Vector spaces: definition of a vector space, dimension and basis, linear dependence, representation of vectors in a basis, coordinate system, products of vectors, projection of vector, GramSchmidtova orthogonalization. Matrices and determinant: concept of matrix, linear combination of matrices, transposition and Hermitian adjoint, matrix representation of vectors and operators, determinant, Laplace expansion, properties of determinant, permanent. Rank of matrix and inverse matrix: inverse matrix, elementary operations with matrices, rank of matrix, computation of rank, computation of inverse matrix. Systems of linear equations: homogeneous and nonhomogeneous system, vector and matrix notation, solution of the system, geometrical interpretation of the solution, GaussJordan elimination, Cramer's rule, LU decomposition. Eigenvectors and eigenvalues: eigenvalue equation, eigenvectors, degeneracy, matrix diagonalization, eigenvalue equation in chemistry. Operators: concept of operator, basis properties of operators, Dirac's braket notation, linear operators, hermitian operators, Schrödinger equation. Symmetry of molecules: concept and importance of symmetry, symmetry elements and operators, point groups, classification of molecules, orientation of molecule in coordinate system, simple application of symmetry in chemistry. Group theory: concept of group, group multiplication tables, direct product of groups, matrix representation of group. Point groups representations: reducible representation of point groups, characters of representation, character tables, irreducibile representations and notation, orthogonality theorem, reduction of reducible representation. Symmetry of functions: application of symmetry operators on functions, eigenvectors of symmetrical operators. Symmetry analysis: symmetry degeneration, molecular orbital theory, analysis of normal modes, symmetry selection rules, ligand field theory. Continuous groups: rotation groups, infinitesimal generators, geometry of rotation, quaternions.
EXPECTED COMPETENCES TO BE ACQUIRED:
to explain the basic principles of vector spaces
to use basic vector operations
to explain the properties of matrices and to use elementary operations and transformations of matrices
to find solutions to a system of linear equations
to interpret solutions to a system of linear equations by the usage of vector and geometrical representations
to explain the concept of operators and to know their properties
to determine eigenvalues and eigenvectors of operators
to explain the concept of molecular symmetry
to know and to use symmetry operators
to know basic concepts of group theory
to identify molecular elements of symmetry and to determine the point group of molecular geometry
to know and to use the properties of point groups for the assessment of basic physical and chemical properties of molecules

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