Office Rm. Blocker 614A
E-mail: kuchment@math.tamu.edu, Home Page: http://www.math.tamu.edu/~kuchment
Sections: 699 and 700
Time: Asynchronous Distance Teaching
Linear algebra is a mathematical topic of great elegancy and enormous importance. Proficiency in it is required in any area of mathematics - analysis, algebra, geometry, etc., as well as applications - from physics to biology, economics, engineering, data science, image processing, cryptography, ... you name it. I will be happy to present it to you.
The Math 640 class assumes the prior knowledge of some basics of linear algebra, for instance
from the classes like Math 304, Math 323, or 423.
Although I will spend the beginning couple of weeks reviewing what you were supposed to learn in
that undergraduate class, it will be extremely useful if you do some refreshing before the class starts
(vectors, matrices, linear systems, linear dependence and independence, bases, complex numbers, etc.). Sources: books [GS] (the main one)
and [HR] listed below and notes distributed through eCampus and e-mail.
Among many good books available, I recommend the two books below, both available free of charge to our students on-line through the University library Web site (libcat.tamu.edu). You will find them useful for refreshing your memory, as well as throughout the class. No other books will be required.
[HR] Modern introduction to linear algebra, by Henry Ricardo. 2010, Call number QA184.2 .R53 2010
[GS] Linear algebra and its applications, by Gilbert Strang. 1988, Call number QA184 .S8 1988
Besides these two books, I will be distributing some handouts on more specific topics
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, tensors, and some numerical analysis topics. We will also cover relations to statistics, differential equations, linear programming, data science, etc.
Every week I will post on eCampus video lectures and additional materials when needed, as well as weekly homework.
I will set up a discussion board for the class, where you can exchange opinions, recommendations, questions, etc. (as long as this would not amount to cheating ). I will be monitoring the board and thus will be able to help when needed.
Although the class is asynchronous, you will be able to communicate with me individually by e-mail, or through ZOOM (in the latter case please arrange the time by e-mail).
| Week | Topic | Reference | Work |
| 1 | Systems of linear equations and geometry. Gauss elimination. | [GS] 1.1 - 1.3, {HR} 2.1 - 2.4 | HW1 |
| 2 | Matrix algebra. Vector spaces | [GS] 1.4 - 1.6, [HR] 3.1 - 3.3 | HW 2 |
| 3-4 | Linear independence. Bases. Dimension. Matrices as linear mappings (operators). | [GS] Ch. 2, [HR] 1.4 - 1.5, 5.3 - 5.4 | HW 3 and 4 |
| 5 | Inner products and geometry. | [GS] 3.1 - 3.4, [HR] 1.2 - 1.5 | HW 5 |
| 6 | Determinants | HW 6 | |
| 7 | Various important classes of matrices (operators). Quadratic forms. | HW 7 | |
| 8 - 10 | Eigenvalues and eigenvectors. Singular Value Decomposition. | HW 8 and 9, Project distributed. Due week 12 | |
| 11 | Jordan form | HW 10 | |
| 12 - 13 | Tensors | HW 11 | |
| 14 - 15 | Various applications | HW 12 |
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