In order to pass the course (grade 3) the student should be able to
give an account of important concepts and definitions in the area of the course;
exemplify and interpret important concepts in specific cases;
formulate important results and theorems covered by the course;
describe the main features of the proofs of important theorems;
express problems from relevant areas of applications in a mathematical form suitable for further analysis;
use the theory, methods and techniques of the course to solve mathematical problems;
present mathematical arguments to others.
Higher grades, 4 or 5, require a higher level of proficiency. The student should be able to solve problems of greater complexity, i.e. problems requiring a combination of ideas and methods for their solution, and be able to give a more detailed account of the proofs of important theorems and by examples and counter-examples be able to motivate the scope of various results. Requirements concerning the student's ability to present mathematical arguments and reasoning are greater.
Linear spaces: subspaces, linear span, linear dependence, basis, dimension, change of bases. Matrices: rank, column space and row space. Determinants of arbitrary order. Linear transformations: their matrix and its dependence on the bases, composition and inverse, range and nullspace, the dimension theorem. Euclidean spaces: inner product, the Cauchy-Schwarz inequality, orthogonality, ON-basis, orthogonalisation, orthogonal projection, isometry. Quadratic forms: diagonalisation, Sylvester's law of inertia. Spectral theory: eigenvalues, eigenvectors, eigenspaces, characteristic polynomial, diagonalisability, the spectral theorem, second degree surfaces. Systems of linear ordinary differential equations.
Lectures and problem solving sessions.
Written examination at the end of the course. Moreover, compulsory assignments may be given during the course.