Linear Algebra: Problems and Methods

The goal of the course is explain the fundamental ideas of linear algebra and how to use them to find easy solutions of hard problems.
3-4 hours / week
Certificate Stepik

About this course

Linear algebra is an area of mathematics which studies linear functions and linear equations. If this sounds too simple, notice that the arguments and values of linear functions can be not only numbers, but also vectors, and this leads to very beautiful structures which we are going to study in this course. Moreover, linear algebra allows to study some nonlinear objects, such as quadratic forms!

Linear algebra has applications in almost any field one can imagine: physics, computer science, chemistry, economics, engineering and many, many others. It is also deeply connected to the other areas of mathematics, such as calculus, probability, group theory, differential equations etc. Ultimately, linear algebra is very exciting on its own: it allows to use geometry when algebra gets very complicated, and vice versa!

What is the best way to study linear algebra? As many other areas of mathematics, linear algebra has some building blocks:

  • key concepts,
  • main results,
  • common tricks;

The number of such building blocks is many times less than the number of pages in a typical textbook on linear algebra. At the same time, once you know these building blocks, you are ready to use linear algebra!

The goal of this course is to help you to find and understand the building blocks of linear algebra. Moreover, it is important to develop your personal vision and intuition about them. This cannot be achieved by solving 1000 similar typical exam problems. In this course you will be provided with a sort of "minimal linear algebra kit", so that you can manage the saved time yourself. In particular, you are encouraged to discuss the lessons here at stepic, to participate in competitions and other events at your campus, and to surf web resources such as math.stackexchange.com, mathoverflow.net, dxdy.ru (in Russian), arxiv.org and of course Wikipedia. If you are enrolled in a traditional course of linear algebra, you will benefit from this course too, since it is not designed to substitute the traditional courses.

The first lesson is available without registration here.

Who is this course for

High school graduates, first year students at universities, anyone who would like to improve his/her background in linear algebra.

Familiarity with the standard vectors (directed arrows) on the plane (or in the space); real, rational and complex numbers; how to solve systems of 2 linear equations with 2 unknowns; how to find roots of a polynomial, etc. Curiosity, ability to work independently, interest in competition level problems. Interest (and some background) in physics is a plus.

Meet the Instructors

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Nikolay Gusev
Moscow, Russia
PhD in Mathematics (Steklov Mathematical Institute), researcher.
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Paolo Bonicatto
Trieste, Italy
I am currently a first year PhD student at International School in Advanced Studies (ISAS), Trieste - Italy. I am studying Mathematical Analysis, Modeling and Applications. My research interests are Partial Differential Equations (mostly hyperbolic ones) and Geometric…
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Stefano Modena
Trieste, Italy
PhD student at SISSA (ISAS).
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Andrey Gavrilyuk
Winner of the International Mathematical Olympiad in 2005. Olympiad participants trainer. Senior lecturer at Faculty of Computer Science, Higher School of Economics, Moscow. PhD in Mathematics.
I do mathematics and teach mathematics, travel a lot and meet new people. It led me to pshycology and neurosciences research.

Course content

Linear Spaces
Linear Transformations
Inner Product Spaces
Advanced topics I
Advanced topics II



Learners' reviews

Very good course with good math, I liked it very much and helped a lot to refresh and improve math in linear algebra.
This is really very well constructed course. With right amount of efforts and struggling you definetly can get the maximum from it and to learn (or maybe, what is more suitable, refresh) linear alebgra more fundamentally than on other courses I've seen. The main problem of the course is that, especially in the last steps, it looks like a dead valley with tumbleweeds. There is noone to check your solutions or to help you with exercises: I was reviewing some solutions older than 3 years. But nevertheless, you should definitely try it, if you want to have a good picture of linear algebra in your mind.
The best linear algebra course I have seen so far. Starting from very basics soon you'll get to super advanced subjects. Be ready to struggle on your own, since certain subjects require you to do your own research. There are even tasks to prove theorems. That is what many courses lack, but not this one. The main disadvantage of the course is the cross review system, but I understand that one can't create an automatic proof verification system, so we had to rely on peer review.

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