corso|33503

Program A) ELEMENTARY ALGEBRA - Integers/polynomials (ring structure), Euclidean division, unique factorization, greatest common divisor, Euclidean algorithm - Equivalence relations, quotient sets, Z/n and Q, Fermat's little theorem, field structure on Z/p (and on Q,R) - Complex numbers, field structure, polar representation, fundamental theorem of algebra, factorization of complex and real polynomials into irreducible polynomials. B) LINEAR ALGEBRA - Vector spaces, systems of linear equations, Gauss elimination algorithm, interpretation of a matrix as a linear map, composition and product of matrices, determinant of a square matrix, Binet's theorem, inverse of a matrix. - Vector spaces, linear combinations and spans, linear independence, sets of generators, bases and dimension. - Vector subspaces, intersections of subspaces, sums and direct sums of subspaces, Grassmann's formula. - Linear transformations, kernel and image, rank and rank theorem, Rouché-Capelli theorem. - Coordinates, changes of coordinates, representation of linear transformations by matrices. - Eigenvalues ​​and eigenvectors of a linear endomorphism, characteristic polynomial, eigenspaces, diagonalizability. C) NOTES ON GROUP THEORY - Definition and examples of groups: cyclic groups, invertible groups in a ring, invertible matrices, permutation groups, transformation groups. - Subgroups and Lagrange's theorem, conjugacy classes and class formula. - Homomorphisms of groups, kernel and image, normal subgroups.

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G. M. Piacentini Cattaneo, Algebra, un approccio algoritmico, Zanichelli

writing and oral exam

Sets, partitions, applications, equivalence relations, order relations, permutations. Natural numbers, the principle of induction. Classes remainder modulus an integer. Congruences and equations in Zn. Algebraic structures: Groups, rings and fields. Rings of polynomials. Euclid's algorithm. Systems of linear equations: Gauss algorithm, determinant of a square matrix. Inverse matrix. Rank of a matrix: Cramer's theorem and Rouche-Capelli's theorem. Solving homogeneous linear systems. Vector spaces: linear dependence and independence, basis. Matrices. Linear maps and their representation: changes of basis, diagonalization of a linear operator. Characteristic polynomial and relative invariance. Elements of group theory: Cyclic groups, period of an element of a group. Classification of cyclic groups. Lateral classes form a subgroup. Lagrange's theorem, normal subgroups. The fundamental theorem of homomorphism between groups.

Lang, Linear Algebra

Three lessons a week, a main theoretical lesson and two containing applications, examples and exercises.

The exam consists of a written test and an oral test. There will be five sessions: two between January and February, two between June and July and one in September. For official information on dates, times and places of exam sessions (written, oral and recorded), always refer to the calendar published by the teaching secretariat on the Computer Science degree course web pages.

1 ongoing test to obtain a bonus, and 5 exam sessions with written exam and optional oral exam. The complete modality is written here https://sites.google.com/uniroma1.it/modalita-esame/home-page