Linear systems and matrices. Vectorial spaces. Symmetric matrices and spectral theorem. Orthogonality. Quadratic forms. Lines, planes and hyperplanes in R^n.
Some notes prepared by the professor on some parts of the content of the course will be available.
Learning Objectives
To get acquainted with real linear algebra with elements of geometry.
To develop skills in logic deductive reasoning. To express mathematical thoughts rigorously.
Prerequisites
Elementary algebraic manipulation on number fields. Basic euclidean geometric notions.
Teaching Methods
Lessons and exercise lessons. Homeworks for self-assesment.
Further information
The course has a moodle page on e-l. unifi
Type of Assessment
Two possibility for the exam: two intermediate written exams on two sections of the course and an oral exam; one written exam on the whole content of the course and an oral exam.
The written part of the exams is essentially about the techniques of linear algebra. In particular it is required to be expert in the use of the Gauss reduction algorithm to face the typical problems in linear algebra.
The oral part of the exam is about definitions and main theorems.
Some simple proofs are required.
The homeworks are intended as a way to follow properly the course. If the exam is passed, having done with good results the homeworks is computed as an increase of 1 or 2 on the final vote.
Course program
Sets. Functions and operations. Images and inverse images. Fields and vectorial spaces. Linear systems and Gauss algorithm. Martices. Homogeneous and not homogeneous systems. Linear independence and basis. Dimension of a vectorial space. Subspaces. Subspace generated by a set of vectors. Square, triangular, diagonal and symmetric matrices. Transpose of a matrix. Rank of a matrix. Scalar product of two vectors. Angle between two vectors. The geometric universe R^n.Parametric equation of a line. Equation of a hyperplane. Norm of a vector. Cauchy-Schwarz inequality. Parallelism and orthogonality. Parallel and othogonal lines and planes. Linear applications and matrices. Kernel and image of a linear application. Determinant. Rank of a matrix through its minors.Inverse of a matrix and its computation by Gauss algorithm and by minors. Eigenvalues and eigenvectors. Spectral theorem. Quadratic forms.