Portale di Ateneo - En.Unibs.it

* Naive Set theory

Correspondences and maps. Relations on a set. Definition of the main algebraic structures.

* Vector Spaces

Definition of a vector space. Linear dependence and independence. Generators. Finitely generated vector spaces: Steinitz Lemma, bases and dimension of a vector space. Subspaces of a vector space. Intersection, sum and direct sum of subspaces. Grassmann formula.

* Matrices and linear systems

Vector space of matrices. Product of matrices. Rank and determinant. Linear systems and their solvability: Rouche-Capelli and Cramer theorems. Eigenvalues ​​and eigenvectors of a matrix. Diagonalization.

* Scalar products and quadratic forms

Bilinear forms and scalar products. Orthogonal and orthonormal bases. Gram-Schmidt orthogonalization. Orthogonal matrices. Orthogonally diagonalizable matrices.

* Affine and Euclidean spaces

Affine spaces: definitions, translations, subspaces, parallelism. Coordinatization of an affine space and analytic geometry in the plane and in three-dimensional space. Euclidean spaces: distances, orthogonality, circles, spheres, surfaces of revolution and fundamental loci.

* Projective spaces

Projective embedding of an affine geometry: projective subspaces, homogeneous coordinates and representation in homogeneous coordinates of the subspaces. Complexification.

* Real algebraic curves and surfaces

Order of a curve, simple and singular points. Conic, projective classification, polarity, affine and metric classification, canonical forms. Quadrics: affine classification, cones and cylinders, plane section investigation.