# GEOMETRY

## Learning outcomes of the course unit

The course aims, by means of frontal lessons, to provide knowledge and techniques of linear algebra for the purpuse of providing tools for resolving linear systems, diagonalising matrices and simply describing the behaviours of geometric bodies in the plane and in space.Applying knowledge and understandingThe student will be able to: i) solve systems of linear equations, ii) simple exercises of analytic geometry in space; operate on vectors and matrices; iii) diagonalize operators and matrices.Making judgments: the student must be able to understand the rightness of the results obtained by himself or by others.Communications skills:Through the frontal class and assistance of the teacher, the student acquires scientific vocabulary. At the end of the course, the student is expected to be able to communicate mathematical arguments.Learning skills:The student who has attended the course will be able to deepen is knowledge of linear algebra and vector spaces.

## Course contents summary

Vector and matrix calculus. Determinant and rank of a matrix. Linear systems. Real and complex vector spaces. Bases and dimension. Sum and direct sum of subspaces: Grasmann relation. Linear applications and associated matrices. and eigenvectors. Diagonalizability of a matrix. Bilinear forms and scalar products. Orthonormal bases. Real symmetrical matrices: diagonalizability. Orthogonal matrices and isometries. Coordination in the plane and in the space. Parametric and cartesian representation of stright lines and planes.Parallelism and orthogonality.

## 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

Marco Abate, Chiara De Fabritiis “Geometria analitica con elementi di algebra lineare", Francesco Capocasa e Costantino Medori ‘’Corso di Geometria e Algebra Lineare’’

## Teaching methods

Privileged education mode is the frontal lesson that offered arguments from a formal point of view, accompanied by significant examples, applications and exercise

## Assessment methods and criteria

Verification of learning takes place through a written test and an oral. In the written examination through the exercises proposed by the student must demonstrate that they possess the basic knowledge of linear algebra and analytical geometry. In the oral examination the student must be able to conduct its own demonstrations relating to the themes of the course using an appropriate language and mathematical formalism. We also apply two midterms written exams.