58414 - Algebra and Geometry

Course Unit Page

Academic Year 2018/2019

Learning outcomes

At the end of the course the student has the essential elements of linear algebra and elementary geometry.

Course contents

Introduction to linear systems. Matrices. Gauss algorithm. Solution of (parametric) linear systems. R-vector spaces:definition and examples. The vector space R^n; the vector space of the mx n matrices with real entries. Vector subspaces. Examples and counterexamples. The vector space R[x] of the polynomials in a real variable. Linear combinations and generators of a vector space. Finitely generated vector spaces: examples and counterexamples. Intersection, union and sum of subspaces. Grassmann formula. Linear independence. Basis of a vector space. Existence of a base of a finitely generated vector space. Dimension of a vector space. Coordinates of a vector with respect to a base. Direct sum of vector subspaces. Linear applications between vector spaces: definition, examples and counterexamples. Construction of linear applications, conditions of existence and / or uniqueness. Study of a linear application: kernel and image. Injectivity and surjectivity. Preimage of a vector by a linear application. Linear varieties. Matrices associated to a linear application. Rank of a matrix. Rouche'-Capelli Theorem. Product of matrices, composition of linear applications. Invertible matrices and calculation of the inverse of a matrix. Change of bases. Conjugated matrices. The determinant and its properties. Eigenvalues and eigenvectors of an endomorphism. Autospaces and their properties. Characteristic polynomial. Algebraic and geometric multiplicity of an eigenvalue and relation between them. Diagonalizable matrices: definition, examples, counterexamples. Diagonalizzability of a matrix on R: necessary and sufficient conditions. Study of the diagonalisability of a matrix (dependent on parameters). Elements of affine geometry: affine subvarieties. Lines and planes in R3, their parametric and Cartesian representation. The scalar product in Rn. Orthogonal bases. Gram-Schmidt ortonormalization procedure. Orthogonal complement of a subspace. Decomposition of Rn into the direct sum of a subspace and its orthogonal complement. Metric on subvarieties. Distance between affine subvarieties.

Readings/Bibliography

Notes of the course will be available online.

Students can also refer to the following books:

1) M. Barnabei, F. Bonetti: Sistemi lineari e Matrici (Pitagora Editrice, Bologna, 1992) ; Spazi Vettoriali e Trasformazioni Lineari (Pitagora Editrice, Bologna,1993)

2) M. Abate: Algebra Lineare ( McGraw-Hill, 2000).

3) P. Maroscia: Introduzione alla geometria e all'algebra lineare (Zanichelli, 2000)

4) M. Abate, C. de Fabritiis: Esercizi di Geometria (McGraw-Hill, 1999).

Teaching methods

The course consists of 60 hours of frontal teaching during which the topics will be presented through examples, counterexamples and numerous exercises. The solution of exercises of various difficulty levels will be explained to the students and they will be given exercises to be solved independently. The correction of these exercises will be done later in the class. Some hours will be devoted to the discussion of the students' questions: students will be invited to express any doubt in the class and the resolution of these doubts will be discussed collegially. The structure of a demonstration will be explained to the students through some relevant theorems, albeit with elementary content.

Assessment methods

The exam consists of a written test and an oral exam. Only those who have passed the written test can take the oral exam. The written test is divided into two parts. The first part contains two elementary statements. The student must determine whether these statements are true or not and briefly explain why. The second part of the test will be corrected only if the first part is completely correct. The second part of the task consists of three exercises. The resolution of the task will be made available online on the teacher's webpage.

Teaching tools

All lessons will be held on the blackboard.

Office hours

See the website of Nicoletta Cantarini

See the website of Fabrizio Caselli