* 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.