* Load is given in academic hour (1 academic hour = 45 minutes)
Course goals are to acquire theoretical and practical knowledge in linear algebra, in particular in understanding algebraic structures of linear operators, and solving the eigenvalue problem and diagonalization problem for (hermitian) 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 Linear algebra 2, the student will be able to:
* prove the rank-nullity theorem
* determine regular operators by using rank, nullity and determinant
* calculate the inverse of regular matrix by Gauss-Jordan algorithm
* explain properties of coordinatization in a given basis of vector space
* calculate the change of operator matrix after the change of coordinates
* explain the algebraic structure of algebra of linear operators
* prove that eigenvalues of a linear operator are zeros of the characteristic polynomial
* determine at least one solution of a system of differential equations Y'(t)=AY(t) for a given eigenvalue of operator A
* prove the diagonalization theorem for hermitian operators
* explain the basic properties of unitary operators
* determine the axis and angle of rotation A from the group SO(3)
Linear maps between Rn and Rm and matrices. Image and kernel of a linear map.
Composition of linear maps and matrix multiplication.
Regular operators. Inverting matrices with Gauss-Jordan operations.
Matrices of linear operators and change of bases .
Vector space of operators from Rn to Rm. Algebra of operators on Rn.
Binet-Cauchy theorem. Determinants of linear operators.
Characteristic polynomial, eigenvalues and eigenvectors of linear operators.
Operator spectrum of A and solutions of a system of linear differential equations y'=Ay.
Nilpotent and semisimple operators. Jordan decomposition (without proofs).
Hermitian adjoint of an operator. Quaternions.
Unitary operators. Rotations and reflections in R3 and Rn.
Diagonalization theorem for normal operators.
Hermitian operators and quadratic forms.
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.
- N. Elezović, Linearna algebra, Element, Zagreb 1995.
- D. Bakić, Linearna algebra, Školska knjiga, Zagreb, 2008.
- K. Horvatić, Linearna algebra, PMF-Matematički odjel i LPC, Zagreb
Linear Algebra 1