Main book: E. Rubei "Geometria e Algebra Lineare" Pearson
Another book which can be used in addition is
Artin "Algebra" Boringhieri
Learning Objectives
Knowledge and comprehension of fundamental theory of linear algebra and analytic geometry. Skill to apply theory to solve autonomously basic problems. Skill to present theory in oral form and to write mathematics correctly.
Prerequisites
a bit of logic. knowledge of the mathematical language, in particular about sets and functions; trigonometry
Teaching Methods
total number of hours: 300, 12 cfu (ects)
front hours:116 (theory+exercises)
Further information
Office hours: see
the web pages of the teachers and moodle page
Type of Assessment
Written test with questions about theory and exercises + oral test,
both to check the knowledge and comprehension of fundamental theory of linear algebra and analytic geometry, the skill to apply theory to solve autonomously basic problems. the skill to present theory in oral form and to write mathematics correctly.
Course program
Introduction to groups. Fields, in particular complex numbers field. Equivalence relations. Matrices. Linear systems, Gauss' method, structure theorem. Vector spaces, subspaces, generators, linear independence, bases, sum of subspaces, Grassmann's formula, linear maps, kernel and image, isomorphisms. matrices associated to linear maps, the space of the linear maps between two vector spaces; the dual space. Determinant and rank. Cramer's theorem. Rouché-Capelli theorem. Eigenvalues and eigenvectors, characteristic polynomial, algebraic multiplicity, geometric multiplicity, diagonalizability, necessary and sufficient criterion for diagonalizability; eigenvalues of polynomials of matrices. Bilinear forms, associated matrices, Gram-Schmidt theorems, signature; quadratic forms; Cauchy-Schwartz theorem triangular inequality (only the statements). Hermitian matrices, normal matrices, unitary matrices, hermitian forms. Spectral theorems. Vector product. Affine and metric geometry, parallelism, orthogonality, lines and planes in the space, conics. Exponential of a matrix.