GEOMETRY
Course objectives
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.
Channel 1
ERNESTO SPINELLI
Lecturers' profile
Program - Frequency - Exams
Course program
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.
Prerequisites
Knowledge of basic arguments of Mathematics given in secondary school.
Books
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)
Frequency
Recommended
Exam mode
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.
Bibliography
Any text of Linear Algebra
Lesson mode
Lectures and exercises.
Channel 2
ALBERTO DE SOLE
Lecturers' profile
Program - Frequency - Exams
Course program
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, rankÐnullity 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.
Prerequisites
No prerequisites
Books
Marco Abate e Chiara de Fabritiis, Geometria Analitica con elementi di Algebra Lineare, III edizione (2015), ed. Mc Graw Hill Educational
Marco Manetti. Algebra lineare.
Frequency
In class, presence not mandatory
Exam mode
To pass the exam, a grade of not less than 18/30 must be obtained. The student must
to demonstrate to have acquired a sufficient knowledge of the arguments of both
parts 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
have acquired an excellent knowledge of all the topics covered during the course and
be able to link them in a logical and consistent way.
Lesson mode
Traditional lectures
FABIO BERNASCONI
Lecturers' profile
Channel 3
RICCARDO SALVATI MANNI
Lecturers' profile
Program - Frequency - Exams
Course program
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.
Prerequisites
none
Books
Marco Abate e Chiara de Fabritiis, Geometria Analitica con elementi di Algebra Lineare, III edizione (2015), ed. Mc Graw Hill Educational
Teaching mode
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 an oral test (consisting of
discussion of the most relevant topics illustrated in the course). The written test will last about three
hours and can be replaced by two intermediate tests, both lasting two hours, the first one
of which will take place in the middle of the course and the second immediately at the end of the course.
Frequency
in presence
Exam mode
To pass the exam, a grade of not less than 18/30 must be obtained. The student must
to demonstrate to have acquired a sufficient knowledge of the arguments of both
parts 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
have acquired an excellent knowledge of all the topics covered during the course and
be able to link them in a logical and consistent way.
Bibliography
Any text in Linear Algebra
Lesson mode
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 an oral test (consisting of
discussion of the most relevant topics illustrated in the course). The written test will last about three
hours and can be replaced by two intermediate tests, both lasting two hours, the first one
of which will take place in the middle of the course and the second immediately at the end of the course.
Channel 4
MARCO MANETTI
Lecturers' profile
Program - Frequency - Exams
Course program
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.
Prerequisites
none in particular
Books
Marco Abate e Chiara de Fabritiis, Geometria Analitica con elementi di Algebra Lineare, III edizione (2015), ed. Mc Graw Hill Educational
Teaching mode
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 an oral test (consisting of
discussion of the most relevant topics illustrated in the course). The written test will last about three
hours and can be replaced by two intermediate tests, both lasting two hours, the first one
of which will take place in the middle of the course and the second immediately at the end of the course.
Frequency
Not mandatory
Exam mode
To pass the exam, a grade of not less than 18/30 must be obtained. The student must
to demonstrate to have acquired a sufficient knowledge of the arguments of both
parts 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
have acquired an excellent knowledge of all the topics covered during the course and
be able to link them in a logical and consistent way.
Lesson mode
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 an oral test (consisting of
discussion of the most relevant topics illustrated in the course). The written test will last about three
hours and can be replaced by two intermediate tests, both lasting two hours, the first one
of which will take place in the middle of the course and the second immediately at the end of the course.
SAMOUIL MOLCHO
Lecturers' profile
- Lesson code1015375
- Academic year2024/2025
- CoursePhysics
- CurriculumFisica
- Year1st year
- Semester1st semester
- SSDMAT/03
- CFU9
- Subject areaDiscipline matematiche e informatiche