1. komponenta

Lecture type	Total
Lectures	45
Exercises	60

* Load is given in academic hour (1 academic hour = 45 minutes)

COURSE AIMS AND OBJECTIVES: Introduction to the theory of linear operators on finite dimensional vector spaces. Basic notions on the theory in unitary spcaes.

COURSE DESCRIPTION AND SYLLABUS:
1. Linear operators in applications. Examples. Matrix representation of linear operator.
2. Image and null--space of linear operators. Rank. Injective operators. Isomorphism.
3. Space of linear operators L(V,W). Operator algebra L(V). Dimension of the space of operators. Dual space and dual basis. Linear functionals.
4. Matrix representation of linear operators. Rank of the matrix. Change of basis. Similar matrices. Invariants.
5. Eigenvalues and eigenvectors. Examples. Cgaracteristic polynomial. Gershgorin circles.
6. Invariant subspaces. Multiplicities of the eigenvalues. Diagonal and triangular form (Shur theorem). Hamilton-Cayley theorem. Minimal polynomial. Regular operators. Nilpotent operators. Jordan normal form.
7. Unitary spaces. Cauchy-Schwarz inequality. Norm. Orthonormal bases.
8. Gram-Schmidt orthogonalization. Orthogonal projection. Least squares approximations.
9. Riesz theorem. Adjoint operator and its matrix representation.
10. Hermitian operators. Unitary operators. Normal operators. Diagonalization of normal operators.
11. Symmetric operators and correspondig quadratic forms.

Enrollment :
Attended : Linear algebra 1

Examination :
Passed : Linear algebra 1

Mandatory course - Regular study - Mathematics