## LINEAR ALGEBRA AND GEOMETRY

No specific prerequisites are needed. Nevertheless, a good knowledge of the principal High School topics in Mathematics is very helpful. In particular: an appropriate manuality when dealing with polynomial expressions, when dealing with trigonometric concepts and formulas and when solving I- and II-degree polynomial equations and inequalities.

The exam consists of a written test with theoretical questions and exercises. During the semester in which the lessons are held, some OPTIONAL AND NON REPEATABLE partial tests are held and their outcome possibly contributes to the final evaluation.

The course aims to provide the fundamental principles of linear algebra and the applications of these principles in the fields of analytical geometry and discrete dynamical systems.

We expect:

- the understanding of the essential properties of analytic geometry in the space

- the understanding of the rudiments of vector spaces theory

- the ability to work with matrices

- the ability to solve linear systems of equations

- the ability to apply the appropriate concepts of linear algebra to the modeling and solution of problem of other nature

N.B. The following order does not necessarily correspond to the order in which these topics will be treated during the course.

0. PROPOSITIONAL LOGIC, SETS, FUNCTIONS AND ALGEBRAIC STRUCTURES (SHORT INTRODUCTION)

Overview of propositional logic. Set theory recalls. Numerical sets and induction principle. Functions. Groups, rings and fields.

1. THE FIELD OF COMPLEX NUMBERS

Roots of -1. Algebraic representation. Gauss plane, Cartesian representation. Trigonometric or polar representation, de Moivre's theorem. Exponential representation. Fundamental theorem of algebra.

3. VECTOR SPACES

Vector spaces on a field. Generator systems. Linear independence. Steinitz exchange lemma. Bases. Dimension. Subspaces and operations.

4. LINEAR APPLICATIONS

Definition, kernel, image. Rank. Invertibility. Matrix of a linear application. Change of bases.

5. ALGEBRA OF MATRICES

Definitions and operations: sum, rescaling and product. Matrices and linear applications. Rank of a matrix. Square matrices. Invertibility.

6. LINEAR SYSTEMS

Definitions. The matrix equation AX = B. Linear systems and linear applications. Homogeneous systems. Resolution of linear systems. Gauss algorithm. Rouché-Capelli theorem. Structure of the solution space.

7. THE DETERMINANT

Definition. Determinant as an alternating multilinear function with value 1 on the identity. Properties of the determinant. First and second Laplace theorem. Binet's theorem. Determinant, rank and invertibility.

Cramer's Rule. Vector product in R^3.

8. EUCLIDEAN SPACES

Scalar products. Norm and distance. Standard scalar product in R^3; scalar product theorem. Orthogonality. Orthonormal bases, Gram-Schmidt orthogonalization. Isometries.

9. EIGENVALUES, EIGENVECTORS AND DIAGONALIZATION

Eigenvalues, eigenvectors and eigenspaces. Characteristic polynomial. Multiplicity. Search for eigenvalues. Endomorphisms and diagonalizability. Diagonalizability by orthogonal matrices. Real spectral theorem.

10. ANALYTICAL GEOMETRY IN THE PLANE AND IN THE SPACE

Affine spaces. Cartesian and parametric equations. Lines in the plane. Lines and planes in R^3. Spheres in R^n.

11. COMPLEMENTARY ARGUMENTS (to be treated depending on the time available and on the needs)

Dual spaces. Singular values decomposition. Discrete dynamic systems.

Quadratic forms. Conics as geometric loci and their classification. Quadrics, definitions and classification. Cayley-Hamilton's theorem. Jordan's canonical form. Hermitian products and complex spectral theorem.

THEORY AND EXERCISES

- Gimigliano Alessandro; Bernardi Alessandra, Algebra lineare e geometria analitica, Editore: CittàStudi http://www.cittastudi.it/catalogo/scienze/algebra-lineare-e-geometria-an...

- Fulvio Bisi, Francesco Bonsante, Sonia Brivio. Esercizi risolti di Geometria e Algebra. Rilasciata con licenza Creative Commons. disponibile all'indirizzo http://smmm.unipv.it/didattica/GeoAlg/book_ex.pdf

o da richiedere al docente

OTHER COMPLEMENTARY/SUBSTITUTE MANUALS

-E. Schlesinger, Algebra lineare e geometria, Zanichelli

- S. Lang. Algebra lineare. Bollati Boringhieri

-D.C. Lay, S.R. Lay, J.J. MacDonald. Linear algebra and its applications. Pearson

-M. Abate, C. De Fabritiis, Geometria analitica

con elementi di algebra lineare. McGraw-Hill Education

The course is divided into frontal lectures and exercitations, held by the teacher. There are also practical lessons in which the student, under the assistance of the teacher, deals with the solution of problems concerning the topics of the course.

- During the lessons further exercises on the topics covered will be provided.

- The final exam and the intermediate tests can be taken in one of the following languages, upon request to the teacher: English, French, Spanish

- The student reception will be held on the days when there are lessons, or by appointment via email.