Geometry 1
A.Y. 2024/2025
Learning objectives
The aim of the course is to present the first results in linear algebra and affine geometry.
Expected learning outcomes
At the end of the course students will have learnt and have been able to apply the notions of vector spaces, basis, linear applications and the techniques of matrix calculus and methods of resolution of linear systems.
Lesson period: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Geometria 1 (ediz.1)
Responsible
Lesson period
First semester
Course syllabus
Vector spaces: definitions and first properties; examples; subspaces; sums and intersections; linear dependence; bases; dimension; Grassman theorem.
Matrices and linear systems: Gauss elimination theorem; matrix algebra; rank; Rouch Rouché-Capelli theorem.
Linear applications: definitions and first properties; examples; kernel and image; rank-nullity theorem; existence and uniqueness theorem; representative matrix; composition of linear applications and matrix multiplication; base change matrix; invertible matrices and isomorphisms; the space of linear maps and its isomrphism with matrix spaces; dual vector space and dual basis.
Determinant: definitions and first properties of multilinear and alternating applications; existence and unicity theorem; Laplace theorem; inverse matrix; Cramer and Kronecker theorems.
Affine spaces: definitions and first properties; affine coordinate system; affine subspaces; parallelism.
Matrices and linear systems: Gauss elimination theorem; matrix algebra; rank; Rouch Rouché-Capelli theorem.
Linear applications: definitions and first properties; examples; kernel and image; rank-nullity theorem; existence and uniqueness theorem; representative matrix; composition of linear applications and matrix multiplication; base change matrix; invertible matrices and isomorphisms; the space of linear maps and its isomrphism with matrix spaces; dual vector space and dual basis.
Determinant: definitions and first properties of multilinear and alternating applications; existence and unicity theorem; Laplace theorem; inverse matrix; Cramer and Kronecker theorems.
Affine spaces: definitions and first properties; affine coordinate system; affine subspaces; parallelism.
Prerequisites for admission
No specific preliminary knowledge is required.
Teaching methods
Frontal lectures about theory and classes of exercises.
Tutoring.
Tutoring.
Teaching Resources
Ariel web page.
Book: E. Sernesi, Geometria I, Bollati Boringhieri.
Book: E. Sernesi, Geometria I, Bollati Boringhieri.
Assessment methods and Criteria
The final examination consists of two parts: a written exam, an oral exam.
- During the written exam, the student must solve some exercises in the format of open-ended and possibly closed answer questions, with the aim of assessing the student's ability to solve problems in linear algebra and affine geometry. The duration of the written exam will be proportional to the number of exercises assigned, also taking into account the nature and complexity of the exercises themselves (however, the duration will not exceed three hours). There will be probably a midterm exam. If successfully passed it allows the student to solve less exercises during the written exams of January and February.
The outcomes of these tests will be available in the ariel web site.
- The oral exam can be taken only if the written component has been successfully passed. In the oral exam, the student will be required to illustrate results presented during the course in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.
The complete final examination is passed if all the parts (written and oral) are successfully passed. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.
- During the written exam, the student must solve some exercises in the format of open-ended and possibly closed answer questions, with the aim of assessing the student's ability to solve problems in linear algebra and affine geometry. The duration of the written exam will be proportional to the number of exercises assigned, also taking into account the nature and complexity of the exercises themselves (however, the duration will not exceed three hours). There will be probably a midterm exam. If successfully passed it allows the student to solve less exercises during the written exams of January and February.
The outcomes of these tests will be available in the ariel web site.
- The oral exam can be taken only if the written component has been successfully passed. In the oral exam, the student will be required to illustrate results presented during the course in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.
The complete final examination is passed if all the parts (written and oral) are successfully passed. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.
MAT/03 - GEOMETRY - University credits: 6
Practicals: 36 hours
Lessons: 27 hours
Lessons: 27 hours
Professors:
Stellari Paolo, Tasin Luca
Shifts:
Geometria 1 (ediz.2)
Responsible
Lesson period
First semester
Course syllabus
Vector spaces: definitions and first properties; examples; subspaces;
sums and intersections; linear dependence; bases; dimension; Grassman
theorem.
