ALGEBRA LINEARE E GEOMETRIA A - C
Academic Year 2026/2027 - Teacher: MONICA LA BARBIERAExpected Learning Outcomes
Expected Learning Outcomes
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.
Course Structure
During the lessons topics and concepts will be proposed in a formal way, together with meaningful examples, applications and exercises. A tutor will carry classroom exercises. The student will be sollicited to carry out exercises autonomously, even during the lessons.
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.
Learning assessment may also be carried out on line, should the
conditions require it.
Required Prerequisites
Attendance of Lessons
The student is required to attend at least 70% of the lessons of the course.
Detailed Course Content
Linear Algebra:
- 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.
- 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.
- 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.
- 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.
- Determinants and properties. Laplace theorems*. Inverse of a square matrix. Binet theorem*. Cramer thoerem. Kronecker theorem*.
- 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.
- Eigenvalues, eigenvectors and eigenspaces of an endomorphism. Characteristic polynomial. Dimension of eigenspaces. Independence of eigenvectors. Simple endomorphisms and diagonalization of matrices.
Geometry
- Linear geometry on the plane. Cartesian coordinates and homogeneous coordinates. Lines and their equations. Intersection of lines. Angular coefficient. Distances. Pencils of lines.
- 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.
- 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.
- 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.
Textbook Information
- P. Bonacini, M. G. Cinquegrani, L. Marino. Algebra lineare: esercizi svolti. Cavallotto Edizioni, Catania, 2012.
- P. Bonacini, M. G. Cinquegrani, L. Marino. Geometria analitica: esercizi svolti. Cavallotto Edizioni, Catania, 2012.
- S. Giuffrida, A. Ragusa: Corso di Algebra Lineare. Il Cigno Galileo Galilei, Roma, 1998.
- Lezioni di Geometria. Spazio Libri, Catania, 2000.
More didactic material is available at https://studium.unict.it/
Course Planning
| Subjects | Text References | |
|---|---|---|
| 1 | Introduction to set theory. introduction to fields and vector spaces. Determinant of a matrix. Rank and reduction of a matrix. Resolution of a linear system. Required time: 9 hours, 7 of theory and 7 of exercises. | Testo 3: cap 1,3 - Testo 1: cap. 1 |
| 2 | Operations with matrices. Required time: 2 hours, 1 of theory and 1 of exercises.Determinant of a matrix. Rank and reduction of a matrix. Resolution of a linear system. | Testo 3: cap 3 - Testo 1: cap. 1 |
| 3 | Vector spaces. Generators and linear independence. Subspaces. Base and components with respect to a base. Dimension of a vector space. Sum and intersection of vector spaces. Extracting a base from a set of generators and expanding a linearly independent set to a base. Required time: 14 hours, 6 of theory and 8 of exercises. | Testo 3: cap. 2 - Testo 1: cap. 2 |
| 4 | Linear applications and their assignment. Studying a linear application. Computation of images and inverse images. Required time: 10 hours, 5 of theory and 5 of exercises. | Testo 3: cap. 4 - Testo 1: cap. 2 |
| 5 | Base change matrices and similar matrices. Operations with linear applications. required time: 2 hours, 1 of theory and 1 of exercises. | Testo 3: cap. 4 - Testo 1: cap. 5 |
| 6 | Eigenvalues, eigenvectors and eigenspaces. Characteristic polynomial. Algebraic and geometric multiplicity of an eigenvalue. Endomorphisms and diagonalization. Required time: 11 hours, 5 of theory and 6 of exercises. | Testo 3: cap. 5 - Testo 1: cap. 6 |
| 7 | Applications under conditions. Restrictions and extensions of linear applications. Required time: 2 hours, 1 of theory and 1 of exercises. | Testo 3: cap. 5 - Testo 1: cap. 7,8 |
| 8 | Generalities on vector calculus. Cartesian coordinates and homogeneous coordinates. Assignment of lines and planes and their equations. Points at infinity. Intersections. Parallelism and orthogonality. Pencils of lines and planes. Distances. Angles. Orthogonal projections. Bisecting lines and planes. Symmetries. Locus of lines. Required time: 14 hours., 7 of theory and 7 of exercises | Testo 4: cap. 1,2,3 Testo 2: cap. 1 |
| 9 | Conics and associated matrices. Change of coordinates in the plane, orthogonal invariants and reduced equations of a conic. Classification of conics. Circles. Tangent lines. Pencils of conics. Required time: 8 hours, 4 of theory and 4 of exercises. | Testo 4: cap. 2 - Testo 2: cap. 2 |
| 10 | Complete study of conics. Conics under conditions. Required time: 3 hours, 1 of theory and 2 of exercises. | Testo 4: cap. 2 - Testo 2: cap. 2 |
| 11 | Quadrics and associated matrices. Irreducible quadrics. Vertices of a quadrics and degenerate quadrics. Conic at infinity. Cones and cylinders. Reduced equations of a quadric. Classification of non degenerate quadrics.Tangency. Conic sections of a quadric. Required time: 7 hours, 4 of theory and 3 of exercises. | Testo 4: cap. 5 - Testo 2: cap. 3 |
Learning Assessment
Learning Assessment Procedures
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 15/30.
NOT PASSED: The student shows poor and fragmented knowledge of the subject, exhibits serious comprehension errors, and is unable to present the content of the subject in an acceptable manner.
18-21: The student shows limited knowledge and a basic understanding of the subject, presents the content unclearly and with little precision.
22-24: The student shows acceptable knowledge and a basic understanding of the subject, presents the content correctly but not in a fully structured way.
25-27: The student shows broad knowledge and an adequate understanding of the subject, presents the content correctly but not in a complete way.
28-29: The student shows in-depth knowledge and a solid understanding of the subject, presents the content clearly and in a fully structured way .
30-30 cum laude: The student shows complete and detailed knowledge and an excellent understanding of the subject, presents the content clearly and in a fully structured way.
To ensure equal opportunities and in compliance with current laws, interested students may request a personal interview in order to plan any compensatory and/or dispensatory measures based on educational objectives and specific needs. Students can also contact the CInAP (Centro per l’integrazione Attiva e Partecipata - Servizi per le Disabilità e/o i DSA) referring teacher within their department.
Examples of frequently asked questions and / or exercises
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.