Università Roma Tre

I modulo

Basics of set theory. Maps between sets.
Basics concepts of logic.
Combinatorics : binomial coefficients, the triangle of Pascal - Tartaglia .
Integers: divisibility, congruences. Numbers in different bases .
Algebraic structures: groups, rings and fields . Polynomials , congruences between polynomials. Finite fields .
Ordered structures: lattices and Boolean algebras .
Elements of graph theory .

I modulo:
Procesi-Rota, Elementi di Algebra e Matematica Discreta, Accademica
Piacentini Cattaneo, Matematica Discreta, Zanichelli

II modulo:
Nicholson, Algebra lineare, McGraw Hill
Procesi-Rota Lezioni di Algebra Lineare e Geometria, Accademica

Sets. Operations between sets. Applications between sets. Permutations. Relations. Equivalence relations. Order relations. Lattices. Algebraic structures. Group of permutations of n elements. Vector spaces. Linear dependence and independence of vectors. Basis. Dimension. Subspaces. Subspace sum. Subspace intersection. Grassmann’s relation. Matrices. Determinants. Inverse matrix. Reduction of a matrix into treads. The rank of a matrix. Linear systems. Theorem of Rouche-Capelli. Cramer's theorem. Method of Gaussian elimination.

F.J. Leon Trujillo, P. Mercuri. Elementi di Algebra Lineare, Ed. Efesto
F.J. Leon Trujillo, P. Mercuri. Elementi di Geometria Affine ed Euclidea, Ed. Efesto
F.J. Leon Trujillo. Esercizi di Algebra Lineare e Geometria, Ed. Efesto

II modulo
Matrices. Addition and scalar multiplication. Diagonal matrices, triangular matrices. Transpose. Symmetric and antisymmetric matrices. Matrix multiplication. Identity matrix. Properties of the product.
Linear systems and matrix multiplication. Homogeneous system associated with a linear system. Elementary operations on rows, the reduction step. Gauss elimination. Rank of a matrix as the number of pivot. Rouché-Capelli theorem.
Echelon form. Inverse of a square matrix, inversion algorithm. the 2 × 2 case. The vector space Rn. Linear combinations. Subspaces of Rn. The subspace Null (A). The subspace spanned by a set of vectors. Linear dependence and independence, relationship with matrix rank. Basis and dimension of a vector space.
Linear maps. Kernel and image. Injections and surjections. Matrix associated to a linear map. Nullity + rank.
Determinants, properties of determinants. Adjunct matrix. Another method to calculate the inverse. Rank of a matrix. Cramer theorem.
Eigenvalues, eigenvectors, eigenspaces. Diagonalization

I modulo:
Procesi-Rota, Elementi di Algebra e Matematica Discreta, Accademica
Piacentini Cattaneo, Matematica Discreta, Zanichelli

II modulo:
Nicholson, Algebra lineare, McGraw Hill
Procesi-Rota Lezioni di Algebra Lineare e Geometria, Accademica

Sets. Operations between sets. Applications between sets. Permutations. Relations. Equivalence relations. Order relations. Lattices. Algebraic structures. Group of permutations of n elements. Vector spaces. Linear dependence and independence of vectors. Basis. Dimension. Subspaces. Subspace sum. Subspace intersection. Grassmann’s relation. Matrices. Determinants. Inverse matrix. Reduction of a matrix into treads. The rank of a matrix. Linear systems. Theorem of Rouche-Capelli. Cramer's theorem. Method of Gaussian elimination.

F.J. Leon Trujillo, P. Mercuri. Elementi di Algebra Lineare, Ed. Efesto
F.J. Leon Trujillo, P. Mercuri. Elementi di Geometria Affine ed Euclidea, Ed. Efesto
F.J. Leon Trujillo. Esercizi di Algebra Lineare e Geometria, Ed. Efesto