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Linear Algebra: Problems and Methods

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To watch this video please visit https://stepik.org/lesson//step/

About the 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.

Instructors

  1. User picture
    Nikolay Gusev
    Moscow, Russia
    PhD in Mathematics (Steklov Mathematical Institute), researcher.
  2. User picture
    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 Measure Theory. In particular, I am working on Linear Transport problems focusing on the existence and uniqueness of solutions to equations with non-smooth coefficients.

    When I was undergrad, I used to write in some Math Forums: I like discussing with interested people about stimulating and tricky problems.
  3. User picture
    Stefano Modena
    Trieste, Italy
    PhD student at SISSA (ISAS).
  4. User picture
    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.

Reviews

Rated:  5
Interesting and useful course. Course has more practical than theoretical profile. Course is perfect for those who has already learnt material to refresh memory using interesting and sometimes complicated tasks. There are some "spoiler hints" from a course team for most complicated tasks which helps you to understand hard things step by step by yourself. It is possible to learn "from zero" using this course, but it is not about watching a lot of videos. There are a lot of text pages with proofs of theorems. Sometimes it is really helpful to use google. But, strictly saying, it is unneccessary (but sometimes it is hard to understand material without it).Be ready to work hard and you will be enjoyed with course like i did! Good Luck!
Play
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Stream TypeLIVE
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00:00
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00:00
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  • 1.75x
  • 1.5x
  • 1.25x
  • 1x
  • 0.75x
  • 0.5x
1
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  • Chapters
Play
To watch this video please visit https://stepik.org/lesson//step/
5 All reviews

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.

Workload:
3-4 hours / week
Language:
English
Certificate:
Yes
Certificate condition: 100 points
With distinction: 180 points

About the 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.

Requirements

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.

Target audience

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

This course is entirely free. All content is available now.