Unit GEOMETRY I
- Course
- Mathematics
- Study-unit Code
- GP006038
- Curriculum
- In all curricula
- Teacher
- Daniele Bartoli
- Teachers
-
- Daniele Bartoli
- Hours
- 63 ore - Daniele Bartoli
- CFU
- 9
- Course Regulation
- Coorte 2022
- Offered
- 2022/23
- Learning activities
- Base
- Area
- Formazione matematica di base
- Academic discipline
- MAT/03
- Type of study-unit
- Obbligatorio (Required)
- Type of learning activities
- Attività formativa monodisciplinare
- Language of instruction
- Italian
- Contents
- Linear algebra.
Elementary analytic geometry. - Reference texts
- K.W. Gruenberg and A.J. Weir, Linear Geometry. GTM, Springer-Verlag, New York, 1977.
- Educational objectives
- Acquisition of the geometrical thought through algebraic tools.
- Prerequisites
- Polynomial factorizations. Solutions of the algebraic equations of the first and second degree. Binomial and trinomial equations. Equations solvable by applying Ruffini's rule. Elementary analytic geometry in the plane. Trigonometry.
- Teaching methods
- Frontal lectures. Almost all results will be rigorously proved with the exception of those concerning projective geometry and algebraic curves. At te same time many exercises will be presented and solved.
- Other information
- Attendance is not mandatory but highly recommended.
- Learning verification modality
- Two hours of class-work with 8 exercises: 4 exercises about linear algebra one of which involving a finite field; 2 exercises about elementary analytic geometry in the space; 2 exercises about the sphere and/or the circle in the space. Then fifteen/thirty minutes of oral examination.
The exercises will not be hard; each of them might be solved in at most 10 minutes by a well prepared student. These exercises are finalized to verify the students' capacity in handling the algebraic-geometric tools treated in the theory.
The purpose of the oral examination is to verify the students' capacity in illustrating the subject with a special attention to a rigorous math language and to the synthesis.
Info about how to support students with disabilities and/or DSA can be found at http://www.unipg.it/disabilita-e-dsa - Extended program
- Linear Algebra. Vector spaces. Linear independence. Steinitz Lemma. Bases. Theorem on the equicardinality of the bases. Dimension. Every independent set is contained in a suitable base. Subspaces. Intersection and sum of subspaces. Grassmann Theorem. Linear applications. Kernel and Image. Fundamental Theorem on the isomorphism between vector spaces. The vector sapce of real matrices of type m x n. Product between matrices. Matrix associated with a linear application. Determinant. Inverse matrix. Rank of a matrix. Linear systems. Rouché-Capelli Theorem. Homogeneus linear systems. The space of all solution of a homogeneous linear system. Cramer Theorem. General algorithm for determining the set of all solutions of a linear system.
Geometry in the plane and in the space. Cartesian coordinates. Oriented segments. Geometric vectors. Parallel and coplanar vectors. Components of a vector. Parametric equations of a line. Equation of a plane. Intersection and parallelism between planes. Cartesian equations of a line. Sheaf of planes. Intersection and parallelism between a line and a plane. Intersection and parallelism between lines. Coplanar lines. Inner product. Distance between two points. Angle between two lines. Distance between a point and a plane. Angle between two planes. Angle between a line and a plane. Distance between a point and a line. Distance between two lines. Projective plane. Homogeneous coordinates. The complex projective plane. Basics on algebraic plane curves. Singular points of an algebraic palne curve. Conics. Conic through five points; conic thorough 4 points A, B, C, T with an assigned tangent t in A; conics through 3 points A. T, T' with an assigned tangent t in T and an assigned tangent t' in T'. Conics as geometric logos. Canonical equations of the conics. Sphere. Circle in the space.