# GEOMETRY

## Learning outcomes of the course unit

The goal of the course is an introduction of the linear algebra with a particular interest in the solution of a system of linear equations.

## Prerequisites

this course is self-contained

## Course contents summary

geometry and linear algebra

## Course contents

Elements of analytic geometry of the 3-dimensional space. Parametric and cartesian equations

Parametric and Cartesian of a straight line. Mutual position of two

lines. Equation of a plane. Scalar product and distance. Wedge product and its fundamental properties.

Real and complex vector spaces. Subspaces: sum and intersection. Linear combination of vectors:

linear dependence / independence. Generators, bases and dimension of

a vector space. Grassmann formula.

Determinants: definition using the formulas of Laplace and

fundamental properties. Binet theorem. Elementary operations

of the row and column of a matrice. Calculation of the inverse matrix.

Rank of

a matrix.

System of linear equations: Gauss-Jordan's theorem and Theorem of Rouche-Capelli

Linear applications. Definition of the kernel and of image, Dimension's theorem, matrix associated to a linear application and rule base change.

Isomorphisms.

Endomorphisms of a vector space: eigenvalues, eigenvectors and

eigenspaces. Characteristic polynomial. Algebraic multiplicity and

geometry of an eigenvalue. Diagonalizable endomorphisms.

Scalar products. Orthogonal complement of a subspace.

Process of Gram-Schmidt orthogonalization. The orthogonal group.

Diagonalization of symmetric matrices: the spectral theorem.

Positivity criterion for scalar products. Outline of the complex case.

## Recommended readings

abate-De Fabritiis

"Geometria e algebra lineare"

## Teaching methods

class

## Assessment methods and criteria

written and oral exam