The aim of the programme is to give some preliminaries and tools for a basic introduction to Linear
Algebra and Analytical Geometry. In this course we look at properties of matrices, systems of linear equations
and vector spaces useful to find real eigenvalues and eigenvectors of applications.
We will learn about classification of plane conics and quadric surfaces, using their invariants and polar coordinates.
We will also solve some problems similar to the ones assigned at the final exam.
Frontal lectures and classroom exercise. The teaching approach is a traditional one. The programme offers personal feedback and attention from tutors in order to help students in their studies.
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