1. Basics of the classical two valued logic, connectives, formulas and their evaluation, principles and methods of mathematical statement proving - direct sequence, contradiction, induction 2. Binary relations between sets and on a set, operations with relations, product of relations, an inverse of a binary relation, properties of binary relations on a set, binary relation of equivalence, derived partition of a set, partial order on a set - Hasse diagram, greatest and lest elements of a partiallly ordered set 3. Set mappings, basic facts and properties - surjectivity and injectivity, mapping product, existence of an inverse mapping, set permutation 4. Binary operations on sets and their properties - commutativity, asociativity, neutral and inverse elements, basic algebraic structures with one binary operations - semigroups, monoids, groups, and with two binary operations - rings, fields and lattices 5. Linear space - construction, linear dependence and independence of vectors, subspaces and their structure, base and dimension 6. Scalar produkt in - length of a vector, angle of vectors, vector orthogonal projection into a subspace, orthogonal complement, Gramm-Schmidt orthogonalization 7. Matrix calculus - matrix types, symbolics, matrix equality, matrix operations, sum, scalar multiple, matrix product, matrix power, matrix transpose, the ring of square matrice 8. Matrix row operations, row space, reduced row echelon form of a matrix, matrix rank and its properties 9. Permutation sign, transpositions, Determinants, properties following from the definition, determinants of matrices in especial forms, Laplace theorem and its consequences 10. Matrix inverse, existence, two ways of calculation - via algebraic complements, via row operations, properties 11. Systems of linear equations - denotations, terminology, Gauss elimination, Cramer rule, using of an inverse matrix, homogeneous systems, solvability, solution space, 12. Spectral matrix analysis - matrix simillarity and its consequences, criterion for matrix simillarity, characteristic polynomial, eigenvalues and their properties, eigenvectors, eigenspace of an eigenvalue, Jordan normal form
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