\contentsline {section}{\numberline {1}Matrici}{1}{section.1}
\contentsline {subsection}{\numberline {1.1}Matrici}{1}{subsection.1.1}
\contentsline {subsection}{\numberline {1.2}Tipo di una matrice}{1}{subsection.1.2}
\contentsline {subsection}{\numberline {1.3}Matrice nulla (zero)}{1}{subsection.1.3}
\contentsline {subsection}{\numberline {1.4}Matrice identica (identita')}{1}{subsection.1.4}
\contentsline {subsubsection}{\numberline {1.4.1}Proprieta'}{1}{subsubsection.1.4.1}
\contentsline {subsection}{\numberline {1.5}Somma}{2}{subsection.1.5}
\contentsline {subsection}{\numberline {1.6}Somma diretta}{2}{subsection.1.6}
\contentsline {subsection}{\numberline {1.7}Prodotto}{2}{subsection.1.7}
\contentsline {subsubsection}{\numberline {1.7.1}Proprieta'}{2}{subsubsection.1.7.1}
\contentsline {subsection}{\numberline {1.8}Prodotto scalare}{3}{subsection.1.8}
\contentsline {subsubsection}{\numberline {1.8.1}Proprieta'}{3}{subsubsection.1.8.1}
\contentsline {subsection}{\numberline {1.9}Matrice diagonale}{3}{subsection.1.9}
\contentsline {subsubsection}{\numberline {1.9.1}Esempio}{4}{subsubsection.1.9.1}
\contentsline {subsection}{\numberline {1.10}Matrice scalare}{4}{subsection.1.10}
\contentsline {subsubsection}{\numberline {1.10.1}Esempio}{4}{subsubsection.1.10.1}
\contentsline {subsection}{\numberline {1.11}Trasposizione}{4}{subsection.1.11}
\contentsline {subsubsection}{\numberline {1.11.1}Esempio}{4}{subsubsection.1.11.1}
\contentsline {subsubsection}{\numberline {1.11.2}Proprieta'}{4}{subsubsection.1.11.2}
\contentsline {subsection}{\numberline {1.12}Matrici triangolari}{5}{subsection.1.12}
\contentsline {subsubsection}{\numberline {1.12.1}Inferiori}{5}{subsubsection.1.12.1}
\contentsline {subsubsection}{\numberline {1.12.2}Superiori}{5}{subsubsection.1.12.2}
\contentsline {subsubsection}{\numberline {1.12.3}Proprieta'}{5}{subsubsection.1.12.3}
\contentsline {subsection}{\numberline {1.13}Matrice simmetrica}{5}{subsection.1.13}
\contentsline {subsubsection}{\numberline {1.13.1}Esempio}{5}{subsubsection.1.13.1}
\contentsline {subsection}{\numberline {1.14}Matrice antisimmetrica}{6}{subsection.1.14}
\contentsline {subsection}{\numberline {1.15}Traccia di una matrice}{6}{subsection.1.15}
\contentsline {subsection}{\numberline {1.16}Permutazioni}{6}{subsection.1.16}
\contentsline {subsubsection}{\numberline {1.16.1}Gruppo delle permutazioni}{6}{subsubsection.1.16.1}
\contentsline {subsection}{\numberline {1.17}Trasposizione}{6}{subsection.1.17}
\contentsline {subsection}{\numberline {1.18}Proprieta'}{6}{subsection.1.18}
\contentsline {subsection}{\numberline {1.19}Parita' di una permutazione}{7}{subsection.1.19}
\contentsline {subsubsection}{\numberline {1.19.1}Proprieta'}{7}{subsubsection.1.19.1}
\contentsline {subsection}{\numberline {1.20}Prodotto dedotto}{7}{subsection.1.20}
\contentsline {subsubsection}{\numberline {1.20.1}Esempio}{7}{subsubsection.1.20.1}
\contentsline {subsection}{\numberline {1.21}Sgn}{7}{subsection.1.21}
\contentsline {subsection}{\numberline {1.22}Determinante}{7}{subsection.1.22}
\contentsline {subsubsection}{\numberline {1.22.1}Proprieta'}{7}{subsubsection.1.22.1}
\contentsline {subsection}{\numberline {1.23}Sistema lineare}{8}{subsection.1.23}
