Description
Overview
This is a page of my course for the $1^{st}$ year students of Master’s Program “Advanced Combinatorics” at MIPT.
What is this course about? Linear algebra is, roughly speaking, a “linear part” of the geometry of finite-dimensional spaces. It studies vector and affine spaces, linear operators and affine transformations, and Euclidean and Hermitian structures. Linear algebra has important applications in pure mathematics, computer science, physics, engineering, and many other areas. However, this subject is very interesting in itself. Its important feature is a beautiful combination of algebra and geometry. Suppose we perform some kind of transformation, such as a reflection or stretching of the space. How do geometric objects change under this transformation? And what does remain unchanged, i.e., what are the invariants of this transformation? Linear algebra provides answers to these and other similar questions.
The course is organized as follows. The duration of this course is approximately 10 weeks. Each week you will watch several videos with the total duration of 40-50 minutes. Each video is related to a particular topic. In addition to the theory, we shall also discuss some exercises. Moreover, weekly seminars (or webinars) and testing will accompany this course. Your final grade will be based on your test results, your homework and the final written exam.
I wish you to experience the process of mathematical discovery during the course and develop the appreciation of linear algebra, see you on the course!
Prerequisities
Groups, rings, fields, matrices, systems of linear equations, Gauss method and elementary transformations, rank, trace, determinant of matrices.
Program
This is a preliminary version of the program. Some small changes are possible during the semester.
- Vector Spaces. Linear Independence. Basis. Dimension. Linear Maps. Coordinates.
- Affine Spaces. Affine Hull. Euclidean Affine Geometry. Affine Transformations and Motions.
- Bilinear Functions (Forms).
- Symmetric Bilinear Functions (Forms). Quadratic Forms. Inertial Law. Orthogonal Basis for Symmetric Bilinear Forms.
- Euclidean and Hermitian Spaces. Gram matrices. Euclidean Affine Spaces. Convex Sets and Convex Polyhedra. The Minkowski-Weyl Theorem.
- Linear Operators. Eigenspaces.
- Diagonalization of Symmetric Operators. Polar Decomposition. 9. Non-Euclidean Geometry: sphere. 10. Non-Euclidean Geometry: hyperbolic Lobachevsky space.
Lectures (Weeks)
- Week1: Preliminaries. Vector Spaces. Linear Independence. Basis. Dimension. Linear Maps. Coordinates.
- Week2:
- Week3:
- Week4:
Week5:
Week6:
Week7:
Seminars & Problems
Problem Set 1. Vector Spaces.
Problem Set 2. Affine Spaces, Maps and Transformations.
Problem Set 3. Linear Maps, Linear Functions and Bilinear Forms.
Problem Set 4. Quadratic Forms.
Problem Set 5. Euclidean and Hermitian Spaces.
Problem Set 6. Convex Polytopes.
Problem Set 7. Linear Operators.
Seminar Results
References
[Ax] S. Axler — Linear Algebra Done Right.
[Vi] E.B. Vinberg — Course in Algebra, 2003, Graduate Studies in Mathematics, Volume 56. AMS.