- Docente: Roberto Pagaria
- Credits: 9
- Language: Italian
- Moduli: Roberto Pagaria (Modulo 1) Nicoletta Cantarini (Modulo 2)
- Teaching Mode: Traditional lectures (Modulo 1) Traditional lectures (Modulo 2)
- Campus: Bologna
- Corso: First cycle degree programme (L) in Mathematics (cod. 6061)
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from Sep 16, 2024 to Dec 18, 2024
Learning outcomes
At the end of the course students will acquire knowledge of the first concepts of linear algebra such as vector spaces and linear applications. They will be able to apply this knowledge to state and prove geometric results with a rigorous language.
Course contents
- Rings and fields: definitions and examples.
- Zero divisors. A field has no zero divisors. The ring Z/n. Z/n is a field if and only if n is prime.
- K-vector spaces: definition and discussion of some properties. The vector space K^n. The vector space of mxn matrices.
- The space of polynomials in a coefficient variable in a field.
- Vector subspaces: definition, examples, counterexamples.
- Intersection and sum of subspaces. Generated subspace.
- Finitely generated vector spaces. Examples and observations on generator systems.
- Exchange theorem and its main consequences. Definition of base and size.
- Set of generators for the sum of two subspaces.
- Grassmann formula
- Direct sum. The transpose of a matrix. Symmetric and antisymmetric matrices.
- Linear applications: examples and first properties. Counterexamples. Linear maps from K to K. Core of a linear map and characterization of injectivity.
- Image of a linear application. Rank theorem and its main consequences.
- Construction of linear applications satisfying given conditions.
- Matrix associated with a linear application.
- Basis change matrix.
- Structure of the counterimage of a vector through a linear application.
- Homogeneous and non-homogeneous, parametric and non-parametric linear systems.
- Rouché-Capelli theorem.
- Product of matrices.
- Product properties of matrices and matrix composition of linear applications. Basic changes. Invertible matrices. Similar matrices: definition and first observations
- Hom(V,W) and isomorphism with the matrix space.
- Projections: characterizations and properties.
- Matrices associated with linear applications: reflections.
- Existence of right and left inverses of a linear map. Elementary operations on a matrix as basis changes in the codomain.
- Dual space. Dual base. Canceller of a subspace.
- The determinant of a matrix: definition and properties
- The determinant: proofs.
- Eigenvalues and eigenvectors of an endomorphism. Eigenspaces, geometric and algebraic multiplicity, invariant subspaces.
- Invariant by similarity. Diagonalizable endomorphisms.
- Eigenspaces relating to different eigenvalues are in direct sum.
- Diagonalizability theorem.
- Cyclic vectors and companion matrices. Existence of cyclic vectors.
- Quotient space.
- Linear applications induced on the quotient.
- Triangulation of an endomorphism.
Readings/Bibliography
M. Manetti. Algebra Lineare, per matematici
Teaching methods
Lectures on the blackboard with open discussions. Individual and collective meetings. Exercise sessions. Many exercises will be provided to the students and appointments will be organized for the correction and discussion of these exercises.
Assessment methods
The exam consists of a written test and an oral test. Students can access the oral test after passing the written test. The written test includes a basic level preliminary part which must be passed for the written test to be considered sufficient.
Teaching tools
A tutoring activity is foreseen.
Office hours
See the website of Roberto Pagaria
See the website of Nicoletta Cantarini