Università Roma Tre

1- Linear systems: matrix associated to a linear system; sum of matrices and multiplication by real numbers; reduced matrices; Gauss-Jordan algorithm.
2- Rows by columns product of matrices; invertible matrices; rank of a matrix; Rouche'-Capelli Theorem.
3- Geometrical vectors. Vector spaces. Subspaces. Generating vectors and linearly independent vectors.
4- Basis of a vector space: dimension of a vector space; Grassmann's formula.
5- Linear applications: Kernel and image of a inear application. Dimension of Kernel and Image of a linear application.
6- Matrix associated to a linear application. Diagonalization of linear operators.

F. Flamini; A. Verra: "Matrici e vettori -Corso di base di geometria e algebra lineare" Carocci ed.

Systems and linear systems of equations. Matrices with real elements. Properties and operations of matrices. Determinant of a square matrix and its properties. Invertible square matrices and their inverse matrix. Theorem of Laplace (only enunciated and applied). Rule of Sarrus (only enunciated and applied). Binet's Theorem (only enunciated and applied). Singular and non singular matrices. Linear dependence and linear independence of the columns (rows) of a matrix. The rank of a matrix. Cramer's theorem (only enunciated and applied). Kronecker's theorem (only enunciated and applied). Rouchè-Capelli's theorem (only enunciated and applied). How to solve linear systems.
Real vector spaces. Key examples. Linear combinations of vectors. Generators of a vector space. Linear dependence and linear independence of vectors. Basis and dimension of a vector space. Coordinates of a vector in a base. The theorem of completing to a base. Reduction in R^n. Change of basis and coordinate transformations. Orientation of R^n. Subspaces of R^n: bases, dimension, parametric equations, codimension, Cartesian equations. Intersection and sum of subspaces. Grassmann's formula (only enunciated and applied). Direct sums of subspaces. Standard scalar product in R^n and its properties. Norm of a vector. Cauchy-Schwarz inequality (only enunciated and applied). Angular measurements. Area of a parallelogram. Projection of a vector onto another. Orthogonal bases and orthonormal bases of space. Gram-Schmidt process (only enunciated and applied). Orthogonal complement of a subspace. Orthogonal projection of a vector onto a subspace. Changes of orthonormal bases. Orthogonal matrices. Definition and examples of linear transformation. Matrix of a linear transformation with respect to two fixed bases. Kernel and image of a linear transformation. Rank nullity theorem (only enunciated and applied). Linear injective, surjective, bijettive transformations. Isomorphisms and invertible matrices. Endomorphisms or operators of R^n. Operators and basic changes: similar matrices. Diagonalizable matrices and operators. Eigenvectors and eigenvalues of an operator. Eigenspaces. Spectrum of an operator. Characteristic polynomial and characteristic equation. How to compute eigenvalues and eigenvectors. Algebraic multiplicity and geometric multiplicity of an eigenvalue. The fundamental theorem on diagonalizability. Transpose of an operator. Symmetric and antisymmetric operators. The spectral theorem (only enunciated and applied). Orthogonal operators. Isometries and orthogonal matrices.

− M. Bordoni, Algebra Lineare, Progetto Leonardo, Bologna
− P. Maroscia, Introduzione alla Geometria e all’Algebra Lineare, Zanichelli