#
Geometry 1

A.Y. 2019/2020

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
(In case of multiple editions, please check the period, as it may vary)

**Assessment methods:** Esame

**Assessment result:** voto verbalizzato in trentesimi

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.

Determinant: definitions and first properties of multilinear and alternating applications; existence and unicity theorem; Laplace theorem; inverse matrix; Cramer and Kronecker theorems.

Eigentheory: endomorphisms of vector space; matrix similarity; matrix diagonalization; eigenvalues and eigenvectors eigenspaces; characteristic polynomial.

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.

Determinant: definitions and first properties of multilinear and alternating applications; existence and unicity theorem; Laplace theorem; inverse matrix; Cramer and Kronecker theorems.

Eigentheory: endomorphisms of vector space; matrix similarity; matrix diagonalization; eigenvalues and eigenvectors eigenspaces; characteristic polynomial.

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 three parts: a written exam, an oral exam.

- During the written exam, the student must solve some exercises in the format of open-ended 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 a midterm exam. If succesfully 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 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 a midterm exam. If succesfully 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: 33 hours

Lessons: 27 hours

Lessons: 27 hours

Professors:
Colombo Elisabetta, Tasin Luca

### 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.

Determinant: definitions and first properties of multilinear and alternating applications; existence and unicity theorem; Laplace theorem; inverse matrix; Cramer and Kronecker theorems.

Eigentheory: endomorphisms of vector space; matrix similarity; matrix diagonalization; eigenvalues and eigenvectors eigenspaces; characteristic polynomial.

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.

Determinant: definitions and first properties of multilinear and alternating applications; existence and unicity theorem; Laplace theorem; inverse matrix; Cramer and Kronecker theorems.

Eigentheory: endomorphisms of vector space; matrix similarity; matrix diagonalization; eigenvalues and eigenvectors eigenspaces; characteristic polynomial.

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 three parts: a written exam, an oral exam.

- During the written exam, the student must solve some exercises in the format of open-ended 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 a midterm exam. If succesfully 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 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 a midterm exam. If succesfully 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: 33 hours

Lessons: 27 hours

Lessons: 27 hours

Professors:
Garbagnati Alice, Tortora Alfonso

Professor(s)

Reception:

friday.8.45-11.45

Office2101, second floor, via C. Saldini 50