Marco Abate, Algebra lineare, McGraw-Hill.
Some notes will be prepared by the professor on some parts of the course.
Learning Objectives
To get acquainted with real linear algebra.
To develop skills in logic deductive reasoning. To express mathematical
thoughts rigorously.
Prerequisites
Elementary algebraic manipulation on number fields. Basic euclidean
geometric notions for teh plane.
Teaching Methods
Lessons and exercise lessons. Homeworks for self-assesment.
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 exam 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 students which receive a sufficient vote in the written part in January (June) session can apply for the oral in February (July).
The homeworks are intended as a way to follow properly the course. If
the exam is passed, having done with good result the homeworks is
computed as an increase of 1 or 2 on the final vote. The written part of the exams contains a teoric question to which it is compulsory to answer correctly in order to be admitted to the oral part of the exam.
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. Norm of a vector. Cauchy-
Schwarz inequality. 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.