EDUCATIONAL OBJECTIVES.
The teaching aims at the acquisition of knowledge and understanding of the terminology, fundamental concepts and demonstration methodologies peculiar to the fields of Geometry and Linear Algebra.

DIDACTIC DELIVERY
NO. 48 ONLINE VIDEO LECTURES (NO. 6 TEACHING UNITS - LASTING TWO HOURS PER CFU)

INTERACTIVE DIDACTICS
N. 2 INTERACTIVE LECTURES PER CFU
N. 5 THEMATIC DISCUSSIONS ON THE DIDACTIC FORUM (TOPIC) AND N. 2 POSTS PER CFU AS PER THE PQA TEACHING GUIDELINES
N. 2 E-ACTIVITIES PER 5 CFU
N. 2 TESTS PER CFU WITH 8 MULTIPLE CHOICE QUESTIONS

COURSE SYLLABUS

Algebraic Structures:
General definitions: operations and properties. Groups. Rings. Fields.
Matrices:
Definitions and properties. Development of determinants: laplace theorem. Rank of a matrix. Rim theorem. Step matrices. Inverse of a matrix. Inverse theorem.
Linear systems:
System of linear equations: definition, associated matrices, compatibility and non-compatibility, number of solutions. Rouché-hair theorem. Cramer's theorem. Gauss elimination method. Basis of solutions of a homogeneous linear system. Discussion of linear systems with parameter.
Vector spaces:
The structure of vector space. Vector subspaces. Linear dependence and independence. Generators. Bases. Steinitz's lemma. Base theorem. Dimension of a vector space. Intersection and sum of subspaces, direct sum. Grassmann's relation.
Euclidean spaces:
Definition of scalar product. Definition of real Euclidean vector space. Definition of norm. Cauchy-Schwarz inequality. Definition of angle. Definition of orthogonal vectors and orthogonal subspace. Orthonormal bases. Components in an orthonormal basis. Orthogonal projections. Gram-Schmidt theorem and procedure.
Linear applications:
Definitions of linear applications (homomorphisms), endo-, epi-, mono- morphisms. Core and image. Dimension theorem.
Diagonalization:
Eigenvalues and eigenvectors: definitions, characteristic polynomial and equation. Eigenspaces and related properties. Algebraic and geometric multiplicity. Simple and orthogonal diagonalization: definitions for matrices and endomorphisms. Main characterization theorem of diagonalization. Spectral theorem.
Analytic geometry in the plane.
Cartesian reference system in the plane. Line equation (algebraic, parametric, symmetric). Parallelism and orthogonality between lines. Conics: definition, classification and canonical form. / Exercises on representations of lines in the plane (construction, membership, conversion between different representations).
Analytic geometry in space.
Cartesian reference system in the plane. Line equation (algebraic, parametric, symmetric). Parallelism and orthogonality between lines. Conics: definition, classification and canonical form. / Exercises on representations of lines in the plane (construction, membership, conversion between different representations).
Analytic geometry in space.
Cartesian reference system in space. Vector product and mixed product. Equation of the plane (parametric and Cartesian). Line equation (parametric, Cartesian, symmetric). Bundles and stars of planes. Conditions of parallelism and perpendicularity in space. Skewed lines./ Exercises on representations of lines and planes in space (construction, membership, conversion between different representations).

INTEGER PROFIT VERIFICATION METHODS
The degree of student learning is constantly monitored through the tools and methods of verification.
In particular, in order to make the verification and certification of learning outcomes feasible, the teacher and tutor will take into account the:
1. automatic tracking of training activities by the system - reporting;
2. didactic and technical monitoring (at the level of quantity and quality of interactions, adherence to didactic deadlines, delivery of scheduled papers, etc.).
3. the formative type tests in itinere, including for self-assessment (e.g. multiple choice tests, true/false, sequence of questions with different difficulty, simulations, concept maps, papers, group projects, etc.)
4. the final profit exam, in which the work done online is taken into account and valued (activities carried out at a distance, quantity and quality of online interactions, etc.).
Assessment, within this framework, takes into account several aspects:
a. the result of a certain number of intermediate tests (online tests, development of papers, etc.);
b. the quality and quantity of participation in online activities (frequency and quality of interventions that can be monitored through the platform);
c. the results of the final test.

Therefore, the data collected will be evaluated by the lecturer for the student's evaluation activity.

ASSESSMENT METHODS AND OBJECTIVES OF THE FINAL EXAMINATION
Access to the exam is subject to the recognition of attendance, which will be sanctioned with an appropriate certificate at the time of booking the exam, which will attest to the performance of the didactic activities of verification in itinere and the level of work done in the various exercises.
The examination will consist of a written test, with optional oral test. The final grade will be expressed in thirtieths, with possible honors depending on the degree of maturity achieved by the student.
The examination is aimed at the educational objectives described below. In particular, the student should.
(a) be able to determine the determinant and rank of a matrix;
(b) be able to find the solutions of a linear system;
(c) be able to solve exercises related to vector and Euclidean spaces;
(d) be able to recognize the equations of lines and planes in 2D and 3D and the various types of conics in the case of 2D geometry.
The written test provides a score for all of the above objectives (a) through (d). Each objective has a variable score. Honors will be awarded in those special cases where special maturity is demonstrated in solving the examination questions.


KNOWLEDGE AND COMPREHENSION SKILLS IN TERMS OF EXPECTED OUTCOMES (DUBLIN DESCRIPTOR NO. 1)

● Acquisition and understanding of mathematical language, concepts of linear algebra and analytic geometry.
● Knowledge and understanding of terminology, fundamental concepts and demonstration methodologies specific to the areas of geometry and algebra, with particular reference to: Matrices and linear systems. Vector and Euclidean spaces. Homomorphisms and Diagonalization. 2D and 3D analytic geometry.

SKILLS IN ORDER TO APPLY KNOWLEDGE AND UNDERSTANDING IN TERMS OF EXPECTED OUTCOMES (DUBLIN DESCRIPTOR #2)
● The student will be able to apply the definitions, theorems and rules studied in solving problems.
● The student will be able to use structures and tools of linear algebra in handling mathematical problems.
● The student will be able to use 2D and 3D elements from an algebraic and geometric point of view.

G. ALBANO, LA PROVA SCRITTA DI GEOMETRIA: TRA TEORIA E PRATICA, MAGGIOLI (2013).
G. ALBANO, C. D'APICE, S. SALERNO, ALGEBRA LINEARE, CUES (2002).