Subject: LINEAR ALGEBRA (A.A. 2020/2021)
Unit Algebra lineare
Basic Mathematics (lesson)
Students should learn some basic concepts of Linear Algebra and should be able to apply them in a logically consistent manner to the solution of standard problems. The capability of autonomously recognizing the most appropriate solving techniques is desirable.
The prerequisites are the knowledge of High School Mathematics, with focus on the following topics. Set operations. The sets of natural numbers, integer numbers, rational numbers, real numbers, complex numbers and their main properties. Polynomial algebra. Algebraic equations and inequalities. Powers, n-th roots and logarithms. Trigonometric functions.
Matrices over a given field of numbers. The ring of square matrices. Echelon matrices and elementary row operations. The determinant of a square matrix. Square submatrices and cofactors. The theorem of Laplace. Invertible matrices.
Vector spaces and subspaces. Linear combinations. Generators. Linear dependence. Bases. Dimension. Sum and intersection of vector subspaces. Direct sums. Grassmann's formula. Linear maps. Kernel and image. A dimensional equation. Isomorphisms. The matrixs associated to a linear map. Change of bases.
The rank of a matrix. Systems of linear equations. The theorems of Rouche`-Capelli and Cramer. Methods for solving a system of linear equations, The method of Gauss-Jordan. Parametric and cartesian representations of vector subspaces.
Linear operators, eigenvalues and eigenspaces. Matrix similarity. The characteristic polynomial. Algebraic and geometric multiplicity. Diagonalization of linear operators. The spectral theorem.
The course treats the basic notions of Linear Algebra and consists in formal lectures covering the theoretical contents and exercise sessions in which the presented techniques are applied to the resolution of exercise/problems of various kinds.
Course assessment consists of a written test involving the resolution of standard exercises, followed by an oral interview on the course topics. The evolution of the sanitary emergency might possibly involve the replacement of the written assessment by a computer based test
- Knowledge and understanding:
at the end of the course a student should have learned the basic features of Linear Algebra.
- Applying knowledge and understanding:
at the end of the course a student should be able to apply this knowledge to standard problems of Linear Algebra.
- Making judgements:
at the end of the course a student should be able to recognize independently some approaches and solving techniques which are typical of Linear Algebra
- Communicating skills:
at the end of the course a student should be able to describe topics in Linear Algebra with an appropriate technical language and a correct mathematical formalism
- Learning skills:
studying the subject matter should stimulate independent learning skills and the capability of treating connected topics in further detail
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S. LIPSCHUTZ, M. LIPSON, Algebra lineare – Terza edizione riveduta e corretta (collana Schaum's), McGraw-Hill Education, Milano, 2003. ISBN: 978-88-3865-076-5
E. SERNESI, Geometria 1 Seconda Edizione, Bollati Boringhieri, Torino, 2000. ISBN: 978-88-339-5447-9
C. BIGNARDI, B. RUINI, F. SPAGGIARI, Esercizi di Algebra Lineare, Pitagora Editrice Bologna, 1996.
PAOLO MAROSCIA, Geometria e algebra lineare, Zanichelli, 2002.