Study-unit GEOMETRY AND ALGEBRA

Course name	Computer science and electronic engineering
Study-unit Code	70099606
Curriculum	Comune a tutti i curricula
Lecturer	Fernanda Pambianco
Lecturers	Fernanda Pambianco
Hours	81 ore - Fernanda Pambianco
CFU	9
Course Regulation	Coorte 2023
Supplied	2023/24
Supplied other course regulation
Learning activities	Base
Area	Matematica, informatica e statistica
Sector	MAT/03
Type of study-unit	Obbligatorio (Required)
Type of learning activities	Attività formativa monodisciplinare
Language of instruction	Italian
Contents	Complex numbers. Vector spaces. Bases and dimension. Linear transformation. Matrices. Determinants. Linear systems. Eigenvalues and eigenvectors. Diagonalization. Geometric vectors. Affine space and parallelism. Euclidean space and orthogonality. Projective space. Algebraic curves and conics.
Reference texts	A. BASILE , L. STRAMACCIA, ALGEBRA LINEARE E GEOMETRIA ED. COM s.r.l. - ROMA or A. BASILE , ALGEBRA LINEARE E GEOMETRIA CARTESIANA ED. COM s.r.l. - ROMA
Educational objectives	Knowledge of mathematical language and a suitable mastery of foundamental concepts of linear algebra and Cartesian geometry.
Prerequisites	No prerequisites except the basic knowledge of arithmetic and algebra.
Teaching methods	The course is organized as follows: -lectures on all subjects of the course -exercises in classroom.
Learning verification modality	The learning outcome assessment method is as follows: by a written test aimed at ascertaining mastery of the use of linear algebra tools and their application to problems of plane and space geometry; from an oral test aimed at ascertaining the knowledge and understanding of theoretical aspects inherent to the topics addressed, as well as the ability to explain the content. For information on support services for students with disabilities and/or DSA visit the page http://www.unipg.it/disabilita-e-dsa
Extended program	Elements of Logic. Relations and Partitions. The field Z_P. Complex numbers. Roots of complex numbers. Vector spaces. Generator systems. Linear dependence. Bases and vector's coordinates. Bases in generator systems. Exchanging theorem and dimension. Linear transformations. The space Hom(V,W). Definition of a linear transformation on the vectors of a basis. Kernel and Image of a linear transformation. Relation between their dimension. Isomorphic vector spaces end their dimension. Vector spaces of matrices. Row-column product. Matrix of a linear transformation. Matrix of a composed linear transformation. Matrix of a bases exchange. Calculus of a matrix determinant. Transpose of a matrix, product of matrices, their determinant. Invertible matrices, their determinant, linear dependence of the columns. Linear systems. Cramer's systems. Rank of a matrix and its determination. Homogenehous linear systems and the space of solutions. General case and theorem of Rouché-Capelli. Eigenvalues and eigenvectors. Characteristic polynomial. Diagonalization. Orientate lines and segments. Cartesian reference systems. The space of geometric vectors. Coordinates of a vector and coordinates of the extreme points of its representative segments. Parallel vectors, complanar vectors and conditions on their coordinates. Affine Space. Parametric representation of lines and planes. Cartesian equation of a plane. Bundles of planes and lines. Cartesian equations of a line. Conditions of parallelism. Exchanges of affine reference systems. Euclidean Space. Definitions of angles. Scalar product. Distance between two points and sphere. Orthogonality conditions. Scalar product spaces. Gram-Schmidt orthogonalization. Projective space. Homogeneous coordinates. Representation of planes and lines in homogeneous coordinates. Coordinates on the complex field. Imaginary points and lines. Algebraic curves. Their order and components. Theorem of Bézout. Simple and singular points. Analytical conditions on singularity. Classification of conics. Bundles of conics. Configuration of basic points and reducible conics in a bundle. Quadrics (sample). Bilinear forms and quadratic forms.