# GEOMETRY

## Learning outcomes of the course unit

Knowledge and understanding

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.

## Course contents summary

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.)

## Recommended readings

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

## Teaching methods

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

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.