Course goals are to acquire theoretical and practical knowledge in linear algebra, in particular in understanding algebraic structures of linear operators, solving the eigenvalue problem and calculating the exponential function for linear operators.
LEARNING OUTCOMES AT THE LEVEL OF THE PROGRAMME:
1. APPLYING KNOWLEDGE AND UNDERSTANDING
1.1. identify the essentials of a process/situation and set up a working model of the same or recognize and use the existing models;
1.3. apply standard methods of mathematical physics, in particular mathematical analysis and linear algebra and corresponding numerical methods;
5. LEARNING SKILLS
5.1. search for and use physical and other technical literature, as well as any other sources of information relevant to research work and technical project development (good knowledge of technical English is required);
LEARNING OUTCOMES SPECIFIC FOR THE COURSE:
Upon passing the course Vector spaces, the student will be able to:
- prove the rank-nullity theorem
- determine regular operators by using rank, nullity and determinant
- explain properties of coordinatization in a given basis of vector space
- calculate the minimal polynomial of square matrix
- prove that eigenvalues of linear operator are zeros of the minimal polynomial
- determine the root subspace of an operator for a given eigenvalue
- calculate the exponetial function exp(tA) for a given operator A
- prove the diagonalization theorem for hermitian operators
- explain the basic properties of unitary operators
Eigenvalues, the characteristic polynomial and minimal polynomial of an operator.
Nilpotent operators. Invariant subspaces and elementary Jordan blocks.
Root subspace decomposition for a linear operator.
Jordan canonical form of an operator.
Convergence of a sequence of vectors. Entire functions of operators.
Operator f(A) as a polynomial p(A). Exponential function of operators.
Geometry of unitary spaces. The projection theorem.
Hermitian adjoint operator. Hermitian, antihermitian and unitary operators.
Diagonalization of normal and Hermitian operators.
Hermitian projectors and the decomposition of identity.
Normal operators on real vector spaces.
Positive operators and the polar decomposition of linear operator.
REQUIREMENTS FOR STUDENTS:
Students are expected to regularly attend lectures and exercises. Furthermore, students are required to pass two colloquiums during the semester, and to achieve at least 40% of the total number of points on them.
GRADING AND ASSESSING THE WORK OF STUDENTS:
The exam consists of two colloquiums and possibly an oral examination. Additional points can be achieved by successful solving homework assignments.