OVERVIEW

The course aims to provide basic technical notions and tools on complex numbers, linear algebra and analytical geometry

LEARNING OUTCOMES

The course aims to provide basic technical notions and tools on complex numbers, linear algebra and analytical geometry.

AIMS AND LEARNING OUTCOMES

The student must learn the concept of number of solutions of a mathematical problem, must know how to work with complex numbers, vectors and matrices, including their diagonalization, must be able to solve equations and linear systems, must know how to work with vector spaces and subspaces, must be able to make a change of coordinates in the plane and in space, as well as knowing how to solve simple problems concerning lines, planes, spheres, circles and conic sections.

PREREQUISITES

• Algebra: factor decomposition, equations and inequalities (first, second degree and fractional);
• Trigonometry: definitions of sine, cosine, tangent, their graphical representations and main formulas;
• Euclidean geometry: basic concepts related to lines and circles, and their graphical representations.

Teaching methods

The course has a duration of 12 weeks (4 months in total) and consists of 5 hours a week (3 hours dedicated to theory and 2 hours for exercises).

SYLLABUS/CONTENT

Program

• Complex numbers and representation in the Gauss plane: solution of particular equations;
• Real/complex coefficient polynomials: decomposition, Fundamental Theorem of Algebra;
• Geometric vectors: equivalence, module, operations and properties. Scalar and vector product;
• Linear systems: elementary operations on equations and Gauss Algorithm;
• Matrices: definitions, operations and properties. Inverse matrix. Definition of determinant, rank and Rouché-Capelli Theorem. Matrix diagonalization: definition of eigenvalue, eigenvector and relative theorems. Spectral theorem for symmetric matrices. Orthogonal matrices;
• Vector spaces and subspaces: definitions and relative theorems;
• Cartesian coordinates and change of coordinates. Lines, planes, parallelism and orthogonality conditions, distances, orthogonal projections and symmetries.
• Spheres and circumferences in space;

Conic sections classification: parabolic, elliptic and hyperbolic type. Canonical equations and theorems on canonical form reduction

RECOMMENDED READING/BIBLIOGRAPHY

Notes and exercises can be found on the website AulaWeb. Suggested books:

• E. Sernesi, Geometria vol. 1, Bollati-Boringhieri;
• D. Gallarati, Appunti di Geometria, Di Stefano Editore-Genova;
• F. Odetti - M. Raimondo, Elementi di Algebra Lineare e Geometria Analitica, ECIG Universitas;
• M. Abate, Algebra Lineare, McGraw-Hill.

Exam Board

VICTOR LOZOVANU (President)

ELEONORA ANNA ROMANO

GIOVANNI ALBERTI

ALESSANDRO DE STEFANI

MATTEO SANTACESARIA

SIMONE DI MARINO (President Substitute)

Teaching methods

The course has a duration of 12 weeks (4 months in total) and consists of 5 hours a week (3 hours dedicated to theory and 2 hours for exercises).

Exam schedule