ALGEBRA
Set theory: correspondences and functions. 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: the vector space of matrices. Product, rank and determinant. Linear systems and their risolubility: Theorems of Rouche-Capelli and of Cramer. Eigenvalues and eigenvectors of a matrix. Diagonalizability and criteria.
Bilinear and quadratic forms: real and symmetric matrices. Orthogonal and orthonormal bases. Process of Gram-Schmidt orthogonalization. Orthogonal matrices. Orthogonally diagonalizable matrices.
GEOMETRY
Affine spaces: definitions, translations, subspaces, parallelism. Coordinatization of an affine space and analytic geometry in the plane and in three-dimensional space.
Euclidean spaces: distance, ortonogality, circles, spheres, surfaces of revolution and fundamental loci.
Projective spaces: projective extension of an affine geometry, projective subspaces, homogeneous coordinates and representation in homogeneous coordinates of the subspaces. Complexification.
Real algebraic curves and surfaces: an order of a curve, simple and singular points. Conic, projective classification, polarity, affine classification, canonical forms.
Quadrics: affine classification, cones and cylinders, flat sections.