## Learning objectives

Knowledge and understanding:

the theory of vector spaces.

Applying knowledge and understanding:

a) solve systems of linear equations;

b) diagonalize matrices;

c) solve easy problems of analytic geometry;

d) recognize the type of a conic and write its canonical form.

Making judgements:

evaluate the correctness of a simple proof.

Communication and learning skills:

properly express themselves with mathematical language.

## Prerequisites

Precourse.

This exam is preparatory to "Analisi matematica 2".

## Course unit content

1. Vectors in tridimensional space.

2. Matrices.

3. Linear system.

4. Vector spaces.

5. Plane and space geometry.

6. Linear applications and diagonalization.

## Full programme

0. Preliminaries: algebraic structures, the field of complex numbers.

1. Vectors in tridimensional space: definition, sum and product, scalar product, cross product and its geometric meaning.

2. Matrices: definition, sum and product, square matrices, determinant, inverse matrix, rank, orthogonal matrices, change of basis matrix.

3. Linear system: definition, compatibility, resolution via Gauss algorithm, Rouché-Capelli Theorem.

4. Vector spaces: definition, linear dependence, subspaces, basis and coordinates, intersection and sum of subspaces, scalar product, orthogonal systems and orthogonal basis, Gram-Schmidt algorithm.

5. Plane and space geometry: lines in the plane, sheaf of lines, intersection of lines, the circle, distance between a point and a line, lines and planes in the tridimensional space, intersection between lines, between a plane and a line and between two planes, sheaf of planes, the sphere, distance between a plane a point.

6. Linear applications and diagonalization: definition and properties, associated matrices, kernel and image, eigenvalues and eigenvectors, endomorphism and square matrices' diagonalizability, Spectral Theorem, scalar product, hermitian product and congruent matrices.

## Bibliography

Pdf notes of the course are given.

Other sources:

G. Catino, S. Mongodi - Esercizi svolti di Geometria e Algebra Lineare, Società editrice Esculapio;

A. Sanini, Elementi di Geometria, Levrotto & Bella;

A. Sanini, Esercizi di Geometria, Levrotto & Bella;

E. Sernesi, Geometria 1, Bollati Boringhieri.

## Assessment methods and criteria

The final exam consists in:

- a written test with 4 exercises to be solved in 2 hours.

- an oral examination, usually taken one week after the test.

The students attending the lectures can choose to divide the written test into two parts to be taken during the course.

If the sanitary emergency persists, the written test could be replaced by a computer based test, with 14 multiple choices questions to be answered in 90 minutes.

## Other information

Lecture attendance is highly recommended.