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Monday, 7 October 2019 to Friday, 31 January 2020

- Written test
- oral exam

**Basics of Mathematics**

Logic and sets; natural, integer, rational and real numbers.

**Analysis**

*Sequences and Series*

Limit and convergence; Cauchy sequences.

*Metric and Topological Spaces*

Open and closed sets; neighborhoods.

*Limits and Continuity*

Limits of real functions; lim sup and lim inf; continuity and Weiestrass theorem.

*Differential Canculus*

Rolle, Cauchy, Lagrange and De L'Hopital theorems; Taylor formula and series.

*Integration*

Riemann integral; fundamental theorem of calculus.

*Functions of Many Variables*

Continuity; partial derivatives; gradient; differential and differentiability; critical points and Hessian.

**Linear Algebra**

Vector spaces over a field, subspaces.

Linearly dependent or independent vectors.

Dimension of a vector space.

First applications to linear systems, homogeneous and not.

Linear maps between vector spaces.

Matrices and linear maps.

Positive definite scalar products. Orthonormal bases and Gram-Schmidt orthogonalization process.

Linear systems: rank and dimension of the space of solutions.

Determinant of a matrix.

Eigenvectors and eigenvalues of an operator. Characteristic polynomial.

Diagonalizable matrices.

Orthogonal matrices.

Jordan's canonical form.

**Educatioanal Goals:**

- become familiar with the abstract language of mathematics (hypothesis, thesis, proof);
- knowledge of basic tools of differential and integral calculus;
- knowledge of basic tools of linear algebra and geometry.

W. Rudin, Principles of Mathematical Analysis, third edition.

S. Lang, Linear Algebra, third edition.