Spring 2022 Course Announcement: MATH 415/815: Theory of Linear Transformations
Since Professor Pitts didn't have an opportunity to introduce MATH 415 during the Spring 2022 Course Preview Event, he has provided much more information about the course here:
Instructor: Professor David Pitts, 213 Avery Hall
Class Time: M-W-F 10:30-11:20
Room: Avery 112 (subject to change)
Goal of Course: To learn about vector spaces and to study linear transformations.
About the Course: Linear algebra is fundamental in most areas of modern mathematics, and this course is one of my favorite courses to teach. The subject is truly beautiful and I particularly like the way you can "take apart a linear transformation to see what makes it tick". While we will not focus much on applications of linear algebra outside of mathematics, its abstraction lends itself to a remarkable array of applications to areas as diverse as: economics, chemistry, genetics, cryptography, and traffic flow.
Our course is a second course in linear algebra, and will be a concept based course suitable for students who have had an introductory course in matrix theory (e.g. Math 314) and who have also had experience in writing proofs (similar to what is found in Math 310 or Math 325 [or Math 309]).
In a first course (e.g. Math 314) on linear algebra, the focus is on matricies, computation, and relatively concrete vector spaces. In our course, we will deal with vector spaces, and conceptual properties of linear transformations on (mostly finite dimensional) real or complex vector spaces. (Much of what we do will also apply for other fields too.) While there will be overlap in subject matter with a first course, our approach will be geared to mathematics and mathematics students.
The course is more proof-based than computational, and many homework problems will ask you for the proof of a statement.
Topics: We'll study vector spaces, subspaces, bases, linear transformations, invariant subspaces, kernel, range, duality, and work our way to an understanding of the Jordan canonical form for a linear transformation. We'll also study inner product spaces and examine the spectral theorem for linear transformations on nite dimensional complex inner product spaces.
Prerequisites: Math 314 and experience with writing proofs, such as that found in Math 310 or Math 325 (or Math 309).
Some Course Structure: I'm expecting to have around 8 homework assignments, two midterms, a final exam, and weekly quizzes.
Text: The required text is Linear Algebra Done Right, 3nd Ed. by Sheldon Axler; published by Springer-Verlag. I will base some of my lectures on the Axler text and also the book Finite Dimensional Vector Spaces by Paul R. Halmos; also published by Springer-Verlag. Many of the exercises in the course will come from Axler's text.
Please contact me at dpitts2@unl.edu if you'd like further information about the course.