GEOMETRIA I

The aim of the programme is to give students 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.

At the conclusion of the course, the students shuold be able to understand the basic notions, to apply their knowledge and understanding. They also should be able to give oral and written presentation of the most important theorems of the contents of the course as well as to work both in collaboration with other people and by themselves, making judgements.

Frontal lectures and classroom exercise. There is no standard way of lecturing: some lectures will be written exclusively on blackboards or sometimes the student receive printed notes. The method used depends also on the sort of material that they are covering.

Together with the professor of Algebra and Analysis, periodical meetings will be organized to give students useful suggestions on how to use and apply the acquired knowledge to the other disciplines.

Linear Algebra

I) Groups, rings, fields. Z, K[x], C.

II) Matrices over a field. Matrices addition, scalar multiplication, abelian group of matrices, matrix multiplication (or product). Properties. Ring of square matrices. Diagonal, triangular, scalar , symmetric, skew-simmetric matrices and transpose of matrix.

III) Vector spaces and their properties over a filed K. Examples: K[x], Kⁿ, K^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*.

IV) 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.

V) 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.

VI) 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.

VII) Real scalar product, hermitian scalar product, Cauchy-Schwarz inequality, Euclidian subspaces and their orthogonal complement. Orthogonal 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. Rulings on a quadric, plane sections of a quadric.

V) Basic notion on affine spaces*

1. S. Giuffrida, A.Ragusa, Corso di Algebra Lineare, Ed. Il Cigno G.Galilei, Roma 1998 (Linear Algebra).
2. C. Ciliberto, Algebra Lineare, Bollati-Boringhieri, 1994 (Linear Algebra)
3. G. Paxia, Lezioni di Geometria, Spazio Libri, Catania, 2005 (Geometria) available at www.giuseppepaxia.com
4. E. Sernesi, Geometria 1,Bollati- Boringhieri 1989 (Geometry)