### Complementi di Matematica per biologi

Course Program

ANALYSIS PART

PRELIMINARIES: - some logic - sets, relations and functions - cartesian product of sets - natural numbers and the induction principle - integers, rationals and basic operations - decimal expression: periodic and non-periodic - real numbers - complex numbers.

METRIC SPACES: open, closed, compact and connected sets

SEQUENCES AND SERIES: convergence criteria

CONTINUITY: basic properties, continuity and compactness-connectedness, Bolzano-Weierstrass theorem

DIFFERENTIABILITY: basic properties, Rolle-Lagrange-Cauchy, de l'Hôpital and Taylor theorems

INTEGRABILITY: integral and indefinite integral, integrable functions "a'la Riemann", fundamental theorem of calculus

FUNCTIONS OF MANY VARIABLES: continuity, directional derivatives, differentiability, second derivatives and Hessian matrix, extremals on an open set, extremals on closed sets

LINEAR ALGEBRA PART

vector spaces on a field, subspaces - linear dependence and independence. Dimension - matrices - first applications to linear systems, homogeneous and not - linear operators between vector spaces. Matrices and linear operators - scalar product - orthonormal basis and Gram-Schmidt orthonormalization process - linear systems: rank and dimension of the space of solutions - determinant of a matrix - eigenvectors and eigenvalues. Characteristic polynomial of an operator - diagonalizable matrices - unitary and orthogonal matrices - Jordan's canonical form

Bibliographic references

S. Lang, "Linear Algebra",

W. Rudin, "Principles of Mathematical Analysis",

Notes "Complementi di Matematica" (pagina www.dida.sns.it --> Classi --> AA 2015-2016)

ExamWritten and oral exam