500473 - GEOMETRY AND ALGEBRA

courses

ID:

500473

Duration (hours):

60

CFU:

SSD:

GEOMETRIA

Year:

2025

Date/time interval

Primo Semestre (29/09/2025 - 16/01/2026)

Course Objectives

This is a basic course on Linear Algebra and Analytic Geometry. Particular emphasis will be given to the fundamental concepts of Linear Algebra and Analytic Geometry as well as to the application of the latter to concrete numerical problems. A tutoring staff, composed by
experienced graduate or undergraduate students, provides an expert help and
support for students attending the course. (also see the webpage
https://elearning.unipv.it/course/view.php?id=9882)

Course Prerequisites

The same mathematics prerequisites for enrollment into the Engineering Faculty.
In particular, the following issues are required
elementary set theory; basic algebra: monomials/polynomials, polynomial division, equations and
inequations (inequalities) of degree 1 or 2, also for fractions of polynomials;
functions; basic trigonometry: goniometric functions, trigonometric equations and inequations,
double- and half-angle formulae etc., laws for right and oblique triangles;
Euclidean basic 2D and 3D geometry, including area and volume formulas for
most of common figures, parallelism and orthogonality between straight lines and/or
planes, parallelograms.

Teaching Methods

The course consist of theoretical lectures, and problem solving / exercises lectures ('esercitazioni'), tightly correlated with one another.
Theoretical lectures aim at giving basic concepts and results, always with some reference examples. Examples and exercises aim at developing computational skills and reasoning capabilities useful to tackle complex problems.
Lectures and exercise sessions at the blackboard, if possible; on-screen lessons by means of a tablet will be the alternative.
Lectures (hours/year in lecture theatre): 22.5
Practical class (hours/year in lecture theatre): 37.5
Practicals / Workshops (hours/year in lecture theatre): 0

Assessment Methods

Written test and oral test. The written test will allow assessing the skills acquired by the student in order to tackle problems on the content of the course. Questions and problems will be graduated in difficulty to determine the real level of knowledge and skills acquired by the student.
The teacher may decide to propose the sudents who passed the written test (mark greater than or equal to 18/30) either the verbalisation of the mark obtained in the written test or an oral exam. In any casde, student may required to undergo an oral test, provided they obtained in the written test a mark greater than or equal to 18/30.
If the mark of the written part is greater tahn 26/30, if no oral test is required (by the student or by the teacher), the final mark will be 26/30. In the oral test focus will be on assessing the level of theoretical knowledge acquired during the course, the clarity with which such knowledge is displayed. and the capability of applying such knowledge.
The final mark is the result of global width and depth of knowledge, as well as of the clarity of speech and of skills shown in solving problems. Such fina mark will not necessarily be the arithmetical mean of the mark obtained in the oral and written tests.

Texts

F. Bisi, F. Bonsante, S. Brivio. Lezioni di Algebra Lineare con Applicazioni alla
Geometria Analitica (available in www.amazon.it website)

Set and functions.
Linear Algebra
Vector spaces, vectors of R^n, linear subspaces; linear span of a set of vectors;
spanning sets and linear independence, basis, coordinates, and dimension.
Operations with matrices, determinant and rank of a matrix, inverse of a matrix.
Linear systems, Rouché-Capelli and Gauss elimination method,
representation of the set of the solutions of a linear system. Linear mappings
between vector spaces, kernel and image, matrix associated with a linear mapping.
Eigenvalues and eigenvectors of a linear operator, diagonalisation of a linear
operator. Inner product in R^n, orthonormal basis, Gram-Schmidt process.
Orthogonal matrices. Real quadratic forms. Spectral theorem: real symmetric
matrices and orthogonal diagonalisation.
Analytic Geometry.

Course Language

Italian

More information

In agreement with project 'didattica innovativa' students with special needs may be offered office hours in the evening or the availability of the notes prepared by the instructor.