Geometry 1
A.Y. 2026/2027
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.
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
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Group 1
Responsible
Lesson period
Second semester
Course syllabus
The course covers the basic concepts of linear algebra:
1. Systems of linear equations: solvability using the Gauss-Jordan method and the structure of solution spaces.
2. Definition of vector spaces and subspaces (over the real and complex numbers);
3. Bases and dimension of a vector space; Grassmann's formula.
4. Linear mappings and matrices; rank of a matrix.
5. Operations on matrices; the determinant of a matrix.
6. Endomorphisms of vector spaces; eigenvalues, eigenvectors, diagonalisation.
7. Bilinear maps.
8. Scalar products and Euclidean spaces.
9. The spectral theorem for the real numbers and an introduction to the complex spectral theorem.
1. Systems of linear equations: solvability using the Gauss-Jordan method and the structure of solution spaces.
2. Definition of vector spaces and subspaces (over the real and complex numbers);
3. Bases and dimension of a vector space; Grassmann's formula.
4. Linear mappings and matrices; rank of a matrix.
5. Operations on matrices; the determinant of a matrix.
6. Endomorphisms of vector spaces; eigenvalues, eigenvectors, diagonalisation.
7. Bilinear maps.
8. Scalar products and Euclidean spaces.
9. The spectral theorem for the real numbers and an introduction to the complex spectral theorem.
Prerequisites for admission
The basic concepts of Mathematics that are usually taught in secondary education.
Teaching methods
In-person lectures and exercise classes.
Tutoring (optional): 2 hours/week
Tutoring (optional): 2 hours/week
Teaching Resources
1) Lecture notes and other materials made available on MyAriel web site for the course.
2) Serge Lang, Linear Algebra.
3) Elsa Abbena, Anna M. Fino, G. Mario Gianella, Algebra lineare e geometria analitica, Vol. 1-2,
2) Serge Lang, Linear Algebra.
3) Elsa Abbena, Anna M. Fino, G. Mario Gianella, Algebra lineare e geometria analitica, Vol. 1-2,
Assessment methods and Criteria
The final exam consists of two parts: a written and an oral one.
The written part consists of closed and/or open questions. The duration
of the written exam will be proportional to the number of questions assigned, having also taken into account the nature and complexity of the exercises themselves.
Part of the written exam can be passed via a mid-term exam which will be offered mid-way thorugh the semester in which the course is taught.
Students passing the written part will be admitted to the oral part of the exam.
During the oral exam, the student will be requested to illustrate some of the results and/or examples presented in the course: the goal is to evaluate the student's knowledge and comprehension of the syllabus, as well as the ability to apply the results illustrated in teh course.
The final examination is passed if all two parts (written, oral) are successfully passed. Final marks are given on a scale from 0 to 30 with integral increments; the passing grade is 18. The final grade will be immediately communicated at the end of the oral examination.
The written part consists of closed and/or open questions. The duration
of the written exam will be proportional to the number of questions assigned, having also taken into account the nature and complexity of the exercises themselves.
Part of the written exam can be passed via a mid-term exam which will be offered mid-way thorugh the semester in which the course is taught.
Students passing the written part will be admitted to the oral part of the exam.
During the oral exam, the student will be requested to illustrate some of the results and/or examples presented in the course: the goal is to evaluate the student's knowledge and comprehension of the syllabus, as well as the ability to apply the results illustrated in teh course.
The final examination is passed if all two parts (written, oral) are successfully passed. Final marks are given on a scale from 0 to 30 with integral increments; the passing grade is 18. The final grade will be immediately communicated at the end of the oral examination.
MATH-02/B - Geometry - University credits: 7
Exercises: 24 hours
Lessons: 40 hours
Lessons: 40 hours
Professors:
Mari Luciano, Svaldi Roberto
Group 2
Responsible
Lesson period
Second semester
MATH-02/B - Geometry - University credits: 7
Exercises: 24 hours
Lessons: 40 hours
Lessons: 40 hours
Professors:
Colombo Elisabetta, Matessi Diego
Professor(s)
Reception:
Office2101, second floor, via C. Saldini 50
Reception:
Please contact me via email to fix an appointment
Math Department "Federigo Enriques"
Reception:
By appointment (to be agreed upon via email)
Room 2102, Dipartimento di Matematica "F. Enriques", Via Saldini 50