Algebra and geometry
Knowledge and comprehension of the main analitic algebra and analitic geometry topics.
Capability to solve linear algebra and linear geometry exercises studied during the course.
Ability to find a solution: solve in a critical way exercises using a quick method giving logical and theorical reasons for the results.
Communicative abilities: ability summurise and to talk clearly about the topics studied.
Ability to learn: Be able to integrate the topics presented with extra resources books in order to further widen their knowledge.
Elementary Algebra. Elements of Euclidean geometry. Elements of analytic geometry in the plane. First elements of mathematical logic : concepts , theorem , demonstration , role of examples and counterexamples.
ons, applications, algebraic structures: group ring field. Field of complex numbers definition- Algebraic shape of a complex number. The complex plane. trigonometric form of a complex number. Operations with complex numbers. Notes on the theory of polynomials on complex field. Fundamental theorem of algebra (1 ECTS) Linear Algebra Vectors Matrices- Linear Systems. Vector spaces on R. Internal and external operations. Subspaces. Subspaces generated by a sequences of vectors. Linear dependence and independence-independent systems - Basis and dimension of a vector space. Changes reference- (1 ECTS) Matrices Determinant of a square matrix and its property- Rank of a matrix- Invertible Matrices. Cramer's rule for solving linear systems, linear representation of subspaces of R^n by using linear systems. (1 ECTS) Liner maps, definition and first properties. Kernel and Immage of a liner map.
Isomorphisms between vector spaces. Matrices and linear applications. Endomorphisms and isomorphisms- Diagonalization of endomorphisms and matrices- definitions and properties - characterizations of endomorphisms and diagonalization matrices - Isomorphism and coordinated representation of subspaces of a vector space by any linear systems in a given frame. (1 ECTS) Analytic geometry in the plane and space Linear dependence in the plane and in the space of geometric vector. Inner product standard- Orthogonal frames. Cartesian orthogonal monometric frame in spaces- Changes of frames. Representation of the line in the space- Direction cosines of a directed line. intersection of two lines and parallelism conditions. Orthogonality between lines. Midpoint and axis of a segment. (1 ECTS) Monometric frame in Cartesian space-Changes of frames- Vector product in the space of geometric vectors. Representation of line-parallelism and orthogonality between planes- Representation of the line in space-Directions a line - Pencil of planes- parallelism and othogonality between lines. Orthogonality and parallelism between lines and planes. Midpoint
[ 1] P. Biondi , P.M. Lo Re ?Appunti di Geometria- E.DI.SU  Seymour Lipschutz , Marc Lipson Algebra Lineare McGraw-Hill-Collana Shaums  A. Alvino, G. Trombetti Elementi di Matematica I- Liguori editore- pag. 192-208  S. Pellegrini, A. Benini, F. Morini Alebra Lineare -esercizi  Nicola Melone Introduzione ai metodi di algebra lineare-CedamDerek J. S. Robinson,A Course in LINEAR ALGEBRA with Applications-World Scientific(2006). K. W. Gruenberg
A.J. Weir, Linear Geometry -Springer- Verlgar New York
Written verification and oral examination.