Math106
Linear Algebra 1
Faculty
Anier Velasco Sotomayor
Lead at the ML Theory group at Cohere for AI Open Science community.
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
- Our main goal is to master the concepts: vector spaces, linear dependence/independence, span and rank, matrices and their arithmetic, the basis, orthogonality, determinants, etc. – discovering them ourselves by solving problems (motivated by practice).
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). Gaussian Elimination.
Seminar: Solving problems on SLEs. Solving SLEs with elimination. Commenting on the code implementation of GE.
Session 3
Lecture: Matrices and matrix operations, inverse matrices.
Seminar: Solving problems on matrix arithmetic.
Session 4
Lecture: LU-decomposition.
Seminar (Lab): Implementing LU-decomposition in code, solving SLEs with the help of it.
Session 5
Lecture: Vector spaces, subspaces, the null-space of Ax=0.
Seminar: Solving problems on that.
Session 6
Lecture: The rank and the row-reduced form. Full solution to Ax = b.
Seminar: Solving problems on that.
Session 7
Lecture/Seminar: Linear independence, basis, dimension, span.
Lecture/Seminar: The four subspaces for Ax=b – the row and column space, the null-spaces.
Session 8
Lecture: Change of basis. Linear transformations in different bases.
Seminar: Solving problems on that.
Session 9
Lecture: Orthogonality. Orthogonal Complements.
Seminar: Solving problems on that.
Session 10
Snow Summit (free)
Session 11
Snow Summit (free)
Session 12
Lecture/Seminar: Least Squares approximation.
Lecture/Seminar: Linear Regression using Least Squares.
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.
Session 15
Final exam
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
Anier started in computer science by doing programming Olympiads in high school, during which he obtained a gold medal in the Cuban National Olympiad and a bronze medal in the Ibero-American Olympiad in Informatics.
He has a BSc. in Computer Science and Data Science from Harbour.Space University. His main professional interest is Machine Learning Theory Research, and he’s an independent researcher.
See full profileApply for this course
Linear Algebra 1
by Anier Velasco Sotomayor
Total hours
45 Hours
Dates
Jan 12 - Jan 30, 2026
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.