Лінійна алгебра

Course description
Linear algebra is definitely a must course for every computer science and business analytics programme. Our aim within this course is to provide students with a good understanding of the main concepts and methods of linear algebra and to help them to develop the abilities and skills of problem-solving using linear algebra.
We will cover the standard basic notions (systems of linear equations; matrices and matrix algebra;
linear vector spaces and subspaces, bases and linear transformations; inner products and orthogonality;
eigenvalues and eigenvectors, matrix factorization) as well as illustrate their application in data science, computer vision, geometry, optimization, difference/differential equations and discuss algorithms and their computational issues whenever possible
Topics:
The course is split into 5 modules 3 topics each
Every topic is covered in a lecture on Wed followed by a consultation, and seminars on Fri
Basic notions for each topic are discussed in the Lesson activity on the cms followed by a graded quiz that
must be completed before the lecture
Each topic also includes self-test
Module 1. Linear systems and matrices
Topic 1 Systems of linear equations
Systems of linear equations
Gaussian and Gauss-Jordan elimination
Existence and uniqueness of solutions
Topic 2 Matrix algebra
Matrices and vectors: basic notions
Matrix multiplication
Elementary matrix transformations and LU factorization
Topic 3 Invertibility and determinants
Invertible matrices
Determinants in dimensions 2 and 3. General definition of the determinant
Properties of determinants, geometric interpretation (volume and cross product), Cramer’s rule
Module 2. Linear spaces and transformations
Topic 4 Linear vector spaces
Linear vector spaces
Linear independence, bases, and dimension
Linear transformations and coordinates
Topic 5 Subspaces
Subspaces. Linear spans
Rank of a matrix
Four fundamental subspaces
Topic 6 Linear transformations between linear spaces
Linear transformations. Matrix of a linear transformation
Change of bases
Applications to difference equations
Module 3. Orthogonality
Topic 7 Orthogonality
Metric linear spaces. Inner products and normed linear spaces
Orthogonal vectors
Orthogonal projections
Topic 8 Orthogonalization
Orthogonal bases and orthogonal transformations
Gram-Schmidt orthogonalization
QR factorization
Topic 9 Some applications
Least square solutions
Application to linear models
Fast Fourier transform
Module 4. Eigenvalues and Eigenvectors
Topic 10 Eigenvalues and eigenvectors
Eigenvalues and eigenvectors
Jordan normal form
Functions of a matrix. Linear difference and differential equations
Topic 11 Symmetric matrices
Eigenvalues and eigenvectors
Bilinear forms and positive definiteness
Cholesky decomposition
Topic 12 Special matrices
Orthogonal and unitary matrices and their eigenvalues
Markov matrices and positive matrices
Sparse matrices
Module 5. Applications
Topic 13 Singular value decomposition
Singular value decomposition
SVD and rank compression
Applications
Topic 14 Project defence
Computer vision
Natural language processing
Topic 15 Project defence
Iterative algorithms
Classification with LA