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Geometry 1

A.Y. 2020/2021

Learning objectives

The course aims to provide students with some knowledge and skills in linear algebra. Starting from the notion of finite dimensional vector space on any field, we arrive at solving the systems of linear equations with the Gauss-Jordan method. Another goal is to study linear and bilinear applications, illustrating the notion of a representative matrix and the related problems of diagonalization. The bilinear applications are used to investigate Euclidean vector spaces (real and complex) and self-adjoint operators, for which the spectral theorem is fully proved.

Expected learning outcomes

At the end of the course, students will have acquired the following skills:

1. they will be able to solve systems of linear equations;

2. they will be able to apply the theory of finite dimensional vector spaces, recognizing vector subspaces and determining their bases;

3. they will be able to study linear applications, determining the representative matrix, the kernel and the image;

4. they will be able to apply some aspects of the theory of diagonalization of endomorphisms and matrices, based on the search for eigenvalues and eigenvectors;

5. they will know how to work in spaces with a positive definite inner product (also called Euclidean spaces) and apply elementary notions of Euclidean geometry;

6. they will know how to recognize self-adjoint operators and will be able to diagonalize them, determining an orthonormal basis of eigenvectors by means of the spectral theorem (real and complex).

1. they will be able to solve systems of linear equations;

2. they will be able to apply the theory of finite dimensional vector spaces, recognizing vector subspaces and determining their bases;

3. they will be able to study linear applications, determining the representative matrix, the kernel and the image;

4. they will be able to apply some aspects of the theory of diagonalization of endomorphisms and matrices, based on the search for eigenvalues and eigenvectors;

5. they will know how to work in spaces with a positive definite inner product (also called Euclidean spaces) and apply elementary notions of Euclidean geometry;

6. they will know how to recognize self-adjoint operators and will be able to diagonalize them, determining an orthonormal basis of eigenvectors by means of the spectral theorem (real and complex).

**Lesson period:**
Second semester

**Assessment methods:** Esame

**Assessment result:** voto verbalizzato in trentesimi

Course syllabus and organization

### CORSO A

Responsible

Lesson period

Second semester

**Course syllabus**

GEOMETRY 1

The course covers the basics of linear algebra:

-- definition of vector spaces and subspaces (over the real numbers and the complex numbers);

-- notion of basis and dimension of a vector space. The Grassmann formula;

-- linear maps and associated matrices. The rank of a matrix;

-- linear systems. Structure of the space of solutions. The Gauss--Jordan method;

-- determinant. Invertible matrices;

-- characteristic polynomial, eigenspaces and eigenvectors of an endomorphism;

-- scalar and hermitian products, orthogonal bases, the Gram--Schmidt method;

-- notions of angle and length via scalar products;

-- the spectral theorem (over the reals and the complex numbers);

-- bilinear forms. The theorems of Sylvester and L

The course covers the basics of linear algebra:

-- definition of vector spaces and subspaces (over the real numbers and the complex numbers);

-- notion of basis and dimension of a vector space. The Grassmann formula;

-- linear maps and associated matrices. The rank of a matrix;

-- linear systems. Structure of the space of solutions. The Gauss--Jordan method;

-- determinant. Invertible matrices;

-- characteristic polynomial, eigenspaces and eigenvectors of an endomorphism;

-- scalar and hermitian products, orthogonal bases, the Gram--Schmidt method;

-- notions of angle and length via scalar products;

-- the spectral theorem (over the reals and the complex numbers);

-- bilinear forms. The theorems of Sylvester and L

**Prerequisites for admission**

The basic mathematics knowledge usually taught in secondary school

**Teaching methods**

Traditional: lessons and exercise classes.

Tutoring: 2 hours a week

Tutoring: 2 hours a week

**Teaching Resources**

1) Lecture notes are available on the teacher's web page.

2) Serge Lang - Algebra lineare - Bollati Boringhieri

2) Serge Lang - Algebra lineare - Bollati Boringhieri

**Assessment methods and Criteria**

The final examination consists of two parts: a written exam and an oral exam.

During the written exam, the student must solve some exercises in the format of open-ended questions,. 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 .

In place of a single written exam given during the first examination session, the student may choose instead to take two midterm exams.

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 two parts (written, 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 questions,. 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 .

In place of a single written exam given during the first examination session, the student may choose instead to take two midterm exams.

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 two parts (written, 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: 7

Practicals: 24 hours

Lessons: 40 hours

Lessons: 40 hours

Professors:
Bertolini Marina, Mastrolia Paolo

### CORSO B

Responsible

Lesson period

Second semester

**Course syllabus**

GEOMETRY 1

The course covers the basics of linear algebra:

-- definition of vector spaces and subspaces (over the real numbers and the complex numbers);

-- notion of basis and dimension of a vector space. The Grassmann formula;

-- linear maps and associated matrices. The rank of a matrix;

-- linear systems. Structure of the space of solutions. The Gauss--Jordan method;

-- determinant. Invertible matrices;

-- characteristic polynomial, eigenspaces and eigenvectors of an endomorphism;

-- scalar and hermitian products, orthogonal bases, the Gram--Schmidt method;

-- notions of angle and length via scalar products;

-- the spectral theorem (over the reals and the complex numbers);

-- bilinear forms. The theorems of Sylvester and Lagrange.

The course covers the basics of linear algebra:

-- definition of vector spaces and subspaces (over the real numbers and the complex numbers);

-- notion of basis and dimension of a vector space. The Grassmann formula;

-- linear maps and associated matrices. The rank of a matrix;

-- linear systems. Structure of the space of solutions. The Gauss--Jordan method;

-- determinant. Invertible matrices;

-- characteristic polynomial, eigenspaces and eigenvectors of an endomorphism;

-- scalar and hermitian products, orthogonal bases, the Gram--Schmidt method;

-- notions of angle and length via scalar products;

-- the spectral theorem (over the reals and the complex numbers);

-- bilinear forms. The theorems of Sylvester and Lagrange.

**Prerequisites for admission**

The basic mathematics knowledge usually taught in secondary school

**Teaching methods**

Traditional: lessons and exercise classes.

Tutoring: 2 hours a week

Tutoring: 2 hours a week

**Teaching Resources**

- Lecture notes are available on the teacher's web page

- S. Lang _ Linear Algebra - Springer -1966

- S. Lang _ Linear Algebra - Springer -1966

**Assessment methods and Criteria**

The final examination consists of two parts: a written exam and an oral exam.

During the written exam, the student must solve some exercises in the format of open-ended questions,. 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 .

In place of a single written exam given during the first examination session, the student may choose instead to take two midterm exams.

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 two parts (written, 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 questions,. 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 .

In place of a single written exam given during the first examination session, the student may choose instead to take two midterm exams.

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 two parts (written, 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: 7

Practicals: 24 hours

Lessons: 40 hours

Lessons: 40 hours

Professor(s)

Reception:

by appointment (by e-mail)

Math. Dept. - via C. Saldini 50 - Milano