# ALGEBRA AND GEOMETRY

## Learning outcomes of the course unit

The aim of this course is to provide students with essential tools in Algebra, Linear Algebra and in Euclidean Geometry in the space; students are required also to apply their knowledge and understanding to problems concerning the spatial structure of real environment, graphics and computer science.

## Course contents summary

This course is an introduction to different aspects of Algebra, Linear Algebra and Geometry.

The first part is devoted to Euclidean Geometry in the space (vectors, lines, planes), while the second part of the course is devoted to matrices and linear systems. In the third part we study vector spaces, linear maps and the diagonalization of linear operators. The course ends with group theory.

## Course contents

Euclidean Geometry in the space.

1. Vectors and its operations. Coordinates. Scalar product. Distances and angles. Vector product in R3.

2. Three dimensional analytic geometry. Parametric and Cartesian equations of a line. Mutual position between two lines in the space; skew lines. Equation of a plane. Quadric surfaces.

Vectors, matrices, linear systems.

3. The n-dimensional space Rn and its properties.

4. Matrices and their properties. Determinants: Laplace expansion and basic properties. Binet theorem. Row and column elementary operations on matrices. Computation of the inverse matrix. Rank of a matrix.

5. Linear systems: Gauss-Jordan method and Rouché Capelli theorem.

6. Linear subspaces of Rn. Linear combinations of vectors: linear dipendence/indipendence. Generators, bases and dimension of a vector subspace.

Linear maps.

7. Linear maps. Definition of kernel and image. Matrix representation of a linear map. Isomorphisms and inverse matrix.

8. Eigenvalues, eigenvector and eigenspaces. Characteristic polynomial. Algebraic and geometric multiplicity. Diagonalizable operators.

Group Theory

9. Natural numbers; integers and their properties. Rational numbers; complex numbers and their properties.

10. Groups, subgroups, homomorphisms; kernel and image. Examples. The fields Zp.

## Recommended readings

ALESSANDRINI, L., NICOLODI, L., GEOMETRIA E ALGEBRA LINEARE, CON ESERCIZI SVOLTI, ED. UNINOVA (PR) 2012.

Notes by the teacher.

## Teaching methods

In the lectures we shall propose formal definitions and proofs, with significant examples and applications, and several exercises. Exercises are an essential tool in Linear Algebra; they will be proposed also in addition to lectures, in a guided manner.

## Assessment methods and criteria

Learning is checked by a written exam and an oral interview. The student can also perform two written exams during the course, to avoid the final written exam.

In the written exam the student must exhibit basic knowledge related to Linear Algebra, Euclidean Geometry in the space and Group Theory. In the oral interview, the student must be able to prove properties of the studied structures, using an appropriate geometric and algebraic language and a proper mathematical formalism.