LINEAR ALGEBRA AND GEOMETRY M - Z

Skill development in the solution of problems through the use of linear algebra tools , linear analytic geometry, and polynomials of II degree in many variables . Ability of description and calculation of geometric objects defined by equations or inequalities.

Linear algebra.

Groups, fields rings. complex numbers. Polynomials with coefficients in a field. Matrices with coefficients in a field and their operations. Vector spaces. Generators, independent sets, bases. Dimension of vector spaces. Subspaces, generators of subspaces, intersections and sums, calculus of dimension. Direct sums. Linear systems. Gauss method for solving linear systems. Description of subspaces by generators or by means of equations with respect to a base. Linear maps, associated matrices with respect to bases. Rank of matrices. Kernel and image of linear applications, theorem of dimensions and their calculation. Determinants and their calculation. Calculating the rank of a matrix by its minors. Endomorphisms of a vector space, eigenvalues, characteristic polynomial of an endomorphism, eigenvectors, eigenspaces and their calculation. Diagonalizable endomorphisms. Criterion for diagonalizability. Bilinear forms on a vector space. Scalar products and hermitian products of complex vector spaces. Schwarz inequality and distance associated with a scalar product defined positive. Orthogonal bases. Gram Schmidt method for calculating orthonormal bases in the positive definite case. Orthogonal projections. Quadratic forms and their associated scalar products. Spectral theorem for self-adjoint endomorphisms with respect to a positive defined scalar product.

Geometry.

Geometric vectors, standard scalar product and cross product in dimension 3. Distances and angles between vectors. Lines in the plane and their equations. pencils of lines. Lines and planes in space and their equations. Pencils of planes. Loci defined by equations or inequalities. Calculations of angles, distances, areas and volumes for some geometric figures. Rigid transformations of the plane and space and their matrix formulas. Orthogonal projections and reflections. Homogeneous coordinates in the plane and in space, proper and improper points. Real conics in the plane and their classification, equations in reduced form. Tangents to conic and polar lines. Vertices. symmetry centers and symmetry axes. Conics as geometrical loci, foci and directrices. Method of pencils of conics for the solution of some problems. Quadrics in space. Cones, cylinders, spheres. Tangent planes to a quadric. Outline of the classification of real quadrics. Conics in 3 dimensional space.

1. S. Giuffrida, A. Ragusa, Lezioni di Algebra Lineare, Il Cigno.

2. G. Paxia, Lezioni di Geometria, Spazio Libri.