Course program
Linear algebra:
0)Definition of a field ed examples: Q,R,F_2.
1)n-tuples of real numbers: addition, scalar multiplication, dot product, definitions and basics.
2)Matrices over real numbers: definitions, addition, scalar multiplication, matrix multiplication, basic properties.
3) Systems of linear equations: definitions, homogeneous systems, basics, Gaussian elimination.
4)Vector spaces: definitions, examples, basics, subspaces, generators, linear independence, Steinitz exchange lemma, existence of basis. Subspaces, Grassmann Formula. Subspaces of R^n.
5)Determinant and inverse of a square matrix: defintions, basics, Cramer's rule.
6)Matrix rank: definitions, basics, Rouché-Capelli theorem. Comparison of the different approaches to study a system of linear equations.
7)Row and column spaces of a matrix: rank theorem, further analysis of the theory of the linear systems in the light of the theory of vector spaces.
8)Linear applications: definition, basics, matrix representation, kernel and image, rank–nullity theorem, isomorphisms, Theorem: every n-dimensional vector space over the real numbers is isomorphic to R^n.
9)Diagonalization.
10)Inner product: definition, basics, norm, orthogonality.
Analytic geometry:
1)Analytic geometry in dimension 2: coordinate system, parallelism, dot product, equations of a line, intersection, parallelism and orthogonality of two lines, sheaves of lines, distance between two points and between a point and a line, change of coordinates.
2)Quadratic forms and basics of conics.
3)Analytic geometry in dimension 3: coordinate system, cross product, equations of a plane, intersection, parallelism and orthogonality between planes, sheaves and bundles of planes, equations of a line, reciprocal position of two lines, sheaves and bundles of lines, parallelism and orthogonality between a line and a plane, the angle between two lines, a line and a plane and two planes, distance between two points, a point and a plane, a point and a line, between two lines, between a line and a plane, the area of a parallelogram.
Prerequisites
Basic knowledge of the real numbers. Basic set theory. Foundations of logic. Relations and functions. Basic algebra. Polynomials over the reals. Equations and inequalities of first and second degree. Foundations of Euclidean geometry, equivalance of triangles, similarity. Foundations of trigonometry.
Books
For Linear Algebra:
A. Bernardi, A. Gimigliano, Algebra lineare e geometria analitica, Città Studi Edizioni
For Geometry:
L. Francisco e P. Mercuri, Elementi di Geometria Affine ed Euclidea, Edizioni Efesto.
Lecture notes of the Professor.
Teaching mode
The lectures are focused about the theoretical part as well as getting through the most difficult exercices.
Frequency
The lectures are given in person, but they are also live streamed on the zoom platform.
Exam mode
The written part consists of exercices and theoretical questions. In the oral part, the written exam is discussed and the professor can ask theorems with their proofs.
Lesson mode
The lectures are focused about the theoretical part as well as getting through the most difficult exercices.