\contentsline {subsection}{\numberline {1.24}Sotto matrice (minore)}{9}{subsection.1.24}
\contentsline {subsubsection}{\numberline {1.24.1}Proprieta'}{9}{subsubsection.1.24.1}
\contentsline {subsubsection}{\numberline {1.24.2}Esempio}{9}{subsubsection.1.24.2}
\contentsline {subsection}{\numberline {1.25}Minore complementare}{9}{subsection.1.25}
\contentsline {subsection}{\numberline {1.26}Minore orlato}{9}{subsection.1.26}
\contentsline {subsection}{\numberline {1.27}Minore principale}{9}{subsection.1.27}
\contentsline {subsection}{\numberline {1.28}Complemento algebrico}{10}{subsection.1.28}
\contentsline {subsection}{\numberline {1.29}Teorema di Laplace I}{10}{subsection.1.29}
\contentsline {subsection}{\numberline {1.30}Teorema di Laplace II}{10}{subsection.1.30}
\contentsline {subsection}{\numberline {1.31}Teorema di Laplace generalizzato}{10}{subsection.1.31}
\contentsline {subsection}{\numberline {1.32}Caratteristica}{11}{subsection.1.32}
\contentsline {subsubsection}{\numberline {1.32.1}Proprieta'}{11}{subsubsection.1.32.1}
\contentsline {subsection}{\numberline {1.33}Rango di una matrice}{11}{subsection.1.33}
\contentsline {subsection}{\numberline {1.34}Matrice aggiunta}{11}{subsection.1.34}
\contentsline {subsubsection}{\numberline {1.34.1}Proprieta'}{12}{subsubsection.1.34.1}
\contentsline {subsection}{\numberline {1.35}Matrice inversa}{12}{subsection.1.35}
\contentsline {subsubsection}{\numberline {1.35.1}Proprieta'}{12}{subsubsection.1.35.1}
\contentsline {subsection}{\numberline {1.36}Uso di sistemi vettoriali}{13}{subsection.1.36}
\contentsline {subsection}{\numberline {1.37}Matrice invertibile}{14}{subsection.1.37}
\contentsline {subsubsection}{\numberline {1.37.1}Proprieta'}{15}{subsubsection.1.37.1}
\contentsline {section}{\numberline {2}Sistemi lineari}{15}{section.2}
\contentsline {subsection}{\numberline {2.1}Sistema lineare $m\times n$}{15}{subsection.2.1}
\contentsline {subsection}{\numberline {2.2}Cramer}{15}{subsection.2.2}
\contentsline {subsection}{\numberline {2.3}Regola di Cramer}{16}{subsection.2.3}
\contentsline {section}{\numberline {3}Spazi vettoriali}{17}{section.3}
\contentsline {subsection}{\numberline {3.1}Vettore nullo}{17}{subsection.3.1}
\contentsline {subsection}{\numberline {3.2}Legge di annullamento}{17}{subsection.3.2}
\contentsline {subsection}{\numberline {3.3}Vettore opposto}{18}{subsection.3.3}
\contentsline {subsection}{\numberline {3.4}Sottospazio vettoriale}{18}{subsection.3.4}
\contentsline {subsubsection}{\numberline {3.4.1}Proprieta}{18}{subsubsection.3.4.1}
\contentsline {subsubsection}{\numberline {3.4.2}Esempio}{18}{subsubsection.3.4.2}
\contentsline {subsection}{\numberline {3.5}Forma lineare}{18}{subsection.3.5}
\contentsline {subsubsection}{\numberline {3.5.1}Proprieta}{19}{subsubsection.3.5.1}
\contentsline {subsection}{\numberline {3.6}Somma di sottospazi vettoriali}{19}{subsection.3.6}
\contentsline {subsubsection}{\numberline {3.6.1}Proprieta}{19}{subsubsection.3.6.1}
\contentsline {subsection}{\numberline {3.7}Somma diretta sottospazi vettoriali}{19}{subsection.3.7}
\contentsline {subsubsection}{\numberline {3.7.1}Proprieta}{20}{subsubsection.3.7.1}
\contentsline {subsection}{\numberline {3.8}Alcuni sottospazi}{20}{subsection.3.8}
