# Mathematical Analysis and Linear Algebra

## Scope of questions for State exams of Bachelor's degree program

Branch: Mathematical Physics

Subject: Mathematical Analysis and Linear Algebra

Subjects regarding the questions:

• 01MA1-2 Mathematical Analysis 1-2
• 01MAA3-4 Mathematical AnalysisA 3-4
• 01LA1-2P Linear Algebra 1-2
• 01FKO Functions of One Complex Variable
• 01DIFR Differential Equations

1.  Differential calculus of real variable -derivative, its application for functions, theorems on increment of functions.

2. Riemann integral in R, definition, sufficient conditions of existence, Newton's formula, substitution, per partes, mean value theorems.

3. Infiniteseries, convergence criteria, series reordering, product series.

4. Power series, properties of power series, Taylor polynomial, Taylor series, development of basic functions into Taylor series.

5. Derivatives in Rn, partial derivatives, gradient, function increment theorems, extrema of functions of several variables, constrained extrema. Linear differential equations of the n-th order and systems of linear differential equations of the first order.

6. Lebesgue integral -definitions, measurable sets and functions, Fubini's theorem, substitution theorem, continuity of integral, conversion theorems (integral and series, integral and limit, integral and derivative).

7. Derivatives in complex domain, holomorphic and analytic functions, Cauchy theorem and Cauchy formula, Laurent series and singularities, residue theorem.

8. Linear mappings and its matrices, systems of linear algebraic equations, Frobenius theorem.

9. Hermite and quadratic forms, scalar product and orthogonality, inequalities.

10. Linear operators and square matrices, determinant, eigenvalues, diagonalizability, normal operators.