28570 - Geometry and Algebra T-A

Learning outcomes

Knowledge of the fundamentals and methods of Linear Algebra and Analytic Geometry, both at theoretical and operational level

Course contents

The course is structured around the following topics:

1. Real and complex numbers. Vectors in n-dimensional space. Polynomials, factorization and roots.
2. Linear systems and matrices (I): sum and scalar multiplication in Euclidean space. Linear equations. Elementary row operations on a system of linear equations. Gauss algorithm. Matrix associated with a linear system.
3. Linear systems and matrices (II): definition of reduced row echelon matrix. Parametric form of the solutions of a linear system.
4. Matrices: sum of matrices and scalar multiplication. Row-by-column product and matrix multiplication. Powers of matrices. Invertible matrices. Gauss algorithm and computation of the inverse.
5. Vector spaces: definition and linear combinations. Examples: Euclidean space, matrices, polynomials. Vector subspaces: definition and properties. Sum and direct sum of vector subspaces.
6. Families of vectors, linearly independent families and generating families. Bases of a vector space. Dimension and coordinates.
7. Linear maps between vector spaces. Connection between linear maps and matrices. Image and kernel. Rank theorem. Injectivity and surjectivity of linear maps.
8. Matrix of a linear map with respect to a fixed basis. Change of basis matrix.
9. Determinants: definition, properties and computation.
10. Eigenvalues and eigenvectors of linear maps and matrices. Definition of diagonalizable matrix. Eigenspaces.

Readings/Bibliography

Recommended textbook and materials:

• Rita Fioresi, Marta Morigi, Introduzione all'algebra lineare. Zanichelli
• Francesco Bottacin, Algebra Lineare e Geometria. Società Editrice Esculapio

Lecture notes are the main reference for the course. They will be available on the web page of the course on "Virtuale", together with exercise sheets and past exams.

Teaching methods

The course consists of 60 hours of in-person lectures, during which the topics will be presented through examples, counterexamples, and exercises. Students will be shown the solutions to exercises of various levels of difficulty and will be assigned exercises to solve independently. The structure of a mathematical proof will be explained to the students through some significant theorems, although they are of elementary content.

Assessment methods

It is recommended (but not mandatory) to:

• Attend lectures regularly,
• Complete the assigned exercises,
• Solve many additional exercises from the textbooks,
• Take advantage of office hours to clarify any doubts.

The exam consists of a written test divided into two parts. In the first part, the student must answer 4 simple questions, each worth 2 points. Only the tests in which the student correctly answers at least 3 questionswill be evaluated. In the second part, the student must solve 4 or 5 structured exercises covering the topics covered during the course.

Finally, an optional oral exam is available for students who have obtained a score of at least 27 in the written test. The final grade for those who take the oral exam may increase or decrease compared to the written test, even significantly.

Office hours

See the website of Luca Marchese