg, Insel, and Spence (Fourth Edition).
– Additional Reference 1: A First Course in Linear Algebra by Robert A. Beezer, available
free online as a .pdf file.
– Additional Reference 2: Linear Algebra by Jim Hefferon, available free online as a .pdf file,
equipped with solutions.
• Grader: Hao Sun
– E-mail: haosun3 at illinois dot edu
2. List of topics
A preliminary list of topics that will be covered includes:
• Vector spaces, subspaces and linear transformations, change of basis.
• Solutions of linear systems of equations, raw reduction and echelon form.
• Matrices, determinants and invertibility.
• Eigenvectors, eigenvalues and diagonalization.
• Inner products, orthogonal projections, Gram-Schmidt process.
• Bilinear forms, Jordan canonical forms.
For a more extensive list of topics you may have a look at the Departmental syllabus.
3. What is MATH 416 about?
Welcome to MATH 416! In this course we will explore with rigor the main ideas of linear algebra. By
rigor we mean that there will be proofs for the different statements we will discuss during class. This
explain the adjective abstract for the title of this course, we will treat the topics from the syllabus from a
mathematical perspective, however, we will also spend some time in calculations and possible applications
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MATH 416 Spring 2016
Syllabus
of the material.
There are several moments in mathematics (and in daily life!) when the knowledge of linear algebra
comes out useful and effective. For instance, the study of approximations for real valued functions. An
engineer could be interested in parametrizing a surface of a material required in the production of a new
prototype. It could be the case that the surface obeys a rather involved expression, such as:
f (x, y) = ln(2x + y) + 10.
For practical purposes, it is much easier if the function would be linear, such as:
flin (x, y) = 2x + y − 1,
since he or she would be able to do algebra and perform computations. One important part of mathematics deal with the theory of how to approximate arbitrary functions by linear ones.
The idea of this course is to prepare you for such situations, by studying the theory of vector spaces,
linear transformations, matrices, etc. The first part of the course will be dedicated to review the basics
of vectors, matrices and linear transformations. We will continue by discussing the theory of eigenvector,
eigenspaces and diagonalizability.
The rest of the course will be devoted to introduce vector spaces with inner product, and their associated
theory: orthogonal projections, Gram-Schmidt process, an introduction to spectral theory. We will finish
describing bilinear forms and the canonical Jordan form.
4. Prerequisites
Officially you should have taken MATH 241. There are various reasons to ask this as the minimum
requirement: you should be familiar with real functions in one variable and several variables, as we saw
in the previous examples, linear algebra will be useful to understand them better! This also means you
are familiar with the notion of vectors and matrices. Anyways, we will briefly review the main concepts
needed throughout the course, and there will be time to practice with homework assignments.
MATH 347 is a recommended prerequisite. The idea of this course is to prove theorems and propositions,
so it would be very useful if you have a basic background in mathematical proofs. This course is focused
on understanding why the things work, rather than learning an algorithm by heart and performing large
computations. It is also part of the objectives of this course that you write your answers in a clear and
organized way; you should also try to verify that you are understanding the definitions and the notation
introduced in the lectures/textbook.
5. Homework, work in classroom and Exams
5.1. Homework.
• Generically, there will be one weekly homework assignment that will be posted on the website of
the course, as well as in Moodle.
• The assignment will be collected each Wednesday during the lecture.
• Maybe you have heard this many times, but it is a very good idea to do the assignments! This
is the only way that you know if you are really getting the material covered in the lecture, doing
mathematics is not sitting down and listening to someone else, it is done in paper or blackboard.
• The assignments will be graded with a scale 0 to 100, some selected exercises will be thoroughly
read and graded each time, the others will be spot-checked.
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MATH 416 Spring 2016
Syllabus
• Homework after the due date will not be accepted. If you happen to be absent when homework
assignment is due, you must ask someone turn it in for you during the lecture, or leave your
assignment in my my mailbox in 250 Altgeld before the lecture begins.
• Working in groups is always a good idea, and you may find a big profit by sharing your ideas
and questions to others. However, writing down the solutions is an individual process, so be sure
that you write the solutions to the assignment in your own words. If you happened to work in
an assignment with a partner (s) and you all came up with the same solution , please write the
name (s) of the students involved.
5.2. Exams.
• There will be three midterm exams, 50 minute each, during class:
– Midterm 1: Wednesday February 24.
– Midterm 2: Wednesday March 16.
– Midterm 3: Wednesday April 13.
• The final exam will take place on Friday, May 6, 8-11 AM.
• The midterms are compulsory and there will be no make-up exams. If there is an emergency
absence, supporting documentation will be required. If there is a time/date conflict with the
midterms, please let me know as soon as possible.
6. Grading
The final grade for the course will be based on the following distribution:
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Homework (16%)
Midterm I (18%)
Midterm II (18%)
Midterm III (18 %)
Final exam (30%).
7. Special arrangements
Any student with a disability requesting accommodations is required to register with Disability Resources
and Educational Services http://www.disability.illinois.edu/ and present an accommodation letter from
DRES to the instructor, preferably within the first two weeks of class. All discussions will remain
confidential.
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