## GEOMETRY

Italian

Italian

### 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.

Knowledge and understanding

By means of frontal lesson, the student will be introduced to the basic concepts and techniques of linear algebra and Euclidean geometry.

Applying knowledge and understanding

The 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.

The course is an introduction to the basic notions of linear algebra and geometry. The first part studies Euclidean geometry in 3-space (vectors, lines, planes), while the second part is devoted to the study of vectors, matrices, and linear systems. In the third part of the course we study

vector spaces, linear maps and the diagonalization problem for linear operators and matrices. The course ends with a study of scalar and hermitian products

### 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.)

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’’

M. Abate, C. De Fabritiis, Geometria analitica con elementi di algebra lineare, 2a ed., Mc Graw-Hill, 2010.

### 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.

Due to the coronavirus, any activity that has been described, at least at the begining of the semester, will be given by using on-line platform like teams and elly

During lectures, the material of the course is presented using formal definitions and proofs; abstract concepts are illustrated through significant examples, applications, and exercises. The discussion of examples and exercises is of fundamental importance for grasping the meaning of the abstract mathematical concepts; for this reason, besides lectures, guided sessions to discuss and solve exercises and assignments will be provided within the the “Progetto IDEA”.

### 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.. 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. The 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.

If coronavirus causes significant disruption to the provision of education then the final exam, written and oral, will be a quiz with multiple choice using the platform elly

Course grades will be based on a final exam which consists of a preliminary multiple-choice test, a written exam and an oral interview. Possibly, there will be two intermediate written exams and tests to avoid the final written exam and test. The written exam, through tests and exercises, should establish that students have learned the course materials to a sufficient level. In the colloquium, students should be able to repeat definitions, theorems and proofs given in the lectures using a proper mathematical language and formalism.