GEOMETRIA

The aim of this course is to provide the fundamental concepts and tools for a first introduction to Linear Algebra and Analytic Geometry. Topics include the properties of matrices, systems of linear equations, and vector spaces, with particular attention to the computation of real eigenvalues and eigenvectors. The course also covers the classification of plane conics and quadric surfaces, using invariants and polar coordinates as key techniques. Throughout the course, students will work on problems analogous to those proposed in the final examination, thereby consolidating both their theoretical understanding and problem-solving skills.

Knowledge and Understanding
Understand statements and proofs of fundamental theorems in algebra, analytic geometry, linear algebra, and the geometry of curves.
Demonstrate mathematical skills in reasoning, manipulation, and computation.
Solve mathematical problems which, although not standard, are of a similar nature to those already familiar to students.
Applying Knowledge and Understanding
Prove known mathematical results using techniques different from those already studied.
Construct rigorous proofs.
Develop simple examples.
These skills will be acquired through interactive teaching: students will continuously test their own knowledge, working independently or in collaboration within small groups on new problems proposed during lectures, tutorials, and support sessions.
Making Judgements
Develop an informed autonomy of judgement in evaluating and interpreting the solution of geometry problems.
Construct and develop logical arguments with a clear identification of assumptions and conclusions.
Recognize correct proofs and identify fallacious reasoning.
These objectives will be pursued through exercises carried out during the course and in support activities for the Geometry course, providing opportunities for students to independently develop decision-making and evaluative skills. Interactive teaching methods will ensure that students in the physics program constantly test their understanding, both individually and collaboratively, through problem-solving activities proposed during lectures and support sessions.
Communication Skills
Communicate information, ideas, problems, solutions, and conclusions clearly and unambiguously.
Present, orally and in writing, the most important theorems of linear algebra and analytic geometry in a clear and comprehensible manner.
Work effectively in groups while maintaining defined levels of autonomy.
To achieve these skills, students will engage in frequent opportunities for written work, discussion, and evaluation. The final exam will also provide an additional opportunity to demonstrate the ability to analyze, elaborate, and communicate the work carried out.
Learning Skills
Develop the competencies needed to pursue further studies with a high degree of autonomy.
Acquire learning abilities and a solid standard of knowledge and skills sufficient for admission to courses at the master’s level in physics.
Cultivate a flexible mindset, enabling quick integration into working environments and the ability to adapt easily to new problems.
These learning skills will be developed throughout the degree program, with particular emphasis on the hours dedicated to independent study.

Frontal lectures and classroom exercise. The teaching approach is a traditional one. The program offers personal feedback and attention from tutors in order to help students in their studies. 

49 hours of lectures and 30 hours of exercises are scheduled, corresponding to 9 ECTS credits.
If the course is delivered in a blended or distance-learning format, necessary adjustments may be introduced to ensure that the program described in the syllabus is fully covered.
Attendance is normally mandatory (please refer to the Academic Regulations of the Degree Program for any exceptions).

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.
 

Linear Algebra

1. Matrices over a field. Matrices addition, scalar multiplication, matrix multiplication (or product). Diagonal, triangular, scalar, symmetric, skew-simmetric matrices and transpose of matrix.
2. Vector spaces and their properties over R. Examples: R[x], Rⁿ, R^m,n.. Subspaces. Intersection and sum of vector spaces. Direct sum. Linear combinations. Span, Linear Independence and dependence,Finitely generated vector spaces, Base, Dimension. Steinitz’s Lemma *, Grassmann’s formulas*.
3. Determinants and their properties. Theorems of Binet*,Laplace I*, Laplace II*, Adjunct matrix, Inverse, Rank and Reduction of a matrix. Theorem of Kronecker*. Systems of linear equations. Rouchè-Capelli‘s rule, Cramer’s rule. Solving systems of linear equations.
4. Linear maps between vector spaces and their properties. Kernel and image of a linear map. Injective, surjective maps and isomorphisms. Study of linear maps. Matrices associated to linear maps. Change of base matrix. Similar matrices.
5. Eigenvalues, Eigenvectors and Eigenspaces of a matrix. Characteristic polynomial. Dimension of an eigenspace. Relation between Algebraic multiplicity and geometric multiplicity. Linear Independence of the eigenvectors. Diagonalizable linear maps and diagonalization of a matrix.

Geometry

I) Euclidean (geometric) vectors and their properties. Scalar multiplication, dot (or scalar) product, wedge (or cross) product.

II) Cartesian coordinates. Points, lines , Homogeneous coordinates, Points at infinity (Improper Points), Parallel and orthogonal Lines. Slope of a line. Distances from a point to a line. Pencil of lines. Planes in The space. Coplanar and Skew lines. Pencil of Planes. Angles between lines and planes. Distance from a point to a plane and from a point to a line in the space.

III) Conics and their associated matrices. Orthogonal Invariants. Canonical reduction of a conic*. Irreducible and degenerate conics. Rank of its associated matrix. Discriminant of a conic. Parabolas, Ellipses, Hyperbolas: equations, focus, eccentricity, directrix, semi-maior axis, center. Circumferences, Tangents, and pencils of conics.

IV) Quadrics and its associated matrix. Nondegenerate, degenerate and singular quadric surfaces. Cones and cylinders. Classification.

1) S. Giuffrida, A.Ragusa, Corso di Algebra Lineare, Ed. Il Cigno G.Galilei, Roma 1998 (Linear Algebra).

2) G. Paxia, Lezioni di Geometria, Spazio Libri, Catania, 2005 (Geometry) available at www.giuseppepaxia.com

3) D. Margalit , J. Rabinoff, Interactive linear algebra, available at https://textbooks.math.gatech.edu/ila/

To assess students' knowledge and skills, students are requested to attend a written exam and an oral exam.

Linear algebra

Geometry

Lines and planes in space. Orthogonality between lines and/or planes. Conics and quadrics: classification, tangent lines. Double points of a quadric.