Math113
Linear Algebra 1
Faculty
Maxim Beketov
PhD student in ML & Topology @ HSE
Course length
Duration
Total hours
Credits
Language
Course type
Fee for single course
Fee for degree students
Skills you’ll learn
Overview
Linear algebra – a study of lines, planes, vectors, linear equations, transforms, matrices, tensors – is the heart of many areas of mathematics. It spans from pure (like representation theory), to applied to any natural sciences such as physics & chemistry (linear algebra is the language of quantum mechanics), any numerical modeling (including mechanical and networks engineering) and data processing, statistical inference and machine learning. This course will be the beginning of your journey in this exciting field, covering the required theory and providing you with a lot of practice (including some numerical programming)!
Learning highlights
- Vector spaces
- Linear dependence/independence
- Span and rank
- Matrices and their arithmetic
- Orthogonality
- Determinants
Course outline
15 classes
Session 1
Lecture: Introduction. Vectors, linear combinations, the dot product, vector’s length.
Seminar (Lab): Vectors in Python & NumPy, comparing data with the dot-product.
Session 2
Lecture: Systems of linear equations (SLEs).
Seminar: Solving problems on SLEs.
Session 3
Lecture: (Gaussian) Elimination: the idea of it, elimination in terms of (augmented) matrix notation for SLEs.
Seminar: Solving SLEs with elimination. Commenting on the code implementation of GE.
Session 4
Lecture: Matrices and matrix operations, inverse matrices.
Seminar: Solving problems on matrix arithmetic.
Session 5
Lecture: LU-decomposition.
Seminar (Lab): Implementing LU-decomposition in code, solving SLEs with the help of it.
Session 6
Lecture: Transposes, permutations, symmetric matrices, and a link to LU-decomposition.
Seminar: Solving problems on that
Session 7
Lecture: Vector spaces, subspaces, the null-space of Ax=0.
Seminar: Solving problems on that.
Session 8
Lecture: The rank and the row-reduced form.
Seminar: Solving problems on that.
Session 9
Lecture: (Finally!) Completely solving Ax=b.
Seminar: Solving problems on that (+Gauss-Jordan elimination).
Session 10
Lecture/Seminar: Linear independence, basis, dimension, span.
Lecture/Seminar: The four subspaces for Ax=b – the row and column space, the null-spaces.
Session 11
Lecture: Orthogonal spaces, orthogonality of the four subspaces, projections onto subspaces.
Seminar: Solving problems on that.
Session 12
Lecture: Projection matrices and the least squares approximation.
Seminar (Lab): Numerous least squares approximation applications, (maybe) a bit on the Moore-Penrose pseudoinverse.
Session 13
Lecture: Orthogonal bases and the Gram-Schmidt procedure.
Seminar: Solving problems on that, (maybe) a bit on special orthogonal bases.
Session 14
Lecture/Seminar: Determinants, permutations, cofactors.
Lecture/Seminar: Cramer’s rule for matrix inverse, determinant as volume.
Final exam
Final exam
Course materials
Books
Prerequisites
We will develop the notions of linear algebra from the ground-up, so the only prerequisite is basic arithmetic, and good understanding of real numbers. For the practical “Lab” assignments, basic Python programming skills are required. Prior familiarity with vectors and complex numbers is desirable, but not essential.
Methodology
Our sessions will consist of two parts: a lecture with detailed slides covering the material, followed by a seminar with problems for you to solve (with as much of my help and guidance as required). During the “Lab”-seminars we’ll learn to implement our new knowledge in code.
Grading
I got my BSc and MSc in Physics & Mathematics from MIPT and Skoltech, with a specialisation in theoretical/computational physics (integrable and chaotic quantum systems, to be certain). Upon graduation I temporarily left academia, working as a data scientist for App in the Air (ask me about recommender systems) for a year, as well as teaching 10-11th grades IB DP Mathematics (passionate about teaching). I then realized that what interests me most is geometry & topology, so I worked as a Python developer for an architectural startup Archeads Inc., doing some computational geometry for procedural floor plan generation; and also joined a wonderful laboratory of Algebraic Topology and its Applications at the CS department of Higher School of Economics, where I'm now a PhD student researching topological data analysis and its applications in machine learning.
See full profileApply for this course
Linear Algebra 1
by Maxim Beketov
Total hours
45 Hours
Dates
Jul 12 - Jul 30, 2021
Fee for single course
€1500
Fee for degree students
€750
How to secure your spot
Complete the form below to kickstart your application
Schedule your Harbour.Space interview
If successful, get ready to join us on campus
FAQ
Will I receive a certificate after completion?
Yes. Upon completion of the course, you will receive a certificate signed by the director of the program your course belonged to.
Do I need a visa?
This depends on your case. Please check with the Spanish or Thai consulate in your country of residence about visa requirements. We will do our part to provide you with the necessary documents, such as the Certificate of Enrollment.
Can I get a discount?
Yes. The easiest way to enroll in a course at a discounted price is to register for multiple courses. Registering for multiple courses will reduce the cost per individual course. Please ask the Admissions Office for more information about the other kinds of discounts we offer and what you can do to receive one.