# GEOMETRY 1A

## 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

A briefly introduction to the complex number. 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. Exercises are proposed every week. The exercises are uploadoaded on the plattform Elly. The aim is to invites students to check theirselver the knowledge and ability. . Also the pdf files of the lessons are uploades on the platform Elly every week.

## 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. The student must register himselves on esse3 to do the written and oral exam. The duration of the written test is 1 hora and 45 minutes. There are also two intermediate tests. The student must register on the esse3 to carries out the intermediate tests. Students that passes successfully the intermediate written tests (evaluation of each test must be equal or bigger that 16/30) will receive a score of access (in thirtieths) to the oral exam. These students are expelled from the written exam from january 2019 to febraury 2020. In order to take oral exam, students must register for one of the exam sessions (only for the oral exam, on the ESSE3 web platform). The duration of the intermediate tests is 1 hour and 45 minutes. Student that do not make or do not pass the intermediate written tests will need to take the final exam consisting of a written exam and an oral exam. In this case, students that get an evalutaion equal or better that 18/30, they will admitt to the oral examination. The final evalutaion aries from the arithmetic average oof the written exam (or intermediate written tests) and the oral exam.