Lecture 1 - Preliminaries

Presentation of various algebraic objects with particular emphasis on differences and relationships between them. Definitions of number sets: natural numbers, integers, rational numbers, irrational numbers, real numbers, complex numbers, vectors and matrices. Relationships between the objects: the representation of a set of vectors as a matrix, the representation of a complex number as a vector.

Lecture 2 - Properties of number sets

Divisibility of integers, the congruence modulo, Chinese Remainder Theorem, different number systems.

Lecture 3 - Complex numbers

Algebra of complex numbers, algebraic and geometric representation. Modulus and argument, trigonometric form, de Moivre's formula.

Lecture 4 - Complex functions

Elementary complex functions: polynomials, roots of unity, roots of arbitrary degree, exponential and logarithmic functions. Basic properties of polynomials: divisibility, Bezout's theorem, the Fundamental Theorem of Algebra, solutions of square and cubic polynomials.

Lecture 5 - Algebra of matrices

Types of matrices, matrix operations, matrix of a system of equations. Gaussian elimination method of solving linear systems of equations.

Lecture 6 - Determinants and their applications

Determinant of a matrix: inductive and permutational definitions. Properties of determinants, Laplace's method of computing a determinant, Sarrus's formula. Inverse of a matrix and methods of finding the inverse of a matrix.

Lecture 7 - Linear systems of equations I

Cramer's formulas and their application, rank of a matrix and methods of computing the rank.

Lecture 8 - Linear systems of equations II

Kronecker-Capelli's theorem. The general form of the Gaussian elimination method of solving linear systems of equations.

Lecture 9 - Vectors and vector spaces I

Algebra of vectors, geometry of vector space, angle between vectors, scalar (inner) product, distance.

Lecture 10 - Vectors and vector spaces II

Linear independence, basis of linear space and its properties, coordinates with respect to a basis. Linear subspaces

Lecture 11 - Linear mappings I

Matrix of linear mapping, projection on subspace, eigenvectors and eigenvalues.

Lecture 12 - Linear mappings II

Transition of matrix of linear mapping with change of basis, diagonable matrices, Jordan Theorem.

Lecture 13 - Euclidean Spaces

Dot product, Euclidean space. Orthogonality, orthogonal decomposition, orthogonalisation algorithm for a set of vectors.

Lecture 14 - Volume measures

Determinant as a measure of volume, Gramian determinant. Vector product in E3.

Lecture 15 - Affine spaces

Straight line, plane, affine independence. Analytical geometry in a 3-dimensional Euclidean space.

The main aim of the course is to present the fundamental concepts and objects of algebra, such as number sets, polynomials, matrices, vectors, linear spaces and linear transformations. Special emphasis is placed on the presentation of objects which have applications in information technology and on their practical use.

The course is of a basic character. The entrance requirements are within the secondary school curriculum of mathematics.

The lectures are accessible in an electronic version in the educational system of PJWSTK (EDU). Each lecture contains exercises with enclosed solutions, aimed to check the understanding of relevant material. It is strongly recommended that the exercises are solved independently by each student. Every lecture is followed by a set of obligatory assignments which should be done and submitted by the specified time. The course consists of 15 lectures, each lecture should be worked out in the consecutive weeks of the semester - starting with the first week. The answers to the assignments should be submitted by the end of the adequate week. The students can obtain help from the instructor via e-mail and through the discussion forum in the EDU system.

The student obtains 1 point for each compulsory assignment. To pass the online course the student must score 80% of the total amount of points. The maximum number of points to attain is 60. The grades are based on a 5-step fixed scale. It is required to obtain 40 points to pass the semester. Only students who have passed the online course are allowed to take the final examination. The course is completed by the final examination held during the summer examination period.

David Lay, Linear Algebra Applications.
Seymour Lipschutz, Marc Lipson, Linear Algebra.
Schaum's Outline of Linear Algebra.
Frank Ayres, Modern Abstract Algebra.