Unit GEOMETRY AND ALGEBRA

Course
Computer science and electronic engineering
Study-unit Code
70099606
Curriculum
In all curricula
Teacher
Fernanda Pambianco
Teachers
  • Fernanda Pambianco
Hours
  • 81 ore - Fernanda Pambianco
CFU
9
Course Regulation
Coorte 2022
Offered
2022/23
Learning activities
Base
Area
Matematica, informatica e statistica
Academic discipline
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
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 examination is divided in two phases:

a) A written exam of 1,5 hours. The written test includes the solution of three exercises on topics covered in the program and is designed to verify the ability to correctly apply the theoretical knowledge, the understanding of the exercises proposed and the ability to communicate in writing language. The student must justify explicitly the used techniques with theoretical recalls. Valuation: from 0 to 30.

b) Oral examination of more or less 20 minutes. The oral test is designed to assess the level of knowledge and the understanding reached by the student on the contents listed in the program, this test is also used for verifying the presentation skills of the student. In this oral examination, taking into account the outcome of the previous part, it establishes the final score of the module.
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.
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).
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