Study program
First cycle study programme in mathematics (Bachelor
level)
1st cycle
Study level
Course title
Vector spaces
Course code
MAT01-018
Language of instruction
English
Course objective. The objective of the course is to give a
more general view to the notions and results students
encounter in linear algebra courses during the first two
years of study. By a more abstract approach one should
understand more deeply and more clearly the matter basic
for many modern mathematical disciplines.
Prerequisites. Geometry of plane and space. Linear
algebra I and II.
Course contents.
1. Finite dimensional spaces. Basis and dimension.
Subspaces. Quotient spaces. Dual space.
2. Linear operators. The space L(V,W) and the algebra
L(V). Matrix of a linear operator. Theorem on rank and
defect. Dual operator.
Course description
3. Minimal polynomial and spectrum. Polynomial of a
linear operator. Minimal polynomial. Spectrum.
Characteristic polynomial. Hamilton–Cayley theorem.
4. Invariant subspaces. Projections. Invariant subspaces.
Projections and their algebraic characterisation.
5. Nilpotent operators. Fitting decomposition. Nilindex.
Nilpotent operator. Index of nilpotence. Nilpotent
operators of maximal index. Elementary Jordan cell.
Decomposition of a nilpotent operator.
6. Reduction of a linear operator. The greatest common
divisor of polynomials. Relatively prime polynomials.
Decomposition of the kernel of a polynomial of a linear
operator. Jordan form of a matrix of a linear operator.
7. Unitary spaces. Inner product. Cauchy–Schwartz–
Buniakowsky inequality. Orthonormal bases. Bessel's
inequality. Gram–Schmidt theorem. Theorem on
othogonal projection. Selfadjoint, skewadjoint, unitary
and normal operators. Diagonalisation.
8. Functions of linear operators. Convergence in L(V).
Definition of f(A) for entire function f. Matrix of f(A)
in Jordan basis. Operator f(A) as a polynomial.
Lagrange–Sylwester polynomial. General definition of
a function of a linear operator. Properties of the
mapping ff(A). Spectrum of f(A).
consultative teaching
Form of teaching
Form of assessment
During the semester students can take tests which replace
the written part of the examination.
Number of ECTS
6
Class hours per week
2+2+0
Minimum number of
students
Period of realization
winter semester
Lecturer
Ivan Matić, Assistant Professor