\contentsline {subsubsection}{\numberline {3.8.1}Matrici Sym e Asym}{20}{subsubsection.3.8.1}
\contentsline {subsubsection}{\numberline {3.8.2}Matrici a traccia nulla}{21}{subsubsection.3.8.2}
\contentsline {section}{\numberline {4}Dipendenza lineare}{22}{section.4}
\contentsline {subsection}{\numberline {4.1}Combinazione lineare}{22}{subsection.4.1}
\contentsline {subsection}{\numberline {4.2}Sottospazio generato}{22}{subsection.4.2}
\contentsline {subsubsection}{\numberline {4.2.1}Proprieta'}{22}{subsubsection.4.2.1}
\contentsline {subsection}{\numberline {4.3}Indipendenza lineare}{23}{subsection.4.3}
\contentsline {subsection}{\numberline {4.4}Criterio degli scarti successivi}{24}{subsection.4.4}
\contentsline {subsection}{\numberline {4.5}Base}{24}{subsection.4.5}
\contentsline {subsection}{\numberline {4.6}Insieme minimale di generatori}{25}{subsection.4.6}
\contentsline {subsubsection}{\numberline {4.6.1}Proprieta'}{25}{subsubsection.4.6.1}
\contentsline {subsection}{\numberline {4.7}Insieme massimale di generatori}{25}{subsection.4.7}
\contentsline {subsubsection}{\numberline {4.7.1}Componenti di un vettore}{25}{subsubsection.4.7.1}
\contentsline {subsection}{\numberline {4.8}Lemma di Steinitz}{26}{subsection.4.8}
\contentsline {subsubsection}{\numberline {4.8.1}Equicardinalita' delle basi}{26}{subsubsection.4.8.1}
\contentsline {subsection}{\numberline {4.9}Completemento della base}{26}{subsection.4.9}
\contentsline {subsection}{\numberline {4.10}Dimensione}{26}{subsection.4.10}
\contentsline {subsubsection}{\numberline {4.10.1}Proprieta'}{27}{subsubsection.4.10.1}
\contentsline {subsubsection}{\numberline {4.10.2}Altre proprieta'}{27}{subsubsection.4.10.2}
\contentsline {subsection}{\numberline {4.11}Componenti e basi}{28}{subsection.4.11}
\contentsline {subsection}{\numberline {4.12}Determinante e vettori l.i.}{28}{subsection.4.12}
\contentsline {subsection}{\numberline {4.13}Teorema di Kroneker I}{28}{subsection.4.13}
\contentsline {subsubsection}{\numberline {4.13.1}Corollario}{28}{subsubsection.4.13.1}
\contentsline {subsection}{\numberline {4.14}Equazioni cartesiane di un sottospazio}{29}{subsection.4.14}
\contentsline {subsubsection}{\numberline {4.14.1}Base dalle equazioni}{29}{subsubsection.4.14.1}
\contentsline {section}{\numberline {5}Sistemi lineari}{31}{section.5}
\contentsline {subsection}{\numberline {5.1}Rouche-Capelli I}{31}{subsection.5.1}
\contentsline {subsection}{\numberline {5.2}Rouche-Capelli II}{32}{subsection.5.2}
\contentsline {subsection}{\numberline {5.3}Corollario di Rouche-Capelli}{32}{subsection.5.3}
\contentsline {subsection}{\numberline {5.4}Grassman}{32}{subsection.5.4}
\contentsline {section}{\numberline {6}Applicazioni lineari}{33}{section.6}
\contentsline {subsection}{\numberline {6.1}Nucleo}{34}{subsection.6.1}
\contentsline {subsection}{\numberline {6.2}Immagine}{35}{subsection.6.2}
\contentsline {subsection}{\numberline {6.3}Controimmagine}{35}{subsection.6.3}
\contentsline {section}{\numberline {7}Spazio quoziente}{36}{section.7}
\contentsline {subsection}{\numberline {7.1}Isomorfismo}{37}{subsection.7.1}
\contentsline {subsection}{\numberline {7.2}Teorema dell'isomorfismo}{37}{subsection.7.2}
