58414 - Algebra and Geometry

Learning outcomes

Students will learn the essential elements of linear algebra

Course contents

Linear systems.
Elementary operations on the rows of a matrix.
Gauss reduction method.
R-vector spaces: denition and examples.
The vector space R^n; the vector space of mxn matrices with real entries.
Subspaces. Examples and counterexamples.
The vector space R[x] of polynomials in one variable with real coefficients.
Linear combinations and generators.
Finitely generated vector spaces: examples and counterexamples.
Intersection, union and sum of subspaces.
Linear dependence and independence.
Bases of a vector space. Existence of a basis of a finitely
generated vector space.
Dimension of a vector space.
Coordinates with respect to a base.
Direct sum of subspaces.
Grassmann's formula and its applications.
Linear maps: denition, examples and counterexamples.
Conditions of existence and / or uniqueness.
Study of a linear transformation: kernel and image. Injectivity and surjectivity.
Matrix associated to a linear map.
Rank of a matrix.
Rouche-Capelli Theorem.
Product of matrices, composition of linear maps.
Invertible matrices. How to compute the inverse of a matrix.
Changes of bases.
Conjugate matrices.
The determinant and its properties. Binet theorem.
Eigenvalues and eigenvectors of an endomorphism.
Eigenspaces and their properties.
Characteristic polynomial.
Algebraic and geometric multiplicity of an eigenvalue and the relationship between them.
Diagonalizable matrices: denition, examples, counterexamples; necessary and sufficient conditions.
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

The notes of the course will be available on the teacher website

Teaching methods

Every lecture will be followed by many exercises. Students will be invited to ask questions and to answer the teacher's questions in classroom, and encouraged to practice at home. Meetings for the correction of the exercises will be organized. The teacher will meet students in small groups for discussions.

Assessment methods

The exam consists of a written test and an oral part. In order to access the oral exam students must pass the written exam. The written exam consists of three exercises. The score of each exercise is specified next to its text. Students must take the oral test right after passing the written one.

Teaching tools

The notes of the course as well as the texts of the previous exams will be available online.

Office hours

See the website of Nicoletta Cantarini

See the website of Fabrizio Caselli