The students will get acquainted and learn how to handle basic operations with vector spaces and matrixes. The student will acquire the skills and the confidence to approach linear systems and eigenvalue problems, by (1) recognizing and translating the linear system into matrix language, (2) becoming aware of important aspects such as existence and uniqueness of a solution, (3) being able to tackle the resolution of the system by computational means. The student will acquire an awareness about important aspects at the computational level, such as efficiency and stability of numerical methods.
Prerequisiti
No particular prerequisite is needed for the course, apart the basic minimal mathematical knowledge from the high school.
Metodi didattici
The major part of the classroom lessons will be held at the blackboard. Such lessons will combine theoretical sessions, where new concepts and notions will be introduced, and practical sessions, where exercises will be developed by the lecturer. In addition, a minor part of the lessons will be held by projecting from a laptop computer, in order to show directly some computational example using the software MATLAB.
Verifica Apprendimento
The exam will take the form of a written examination. The final mark will follow the usual range 0-30, 18 representing the minimal value to pass the exam. During the written examination, no support material (such as textbooks, notes or hand-calculators) can be used. The written examination will combine some ``theoretical questions’’ aiming at checking the student’s knowledge on the theoretical aspects of the course (such as stating and explaining a definition or a certain theorem) and some practical exercises (along the lines developed in the practical classroom sessions). The relative weight of the first part will be around 40%, and the second will be around 60.
Testi
The MAIN textbook of the course is:
“Linear algebra and its applications” (J.C. Lay, R.S. Lay, J.J. MacDonald), Pearson
A possible support books on the computational side is:
“Numerical linear algebra” (W. Layton, M.M. Sussman), World Scientific Publishing
Contenuti
Vector spaces: linear independence, basis and dimensions, norm and scalar product. Matrixes: operations, determinant, rank, nonsingular matrixes. Symmetric, positive definite and orthogonal matrixes. Linear applications between vector spaces: kernel, image, dimensional relation, operator norms. Linear systems: direct methods and LU factorization, computational cost. Stability and condition number of linear systems. Least square problems, QR factorization, computational aspects. Eigenvectors and eigenvalues.