Academic Year 2019/2020 - 1° Year

MAT/03 - 6 CFU - 1° Semester

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.

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**Linear Algebra**

- Matrices over a field. Matrices addition, scalar multiplication, matrix multiplication (or product). Diagonal, triangular, scalar, symmetric, skew-simmetric matrices and transpose of matrix.
- Vector spaces and their properties over R. Examples: R[x], R
^{n}, 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*. - 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.
- 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.
- 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