### Description

The course's main goal is to explain the fundamental concepts of linear algebra that are used in data analysis and machine learning. Another goal is to help students improve their practical skills in using linear algebra methods in machine learning and data analysis. You will learn the fundamentals of working with vector and matrix data, as well as how to solve systems of linear algebraic equations and find basic matrix decompositions, as well as a general understanding of their applicability.

**Syllabus :**

1. Systems of linear equations and linear classifier

- Introduction to Linear Algebra
- Linear Algebra and Calculus
- Matrices and Multidimensional Vectors
- Matrix arithmetics
- Properties of matrix operations and some special matrices
- Vectors and matrices in Python
- Systems of linear equations
- Matrix inverse
- Gaussian elimination. The first example
- Elementary row operations
- Gaussian elimination. Main theorem.
- Gaussian Elimination. The algorithm.
- The Inverse matrix with Gaussian elimination
- LU and PLU decomposition

2. Full rank decomposition and systems of linear equations

- Abstract algebra and linear algebra
- Axioms of vector spaces: first application
- Examples of vector spaces
- Subspaces
- Linear combinations and spans
- Basis and linear dependence
- Dimension of a vector space
- Examples of bases
- Linear dependence and rank
- Formula for the solution of a SLAE
- An example of vector representation of the set of solutions
- Rouché–Capelli Theorem
- Full rank decomposition

3. Euclidean spaces

- Coordinates and basis
- Coordinates change example
- Euclidean space
- Geometry and Euclidean spaces
- Orthogonal and orthonormal bases
- Distance and orthogonal projections
- Inconsistent systems and the least squares method
- Linear regression example
- Introduction to support vector machine
- Linear regression and SVM with Python