# GEOMETRIA

Module DIDATTICA FRONTALE

**Academic Year 2022/2023**- Teacher:

**Antonio CAUSA**

## Expected Learning Outcomes

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.

## Course Structure

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.

*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

Elementary Euclidean Geometry. Cartesian geometry. Basic knowledge of polynomial algebra.## Detailed Course Content

**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.

## Textbook Information

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/

## Course Planning

Subjects | Text References | |

1 | Linear spaces: definition and basic properties. Examples: R^n, R^m,n, R[X]. Intersection and sum of subspaces. Generators of a linear space. Linear dependence. Bases of a linear space. Dimension of a linear space. Grassmann formula. | D. Margalit , J. Rabinoff, Interactive linear algebra, available at https://textbooks.math.gatech.edu/ila/ |

2 | 1 Matrices with coefficients in a field. Matrix algebra: sum, scalar product, dot product. Basic properties of the ring of square matrices. Triangular and diagonal matrices, symmetric and antysimmetric matrices. Operations on matrices. | D. Margalit , J. Rabinoff, Interactive linear algebra, available at https://textbooks.math.gatech.edu/ila/ |

3 | Determinant of a matrix. Binet and Laplace theorems on determinants. Inverse of matrix. Gaussian elimination: reduced row echelon form. Rank of a matrix. Kroneker theorem. Linear equations. Cramer theorem, Rouchè theorem. Homogeneous linear systems of equations. | D. Margalit , J. Rabinoff, Interactive linear algebra, available at https://textbooks.math.gatech.edu/ila/ |

4 | Vector spaces homomorphisms. Null space and range space. Injectivity and surjectivity. Rank-nullity theorem. Matrix representation of a linear transformation. Matrix similarity. | 3) D. Margalit , J. Rabinoff, Interactive linear algebra, available at https://textbooks.math.gatech.edu/ila/ |

5 | Endomorphism and their eigenvalues and eigenvectors. Characteristic polynomial of a matrix. Eigenspaces. Endomorphisms and diagonalization. Diagonalizable matrices. | |

6 | Geometric vectors and basic operations with geometric vectors: dot product, cross product, triple product. | |

7 | Cartesian coordinates in plane and space. Orthogonality and parallelism.Planes and lines in space. | G. Paxia, Lezioni di Geometria, Spazio Libri, Catania, 2005 |

8 | Classification of affine conics. Quadratic forms. Canonical forms of a conic. Tangent lines to a conic. | G. Paxia, Lezioni di Geometria, Spazio Libri, Catania, 2005 |

9 | Quadrics in affine and projective spaces. Classification of quadrics. Double points of a quadric. | G. Paxia, Lezioni di Geometria, Spazio Libri, Catania, 2005 |

## Learning Assessment Procedures

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

## Examples of frequently asked questions and / or exercises

Linear algebra

Linear spaces: definition and basic properties. Linear independence, bases of a vector space. Dimension of a vector space. Coordinates of a vector. Linear transformations: definition and basic properties. Matrix associated to a linear transformation. rank-nullity theorem. Eigenvalues and eigenvectors. Spaces endowed with a scalar product. Cauchy-Schwartz inequality. The Spectral theorem.Geometry

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