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## ALGEBRA AND GEOMETRY

CODE 110730 2023/2024 8 cfu during the 1st year of 11634 SCIENZA DEI MATERIALI (L-SC.MAT.) - GENOVA MAT/02 GENOVA 2° Semester AULAWEB

## AIMS AND CONTENT

### LEARNING OUTCOMES

The course aims to provide the fundamental notions of the theory of vector spaces (real and complex) of finite dimension, of linear transformations and of analytic geometry in the plane and in space. The skills to be developed are: solving linear systems, searching for eigenvectors and eigenvalues, diagonalising real symmetric matrices and Hermitian matrices; solve simple analytic geometry problems

### TEACHING METHODS

Mode of teaching: the lectures will be carried out face-to-face, the theoretical topics will be taught, the proofs will be shown and simple consequences to be proven will be assigned, fundamental problems will be solved on the board and additional exercises will be assigned in preparation for the written exam.

### SYLLABUS/CONTENT

Prerequisites which will be furthered: vectors and vector calculations

Matrices: definition and operations with matrices, properties (powers, zero divisors, nilpotent matrices, invertibility), determinant of a square matrix, methods for calculating the determinant, in particular Laplace's theorem and elementary operations. Rank of a matrix, properties, methods for calculating the rank (Kronecker's theorem).

Linear systems: resolution of systems of linear equations with the Gauss method, results about existence and quantity of solutions using the properties ofthe matrix associated to the linear system (Cramer, Rouché-Capelli)

Vector spaces and subspaces, linear dependance and linear independence, fundamentals of vector spaces, in particular orthonormal vectors. Direct sum of subspaces.

Linear functions: definition, properties, kernel and image of a linear function, dimension theorem; matrix of a linear function with respect to a base of the domain and one of the codomain, relation between the properties of the matrix and the properties of the linear function. Isomorphism between the space of linear functions between two vector spaces and a space of matrices. Similitude of matrices and properties of similar matrices.

Eigenvalues and eigenvectors of an endomorphism. Diagonalization of a square matrix.

Geometry in the plane and in the three-dimensional space: vectors and operations with vectors (sum, product, scalar product, vector product); lines and their equations in the plane and in the three-dimensional space; position of a line with respect to another line or to a plane, position of a plane with respect to another plane; orthogonal projection of a point on a line or on a plane, of a line on a plane, and their symmetric. Distance of a point from a plane or from a line. Circles, spheres, cones, cylinders and their equations in the three dimensional space.

A.Bernardi, A. Gimiglino- Algebra Lineare e Geometria Analitica- Città Studi Edizioni.
Supporting PowerPoint slides will be provided by the lecturer.

## LESSONS

### Class schedule

All class schedules are posted on the EasyAcademy portal.

## EXAMS

### EXAM DESCRIPTION

the exam consistent in a 2-3 hour written exam which has a minimum passing grade of 16/30 which allows the progression onto the oral exam. The minimum passing grade is an average of 18/30 out of the written and oral examinations.
During the term the students can take two exams about the topics covered until that time which then replace the written exam. The passing requirements follow the same criteria of the written exam, a minimum average of 16/30 is required between the two exams. The use of the scientific calculator will be allowed, not the use of the graphical calculator.

### FURTHER INFORMATION

Recommendation for the student : regularly attend the lectures, take notes, refer to the notes shared on aulaweb, re-work through the notes, be aware of the ongoing learning by solving the proofs and exercises on your own, ask course mates for clarification if necessary, use the lecturer’s available hours to clarify any doubt.