Physics

ERNESTO SPINELLI

General aim:
to acquire basic knowledge on linear systems, vector spaces, linear maps, symmetric bilinear forms and hermitian forms.

Specific aim
Knowledge and comprehension: successful students will acquire basic notions and results about solvability of linear systems and its geometric interpretation, matrix calculus, vector spaces and linear maps between them, with special focus on endomorphisms of vector spaces and their associate decompositions into eigenspaces.

Applied knowledge and comprehension: successful students will be able to solve linear systems with a finite number of variables, to
recognize mathematical problems that can be codified into vector spaces and linear maps and therefore to solve them;
to manipulate matrices and determine the solvability of a linear system and the invertibility of a linear map by studying its rank and by
computing the determinant of the associated matrix; moreover, he/she will be able to compute the eigenvalues of a linear endomorphism
and to determine the associated decomposition into eigenspaces.

Critical thinking abilities: in this course the student will acquire basic knowledge that will make him/her able to discover analogies between the topics treated in this course and the topics treated in following courses such as Mechanics and Vectorial Analysis.

Communication skills: ability to illustrate the content of the course in the oral exam and in the possible theoretical questions in the written test.

Learning skills: the learnings of these lectures will allow the student to access the study of the following courses of the degree program in Physics.

- General aim: to acquire basic knowledge on linear systems, vector spaces, linear maps, symmetric bilinear forms and hermitian forms.

- Specific aim:
Knowledge and comprehension: successful students will acquire basic notions and results about solvability of linear systems and its geometric interpretation, matrix calculus, vector spaces and linear maps between them, with special focus on endomorphisms of vector spaces and their associate decompositions into eigenspaces.

Applied knowledge and comprehension: successful students will be able to solve linear systems with a finite number of variables, to recognize mathematical problems that can be codified into vector spaces and linear maps and therefore to solve them; to manipulate matrices and determine the solvability of a linear system and the invertibility of a linear map by studying its rank and by computing the determinant of the associated matrix; moreover, he/she will be able to compute the eigenvalues of a linear endomorphism and to determine the associated decomposition into eigenspaces.

Critical thinking abilities: in this course the student will acquire basic knowledge that will make him/her able to discover analogies between the topics treated in this course and the topics treated in following courses such as Mechanics and Vectorial Analysis.

Communication skills: ability to illustrate the content of the course in the oral exam and in the possible theoretical questions in the written test.

Learning skills: the learnings of these lectures will allow the student to access the study of the following courses of the degree program in Physics.

Knowledge of basic arguments of Mathematics given in secondary school.

Basic notions. Sets, functions, fields, polynomials. The field of complex numbers.
Linear systems. Gaussian elimination, structure of solutions.
Vector spaces. Linear combinations, subspaces and affine subspaces, linear independence, bases and generators, dimension, sum and intersection, Grassmann's formula.
Linear maps. Kernel, image, rank–nullity theorem.
Matrices. Algebra of matrices, matrix associated to a linear map, rank, invertible matrices, change of coordinates, determinants.
Diagonalization. Eigenvalues and eigenvectors, characteristic polynomial.
Bilinear forms. Scalar products, Hermitian products, orthogonal bases, Sylvester's law of inertia, Euclidean and Hermitian vector spaces.
Self-adjoint operators, linear isometries, Spectral Theorem.

Marco Abate e Chiara de Fabritiis, Geometria Analitica con elementi di Algebra Lineare, III edizione (2015), ed. Mc Graw Hill Educational

Notes of "Algebra Lineare e Geometria" (from the website https://www.mat.uniroma1.it/didattica/materiale-didattico)

Lectures and exercises

Recommended

The exam aims to evaluate learning through a written test (consisting of the resolution of problems of the same type as those carried out in the exercises) and a possible oral test (consisting of discussion of the most relevant topics illustrated in the course). The written test will last about three hours.
To pass the exam, a grade of not less than 18/30 must be obtained. The student must demonstrate to have acquired a sufficient knowledge of the arguments of the program and to be able to carry out at least the simplest among the assigned exercises. To achieve a score of 30/30 cum laude, the student must instead demonstrate to have an excellent knowledge of all the topics covered during the course and be able to link them in a logical and consistent way.