Università Roma Tre

1. Linear systems: matrix associated to a linear system; the sum of matrices and multiplication by real numbers; reduced matrices; Gauss-Jordan algorithm.
2. Rows by columns product of matrices; invertible matrices; the rank of a matrix; Rouché-Capelli Theorem.
3. Geometrical vectors. Vector spaces. Subspaces. Generating vectors and linearly independent vectors.
4. Basis of a vector space: the dimension of a vector space; Grassmann's formula.
5. Linear applications: Kernel and image of a linear application. Dimension of Kernel and Image of a linear application.
6. Matrix associated to a linear application. Diagonalization of linear operators.
7. Symmetric bilinear forms. Lengths, angles, orthogonality. Orthogonal and orthonormal bases. Gram-Schmidt algorithm.
8. Quadratic forms. Spectral Theorem. Diagonalization and classification of quadratic forms in a Euclidean space. Sylvester bases and Sylvester canonical form. Vector product in a Euclidean space of dimension three.
Analytic geometry in the plane and in space. Cartesian and parametric equations of linear spaces. Proper and improper sheaves of lines and planes. Determination of reciprocal position of linear varieties from their equations. Conics and quadrics.

Flamini-Verra "Matrici e vettori" Carocci ed.