Matrices and linear systems: Gauss elimination theorem; matrix
algebra; rank; Rouch Rouché-Capelli theorem.
Linear applications: definitions and first properties; examples;
kernel and image; rank-nullity theorem; existence and uniqueness
theorem; representative matrix; composition of linear applications and
matrix multiplication; base change matrix; invertible matrices and
isomorphisms; the space of linear maps and its isomrphism with matrix
spaces; dual vector space and dual basis.
Determinant: definitions and first properties of multilinear and
alternating applications; existence and unicity theorem; Laplace
theorem; inverse matrix; Cramer and Kronecker theorems.
Affine spaces: definitions and first properties; affine coordinate
system; affine subspaces; parallelism.
sums and intersections; linear dependence; bases; dimension; Grassman
theorem.
Matrices and linear systems: Gauss elimination theorem; matrix
algebra; rank; Rouch Rouché-Capelli theorem.
Linear applications: definitions and first properties; examples;
kernel and image; rank-nullity theorem; existence and uniqueness
theorem; representative matrix; composition of linear applications and
matrix multiplication; base change matrix; invertible matrices and
isomorphisms; the space of linear maps and its isomrphism with matrix
spaces; dual vector space and dual basis.
Determinant: definitions and first properties of multilinear and
alternating applications; existence and unicity theorem; Laplace
theorem; inverse matrix; Cramer and Kronecker theorems.
Affine spaces: definitions and first properties; affine coordinate
system; affine subspaces; parallelism.
Prerequisites for admission
No specific preliminary knowledge is required.
Teaching methods
Frontal lectures about theory and classes of exercises.
Tutoring.
Tutoring.
Teaching Resources
Ariel web page.
Book: E. Sernesi, Geometria I, Bollati Boringhieri.
Book: E. Sernesi, Geometria I, Bollati Boringhieri.
Assessment methods and Criteria
The final examination consists of two parts: a written exam, an oral exam.
- During the written exam, the student must solve some exercises in the format of open-ended and possibly closed answer questions, with the aim of assessing the student's ability to solve problems in linear algebra and affine geometry. The duration of the written exam will be proportional to the number of exercises assigned, also taking into account the nature and complexity of the exercises themselves (however, the duration will not exceed three hours). There will be probably a midterm exam. If successfully passed it allows the student to solve less exercises during the written exams of January and February.
The outcomes of these tests will be available in the ariel web site.
- The oral exam can be taken only if the written component has been successfully passed. In the oral exam, the student will be required to illustrate results presented during the course in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.
The complete final examination is passed if all the parts (written and oral) are successfully passed. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral exam
- During the written exam, the student must solve some exercises in the format of open-ended and possibly closed answer questions, with the aim of assessing the student's ability to solve problems in linear algebra and affine geometry. The duration of the written exam will be proportional to the number of exercises assigned, also taking into account the nature and complexity of the exercises themselves (however, the duration will not exceed three hours). There will be probably a midterm exam. If successfully passed it allows the student to solve less exercises during the written exams of January and February.
The outcomes of these tests will be available in the ariel web site.
- The oral exam can be taken only if the written component has been successfully passed. In the oral exam, the student will be required to illustrate results presented during the course in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.
The complete final examination is passed if all the parts (written and oral) are successfully passed. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral exam
MAT/03 - GEOMETRY - University credits: 6
Practicals: 36 hours
Lessons: 27 hours
Lessons: 27 hours
Professors:
Garbagnati Alice, Tortora Alfonso
Shifts:
Educational website(s)
Professor(s)
Reception:
Fix an appointment by email
Dipartimento di Matematica "F. Enriques" - Room 2046
Reception:
Appointment via email.
Dipartimento di Matematica "F. Enriques" - Ufficio 0.007