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Syllabus of Math 423, Sections 500 and 200 (honors)

Linear Algebra II, Spring 2017

Instructor Peter Kuchment

Office Rm. Blocker 614A, Telephone (979)862-3257

E-mail: kuchment@math.tamu.edu, Home Page: /~kuchment

Introduction

Linear algebra is a mathematical topic of great elegancy and enormous importance. Proficency in it is required in any area of mathematics - analysis, algebra, geometry, etc., as well as applications - from physics to biology, economics, engineering, ... you name it.

The Math 423 class assumes the prior knowledge of some basics of linear algebra from the classes like Math 304 or Math 323.

Starting with a brief going through the basics, which are assumed to be known: concerning vector spaces, bases, Gauss elimination, and matrices, we will progress to a variety of more advanced topics and applications. These, time permitting, will include duality of spaces and operators, bilinear and quadratic forms, various important matrix decompositions, Euclidean and unitary spaces, various classes of matrices (self-adjoint (Hermitian), normal, orthogonal, unitary, positive definite, etc.), spectra, resolvents, eigenvectors and generalized eigenvectors, normal forms, functions of matrices, some numerical analysis topics. Time permitting, we might also cover relations to differential equations, quantum mechanics, and linear programming.

Students who attend the "stacked" Honors sub-section 200, will have to address somewhat more advanced topics and will get different homeworks, quizzes, midterms, and finals.

This is a rigorous, proof-based course, which means that close attention will be paid to students' ability to write mathematical statements and proofs mathematically and grammatically correctly.

Catalog description

Eigenvalues, similarity and canonical forms, advanced topics to be chosen by the instructor.

Prerequisite

MATH 304 or MATH 323, or instructor's content.

Grading

  1. Participation - 5% of the grade.
  2. Weekly home assignments or quizzes - 20%.
  3. Two mid-term exams - 25% each.
  4. Final exam - 25%
Home assignments and tests will contain three parts. Parts 1 and 2 will be mandatory for the regular section 500, parts 2 and 3 for the honors section 200 (part 1 not required and not counted).

Tentative schedule of the course

Dates # of sessions Topic Weekly HW Exams
2 Overview of Linear Algebra. Sections 1, 9, 10 HW1, Q1
4 Euclidean vector spaces. Sections 2, 3, 5 HW2
4 Bilinear and quadratic forms. Sections 4-7 HW3
February 23rd 1 Exam 1
2 Complex Vector Spaces, Unitary spaces, Section $8. Q2, March 2nd
2 Subspaces HW4
4 Various classes of operators. The spectral theorem. Ch. II + notes. HW5, 6
Variational approaches, SVD, etc. Ch. II + notes. HW 7
1 Relations to quantum mechanices
Functions of matrices and ODEs
April 4th 1 Exam 2
Jordan form. Ch. III HW8
Tensors. Ch. IV
May 4th 12:30 - 2:30 pm Final exam

Academic Calendar

Final Exams Schedule

Supplementary reading:

There are many other great books on advanced linear algebra and its applications, from browsing through which students can get a broad view on a variety of topics, e.g.
  1. P. Halmos, Finite dimensional vector spaces, this wonderfully written classics has been through many editions over the years. Somewhat similar to the next Glazman and Ljubich's book.
  2. I. Glazman and Y Ljubich, Finite-Dimensional Linear Analysis: A Systematic Presentation in Problem Form (Dover Books on Mathematics) Paperback, 2006.
    This is a very unusual book, which through the problems leads students not only through linear algebra, but also "covertly" prepares them for much more advanced topics, such as functional analysis.
  3. R. Bellman, Introduction to matrix analysis, A wonderful classics, with many applications.
  4. T. W. Korner, Vectors, pure and applied, Cambridge Univ. Press, 2013.
    Some other good linear algebra textbooks
  5. An outstanding book on linear algebra using Matlab:
    LLOYD N. TREFETHEN AND DAVID BAU, III, Numerical Linear Algebra, SIAM 1997.
  6. G. Strang, An introduction to linear algebra, Wellesley-Cambridge Press; 5th edition, 2016.
  7. G. Shilov, Linear algebra, Dover 1977.
  8. S. Axler, Linear algebra done right, 3rd edition, Springer 2014.
  9. A classical treatise F. Gantmacher, The theory of matrices. vols. I and II. New York, Chelsea, 1959. Freely available on-line.
  10. A more contemporary treatise R. Horn and C. Johnson, Matrix analysis. One of the older editions available on-line.
  11. Also by F. R. Gantmacher, Applications of the theory of matrices. New York, Interscience, 1959.

Class Announcements, E-Mail Policy and Communications:

Class announcements will be posted on my homepage and emailed to your NEO accounts. It is your responsibility to check the accounts daily.
E-mail (kuchment AT math.tamu.edu) is the preferred way to contact me.
When writing to me, please include your full name and "Math 423". Use your tamu.edu e-mail account to send me e-mails.

Make-up policy:

Make-ups for missed quizzes, home assignments and exams will only be allowed for a university approved excuse in writing. Wherever possible, students should inform the instructor before an exam or quiz is missed. Consistent with University Student Rules , students are required to notify an instructor by the end of the next working day after missing an exam or quiz. If there are confirmed circumstances that do not allow this (a written confirmation is required), the student has two working days to notify the instructor. Otherwise, they forfeit their rights to a make-up.

Late work

Late work will not be accepted, unless there is an university approved excuse in writing. In the latter case student has a week to submit the work.

Grade complaints:

Sometimes the instructor might make a mistake grading your work. If you feel that this has happened, you have one week since the graded work was handed back to you to talk to the instructor. If a mistake is confirmed, the grade will be changed. No complaints after that deadline will be considered.

Students with Disabilities:

The Americans with Disabilities Act (ADA) is a federal anti-discrimination statute that provides comprehensive civil rights protection for persons with disabilities. Among other things, this legislation requires that all students with disabilities be guaranteed a learning environment that provides for reasonable accommodation of their disabilities. If you believe you have a disability requiring an accommodation, please contact Services for Students with Disabilities (Cain Hall, Room B118, or call 845-1637).

Copyright policy:

All printed materials disseminated in class or on the web are protected by Copyright laws. One xerox copy (or download from the web) is allowed for personal use. Multiple copies or sale of any of these materials is strictly prohibited.

Scholastic dishonesty:

Copying work done by others, either in class or out of class, looking on other student?s papers during exams or quizzes, having possession of unapproved information in your calculator/computer/phone, etc., and/or having someone else do your work for you are all acts of scholastic dishonesty. These acts, and other acts that can be classified as scholastic dishonesty, will be prosecuted to the full extent allowed by University policy. In this class, collaboration on graded assignments, either in class or out of class, is forbidden unless permission to do so is granted by the instructor. For more information on university policy regarding scholastic dishonesty, see University Student Rules at http://studentrules.tamu.edu/.
"An Aggie does not lie, cheat, steal, or tolerate those who do." Visit http://www.tamu.edu/aggiehonor and follow the rules of the Aggie Honor Code.

GOOD LUCK IN YOUR STUDIES!!!

This syllabus is subject to change at the instructor's discretion

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Last revised March 1st, 2017