ALGEBRA LINEARE E GEOMETRIA R - Z

Dublin Descriptors

With reference to the aim of the Degree Course in training students in the Industrial Engineering class, the contribution of the course is carried out through the acquisition of the knowledge and skills summarized below.

Knowledge and understanding

Passing the teaching exam implies that the student has acquired:

• the mathematical tools for the representation and study of engineering systems of medium complexity.

Applying knowledge and understanding

Passing the teaching exam implies that the student has acquired the ability to:

• understand and work on engineering problems using appropriate terminology and mathematical formulations;

• share and apply knowledge in multidisciplinary teamwork settings.

During the lessons topics and concepts will be proposed in a formal way, together with meaningful examples, applications and exercises.  The student will be sollicited to carry out exercises autonomously, even during the lessons. The teaching materials will be made available to students on Studium at the beginning and during the course. Should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, in line with the programme planned and outlined in the syllabus.

Linear Algebra:

1. Generalities on set theory and operations. Maps between sets, image and inverse image, injective and surjective maps, bijective maps. Sets with operation, gropus, rings, fields.
2. Vectors in the ordinary space. Sum of vectors, product of a number and a vector. Scalar product, vector product. Components of vectors and operations with components.
3. Complex numbers, operations and properties. Algebraic and trigonometric form of complex numbers. De Moivre formula. nth root of complex numbers.
4. Vector spaces and properties. Examples. Subspaces. Intersection, union and sum of subspaces. Linear independence. Generators. Base of a vector space, completion of a base. Steinitz Lemma*, dimension of a vector space. Grassmann formula*. Direct sum.
5. Generalities on matrices. Rank. Reduced matrix and reduction of a matrix. Elementary matrices. Product of matrices. Linear systems. Rouchè-Capelli theorem. Solutions of linear systems. Homogeneous systems and space of solutions.
6. Determinants and properties. Laplace theorems*. Inverse of a square matrix*. Binet theorem*. Cramer thoerem*. Kronecker theorem*.  
7. Linear maps and properties. Kernel and image. Injective and surjcetive maps. Isomorphisms. L(V,W) and isomomorphism with k^{m,n}. Study of a linear map. Base change.
8. Eigenvalues, eigenvectors and eigenspaces of an endomorphism. Characteristic polynomial. Dimension of eigenspaces. Independence of eigenvectors. Simple endomorphisms and diagonalization of matrices.

Geometry

1. Linear geometry on the plane. Cartesian coordinates and homogeneous coordinates. Lines and their equations. Intersection of lines. Angular coefficient. Distances. Pencils of lines.
2. Linear geometry in the space. Cartesian coordinates and homogeneous coordinates. Planes and their equation. Lines and their representation. Ideal elements. Angular properties of lines and planes. Distances. Pencils of planes.
3. Change of coordinates in the plane, rotations and translations. Conics and associated matrices, orthogonal invariants. Reduced equations, reduction of a conic in canonic form. Classification of irreducible conics. Study of equations in canonic form. Circle. Tangent lines. Pencils of conics.
4. Quadrics in the space and associated matrices. Irreducible quadrics. Vertices and dengerate quadrics. Cones and cylinders. Reduced equations, reduction in canonic form. Classification of non degenerate quadrics. Sections of quadrics with lines and planes. Lines and tangent planes.

The proofs of the theorem signed with * can be ometted.

Teaching’s contribution to the Goals of the Agenda 2030 for Sustainable Development:

Quality Education, Targets 4.3, 4.4, 4.5, 4.6.

1. P. Bonacini, M. G. Cinquegrani, L. Marino. Algebra lineare: esercizi svolti. Cavallotto Edizioni, Catania, 2012.
2. P. Bonacini, M. G. Cinquegrani, L. Marino. Geometria analitica: esercizi svolti. Cavallotto Edizioni, Catania, 2012.
3. S. Giuffrida, A. Ragusa: Corso di Algebra Lineare. Il Cigno Galileo Galilei, Roma, 1998.
4. Lezioni di Geometria. Spazio Libri, Catania, 2000.

More didactic material is available at https://elearning.unict.it/ and https://www.dmi.unict.it/bonacini/didattica/

The examination is written and oral. The written examination, which usually lasts 3 hours, is compulsory to take the oral examination. The minimum mark to pass the written exam is 12/30. 

Note

Exminations may also be carried out on line, should the conditions require the use of this mode.

To ensure equal opportunities and in compliance with current laws, interested students enrolled in the CInAP may agree with the lecturer on any compensatory and/or exempting measures, based on the learning objectives and their specific needs. It is also possible to contact the CInAP reference lecturer (Center for Active and Participatory Inclusion — Services for Disabilities and/or SLD/DSA) of the DIEEI, professors Antonella Di Stefano and Arturo Pagano (https://www.cinap.unict.it/content/referenti).

Linear Algebra exercises:

1. study of a linear application with parameter, determining kernel and image.

2. study of the diagonalization of an endomorphism with parameters, determining, if possible, a bases of eigenvectors.

3. inverse image of a vector, resolution of a linear system with parameter, inverse image of a vector space, image of a vector space.

4. exercises on vector spaces and their dimension, direct sum, operations on linear applications, induced linear applications, restrictions and extensions.

Geometry exercise:

1. linear geometry exercises in the 3-dimensional space: parallelism and orthogonality, distances, orthogonal projections, angles.

2. pencil of conics, complete study of a conic, conics under conditions.

3. classification of quadrics with parameter, quadrics under conditions, intersection of quadrics with planes.

All the topics mentioned in the program can be requested during the exam. Precisely, the proofs of the following theorems and propositions can be requested: 

•     characterization of subspaces
•     First Rouché-Capelli Theorem
•     Intersection of subspaces
•     Direct sum 
•     linear independence criterion
•     characterization of a base
•     Theorem of existence of a basis
•     base completion theorem
•     Theorem on the number of vectors in a basis
•     proposition on image and kenel of a linear application
•     characterization of injective applications
•     Dimension theorem
•     Theorem on the assignment of a linear application
•     Characterization theorem of similar matrices
•     Characterization theorem for 0 eigenvalue
•     Characterization of eigenvalues
•     invariance of the characteristic polynomial
•     Independence of the eigenvectors
•     Simple Endomorphisms with n distinct eigenvalues
•     Algebraic and geometric multiplicity of an eigenvalue 
•     characterization of simple endomorphisms
•     Diagonalizable matrices
•     criterion for parallel vectors 
•     point at infinity and direction of a line
•     sheaf of planes
•     sheaf of lines
•     theorem on the conic for 5 points
•     degenerate conics in a sheaf
•     Number of vertices of a quadric surface 
•     degenerate plane sections of cones and cylinders
•     non-degenerate plane sections of cones and cylinders
•     plane sections of non-degenerate quadrics.

It is also possible that questions may be asked on geometric topics relating, amongst other things, to lines and planes, and the classification of conics and quadric surfaces.