* 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 solving systems of linear equations, and understanding geometric and algebraic structures of sets of solutions of systems of linear equations and their connection to 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:
- solve the system of linear equations by using Gauss method
- determine the matrix of a linear map from Rn to Rm
- reduce a matrix to a row eschalon form by using elementary transformations
- find a basis of a subspace of Rn defined by a spanning set
- find a basis of a subspace of Rn defined by a system of equations
- interpret the determinant as a volume of a paralleletope in u Rn
- prove the Binet-Cauchy theorem
- explain the connection between scalar product and orthogonal projection of vector on a line
- orthonormalize a sequence of vectors by using Gram-Schmidt process
- solve a minimization problem for || Ax - b || with the least squares method
- describe sets O(2), SO(2), U(2), SU(2) of orthonormal bases in R2 and C2
- apply the cross product in solving some geometrical problems in u R3
Real and complex numbers. Systems of linear equations. Triangular systems.
Elementary operations on equations. Gauss elimination. Homogeneous systems.
Vector space Rn. Linear span of vectors. Elementary operations on vectors.
Bases of Rn. Bases and elementary operations.
Linear independence of vectors in Rn. Dimensions of vector spaces.
Kronecker-Capelli's theorem. Rank-nullity theorem. Rank of a transposed matrix.
Norms and inner products on Rn and Cn. Triangle inequality.
Orthonormal bases. Gram-Schmidt orthogonalization process.
Projection theorem. Best approximation theorem. Least squares method.
Determinants. Determinants and elementary operations. Orientation on Rn.
Cramer's rule. Determinant of a transpose matrix. Laplace expansion.
Gram determinant. Cross product in R3.
Lines and planes in Rn. Equations of lines and planes.
Analytic geometry in R2 and R3.
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, Element, Zagreb
- D. Bakić, Linearna algebra, Školska knjiga, Zagreb, 2008.
- K. Horvatić, Linearna algebra, PMF-Matematički odjel i LPC, Zagreb