\contentsline {subsection}{\numberline {7.3}Teorema delle dimensioni}{39}{subsection.7.3}
\contentsline {subsection}{\numberline {7.4}Assegnazione di un'applicazione lineare}{39}{subsection.7.4}
\contentsline {subsection}{\numberline {7.5}Prodotto cartesiano di spazi vettoriali}{41}{subsection.7.5}
\contentsline {subsubsection}{\numberline {7.5.1}Formula di Grassman}{42}{subsubsection.7.5.1}
\contentsline {subsection}{\numberline {7.6}Matrice associata}{42}{subsection.7.6}
\contentsline {subsubsection}{\numberline {7.6.1}Proprieta'}{43}{subsubsection.7.6.1}
\contentsline {subsection}{\numberline {7.7}Equazioni cartesiane del nucleo}{44}{subsection.7.7}
\contentsline {subsection}{\numberline {7.8}Matrici associate di f compost}{44}{subsection.7.8}
\contentsline {subsubsection}{\numberline {7.8.1}Matrici associate di isomorfismi}{45}{subsubsection.7.8.1}
\contentsline {subsection}{\numberline {7.9}Matrici equivalenti}{46}{subsection.7.9}
\contentsline {subsection}{\numberline {7.10}Matrici del cambiamento di base}{46}{subsection.7.10}
\contentsline {subsection}{\numberline {7.11}Relazione tra matrici associate}{48}{subsection.7.11}
\contentsline {subsection}{\numberline {7.12}Matrice invertibile e trasf. elementari}{49}{subsection.7.12}
\contentsline {subsection}{\numberline {7.13}Matrice ortogonale}{50}{subsection.7.13}
\contentsline {subsubsection}{\numberline {7.13.1}Proprieta'}{50}{subsubsection.7.13.1}
\contentsline {section}{\numberline {8}Endomorfismi}{50}{section.8}
\contentsline {subsection}{\numberline {8.1}Matrici simili}{51}{subsection.8.1}
\contentsline {subsection}{\numberline {8.2}Matrici diagonalizzabili}{51}{subsection.8.2}
\contentsline {subsection}{\numberline {8.3}Autovalore e autovettore}{52}{subsection.8.3}
\contentsline {subsection}{\numberline {8.4}Autospazio}{52}{subsection.8.4}
\contentsline {subsection}{\numberline {8.5}Polinomio caratteristico}{52}{subsection.8.5}
\contentsline {subsection}{\numberline {8.6}Molteplicita' algebrica e geometrica}{54}{subsection.8.6}
\contentsline {subsection}{\numberline {8.7}Autovettori linearmente indipendenti}{55}{subsection.8.7}
\contentsline {subsection}{\numberline {8.8}Endomorfismo semplice}{56}{subsection.8.8}
\contentsline {subsection}{\numberline {8.9}Equazioni cartesiane di un autospazio}{57}{subsection.8.9}
\contentsline {subsection}{\numberline {8.10}Test di semplicita'}{58}{subsection.8.10}
\contentsline {subsection}{\numberline {8.11}Polinomio endomorfismo}{58}{subsection.8.11}
\contentsline {subsubsection}{\numberline {8.11.1}Proprieta'}{58}{subsubsection.8.11.1}
\contentsline {subsubsection}{\numberline {8.11.2}Proprieta'}{59}{subsubsection.8.11.2}
\contentsline {subsection}{\numberline {8.12}Ideale dell'anello di T}{60}{subsection.8.12}
\contentsline {subsection}{\numberline {8.13}Endomorfismo singolare (degenere)}{61}{subsection.8.13}
\contentsline {subsection}{\numberline {8.14}Sottospazio invariante}{61}{subsection.8.14}
\contentsline {subsection}{\numberline {8.15}Endomorfismi triangolizzabili}{62}{subsection.8.15}
\contentsline {subsection}{\numberline {8.16}Teorema di Cayley-Hamilton}{66}{subsection.8.16}
\contentsline {section}{\numberline {9}Spazio duale}{66}{section.9}