## 20410009 - ELEMENTS OF GEOMETRY

In the course students are taught the basics of linear algebra and analytic geometry in plane and space. In particular, the essential notions are developed to solve a system of linear equations, to calculate the rank of a matrix and its other invariants. As for the notions of
analytical geometry will pay particular attention to the notion of scalar product and to the study of conics and quadrics
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Programme

Vectors in Euclidean Space. Reference and co-ordered systems. Orthogonal poijections, scalar, vector and mixed product. Parametric and Cartesian equations of lines and planes.
Linear systems. Linear equations, column vectors and matrices. Gauss elimination method: rank of a matrix, Cramer and Rouch ́e-Capelli theorems.

Algebra of Matrices. Sum and product for a scalar. Produced rows by columns. Invertible matrices, transposed matrix and symmetric matrices. The Gauss-Jordano algorithm for calculating the inverse. LU factorization. Product of block matrices.
Vector spaces and linear applications. Examples of vector spaces and linear applications. Generators and bases. Core, image. Linear independence and size; rank of a matrix. Nullity theorem plus rank and Grassmann's formula.
Determinant of a matrix and Gauss moves. Developments of Laplace. Binet's theorem.
Eigenvalues and eigenvectors. The characteristic polynomial of a linear operator. Similar matrices.
Scalar products. Schwartz inequality, orthogonal bases and matrices. Orthogonal projections and Grahm-Schmidt algorithm.

Core Documentation

Enrico Schlesinger, Algebra lineare e geomegtria. Zanichellii, (2017).
Luca Mauri, Enrico Schlesinger, Esercizi di algebra lineare e geomegtria. Zanichellii, (2020).

Type of delivery of the course

The lessons take place in traditional mode, in presence and in the classroom; in case of extension of the containment provisions for the COVID-19 emergency, the lessons will take place remotely through the use of the University Teams platform

Type of evaluation

Written test, Oral exam; partial written tests during the course.

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Programme

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Core Documentation

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Type of delivery of the course

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Type of evaluation

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