Module page
Algebra lineare
| academic year: | 2013/2014 |
| instructor: | Paolo Piccinni |
| degree course: | Mathematics - DM 270/04 (triennale), I year |
| type of training activity: | di base |
| credits: | 9 (72 class hours) |
| scientific sector: | MAT/03 Geometria |
| teaching language: | italiano |
| program: | I-Z |
| period: | I sem (30/09/2013 - 17/01/2014) |
Lecture meeting time and location
Presence: highly recommended
Module subject:
- Preliminari: Insiemi,funzioni,relazioni,induzione matematica, campi(in particolare il campo reale e complesso), polinomi.
- Spazi vettoriali: Gli archetipi e la definizione. Proprietà algebriche elementari. Basi e dimensione.
- Geometria Affine I : Spazi affini, coordinate affini e baricentriche.
- Applicazioni lineari e matrici: Applicazioni lineari, isomorfismi. Matrici, applicazione lineare associata a una matrice.Operazioni elementari sulle matrici. Il duale di uno spazio vettoriale. Cambiamenti di base e coniugio.
- Geometria Affine II : Applicazioni affini. Cambiamenti di coordinate affini. equazioni cartesiane.
- Determinati: Definizione. Applicazioni multilineari alternati. Determinante e volume. Formula di Cramer.
- Forme quadratiche e forme bilineari simmetriche: Dalle forme quadratiche alle forme bilineari simmetriche e viceversa. Diagonalizzazione e congruenza. Segnatura di forme quadratiche reali. Coniche e quadriche Prodotti scalari definiti positivi, spazi euclidei.
- Endomorfismi e loro forma normale: Autovalori, autovettori e autospazi. Polinomio caratteristico. Diagonalizzabilità di endomorfismi. Teorema spettrale.
Type of course: standard
Knowledge and understanding:
Succesful students will be able to understand: the concept of vector spaces and subspaces; the concept of basis and dimension of a vector space; the concept of linear map and of matrices associated to a linear map; the concept of conjugate matrices and of determinant; the concept of eigenvalues and eigenvectors of an endomorphism; the concept of algebraic and geometric moltiplicity of an eigenvalue; the concept of diagonalizable endomorphism; the concept of affine coordinate system.
The student will be also able to explain the basic results concernig the above concepts.
Skills and attributes:
Successful students will be able to utilize: Gauss` elimination algorithm in order to solve systems of linear equations, in order to compute the rank of a matrice, in order to compute the inverse of a square (invertible) matrix, in order to select a basis from a system of generators, in order to compute the determinant of a square matrix; Laplace`s expansion in order to compute a determinant; determinants in order to compute the rank of a matrix, in order to compute the inverse of a square mmatrix, in order to solve systems of linear equations.
The student will be also able to apply criteria that allow to establish when a square matrix is diagonalizable, and to solve some elementary problems in analytic geometry.
Personal study: the percentage of personal study required by this course is the 65% of the total.