# Università degli Studi di Napoli "Parthenope"  ## Teaching schedule

2020/2021
Teaching:
Belonging course:
Course of Bachelor's Degree Programme on CIVIL AND ENVIRONMENTAL ENGINEERING FOR THE MITIGATION OF RISKS
Location:
Napoli
Disciplinary sector:
GEOMETRY (MAT/03)
Credits:
5
Year of study:
1
Teachers:
Cycle:
Annualita' Singola
Hours of front activity:
40

Italian

### Course description

The aim of this course is to learn analitic algebra and analitic geometry topics. A further aim is to apply these techniques in other scientific disciplines.  Learning outcomes (declined compared with the Dublin descriptors)  -Knowledge and understanding. Knowledge of analitic algebra and analitic geometry topics. The student will be able to state and prove basic theorems.  -Ability to apply knowledge and understanding. The ability to understand the problems proposed during the course, the ability to correctly apply the theoretical knowledge. The student will be able to study of the graphs of elementary functions, to solve integration problems of elementary character, to discuss the nature of numerical sequences and series.  -Making judgments. Develop the ability to critically evaluate the problems and propose the most appropriate approach.  -Communication skills. Ability to report and present the results with a logical-deductive and synthetic exposition  -Ability to learn.Ability to develop, outline, summarize the contents.

### Prerequisites

Elementary Algebra. Elements of Euclidean geometry. Elements of analytic geometry in the plane.

### Syllabus

Vector spaces (6 hours). The space Rn. Abstract vector spaces. Subspaces. Linearly dependent and independent vectors. Finitely generable spaces, bases and dimension of a vector space. Grassmann's formula. Scalar product in Rn. Linear matrices and transformations (11 hours). Operations between matrices. Linear applications between vector spaces Linear transformation associated with a matrix. Theorem of representation of linear transformations. Determinant. Binet formula. Rule of Sarrus. Invertible matrices. Rank of a matrix. Scale matrix, pivot. Linear systems (6 hours). Cramer's theorem. Homogeneous systems. Kernel and image of a linear transformation of Rn into Rm. Dimension theorem. Rouchè Capelli's theorem. Gauss method. Diagonalization (7 hours). Similar matrices. Diagonalizable endomorphisms and matrices. Eigenvalues, eigenvectors, characteristic equation, Autospace. Algebraic and geometric multiplicity of an eigenvalue. Characterization theorem for diagonalizable matrices. Differential equations (10 hours) Cauchy problem. Differential equations with separable variables, first order linear equations, local and global existence and uniqueness theorem, second order linear differential equations: general integral, linear equations with constant coefficients, the Lagrange method.

Vector spaces (6 hours). Linear matrices and transformations (11 hours). Linear systems (6 hours). Diagonalization (7 hours). Differential equations (10 hours)

### Teaching Methods

Lectures, exercise sessions

### Textbooks

- J. S. Robinson, A Course in LINEAR ALGEBRA with Applications, World Scientific.

### Learning assessment

The objective of the exam is to check the level of achievement of the above-mentioned training objectives. The exam is divided into two parts: - A written test that aims to evaluate the ability to correctly use the theoretical knowledge acquired during the course to solve mathematical problems. The student who does not show sufficient mastery of the arguments is not admitted to the next test. The expected time is 2 hours. - an oral test in which the ability to link and compare different aspects of the course will be evaluated. The final vote takes into account the evaluation of both tests.