From f39d6e80da6fda43cecf441a63a5fa7322101ab3 Mon Sep 17 00:00:00 2001 From: Hearot Date: Wed, 22 Mar 2023 18:57:22 +0100 Subject: [PATCH] feat(geometria): aggiunge gli appunti del 6/03/23 (teorema degli orlati) --- .../Calculus/Classificazione di funzioni.pdf | Bin .../{PDF => }/Calculus/Eq. differenziali.pdf | Bin .../Calculus/Introduzione agli integrali.pdf | Bin .../{PDF => }/Calculus/Polinomi di Taylor.pdf | Bin .../Calculus/Punti notevoli di Taylor.pdf | Bin .../Esercitazioni/Es. sui punti notevoli.pdf | Bin .../Es. sulle parti principali.pdf | Bin .../esempio.eps | 0 .../esempio2.eps | 0 .../main.pdf | Bin .../main.tex | 0 .../main.pdf | Bin .../main.tex | 0 ...Introduzione a ZFC e alle cardinalità.pdf | Bin .../main.pdf | Bin 0 -> 32937 bytes .../main.tex | 18 +++++ .../Algoritmo di conversione da NFA a DFA.pdf | Bin .../Algoritmo di minimizzazione dei DFA.pdf | Bin .../Introduzione agli NFA.pdf | Bin ...zione agli automi e ai sistemi formali.pdf | Bin .../Introduzione agli 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lineari.pdf | Bin .../Es. sulle app. lineari II.pdf | Bin .../Es. sulle app. lineari III.pdf | Bin .../Es. sulle app. lineari IV.pdf | Bin .../Esercitazioni/Es. sulle app. lineari.pdf | Bin .../main.pdf | Bin 0 -> 149269 bytes .../main.tex | 70 ++++++++++++++++++ ...a scala ridotta e proprietà del rango.pdf | Bin .../Matrici/Metodo di elim. di Gauss.pdf | Bin .../Somme tra spazi/Somma diretta.pdf | Bin .../main.pdf | Bin .../main.tex | 0 .../main.pdf | Bin 0 -> 32906 bytes .../main.tex | 19 +++++ tex/latex/style/personal_commands.sty | 4 +- 56 files changed, 110 insertions(+), 1 deletion(-) rename Analisi I/{PDF => }/Calculus/Classificazione di funzioni.pdf (100%) rename Analisi I/{PDF => }/Calculus/Eq. differenziali.pdf (100%) rename Analisi I/{PDF => }/Calculus/Introduzione agli integrali.pdf (100%) rename Analisi I/{PDF => }/Calculus/Polinomi di Taylor.pdf (100%) rename Analisi I/{PDF => }/Calculus/Punti notevoli di Taylor.pdf (100%) rename Analisi I/{PDF => }/Esercitazioni/Es. sui punti notevoli.pdf (100%) rename Analisi I/{PDF => }/Esercitazioni/Es. sulle parti principali.pdf (100%) rename Analisi I/{PDF => }/Parte teorica/2023-03-17, Successioni per ricorrenza/esempio.eps (100%) rename Analisi I/{PDF => }/Parte teorica/2023-03-17, Successioni per ricorrenza/esempio2.eps (100%) rename Analisi I/{PDF => }/Parte teorica/2023-03-17, Successioni per ricorrenza/main.pdf (100%) rename Analisi I/{PDF => }/Parte teorica/2023-03-17, Successioni per ricorrenza/main.tex (100%) rename Analisi I/{PDF => }/Parte teorica/2023-03-22, Limiti di funzioni e funzioni continue/main.pdf (100%) rename Analisi I/{PDF => }/Parte teorica/2023-03-22, Limiti di funzioni e funzioni continue/main.tex (100%) rename Analisi I/{PDF => }/ZFC e teoria delle cardinalità/Introduzione a ZFC e alle cardinalità.pdf (100%) create mode 100644 Fisica I/2023-03-22, Derivate parziali e integrali di linea/main.pdf create mode 100644 Fisica I/2023-03-22, Derivate parziali e integrali di linea/main.tex rename Fond. di programmazione/{PDF => }/Automi a stati finiti/Algoritmo di conversione da NFA a DFA.pdf (100%) rename Fond. di programmazione/{PDF => }/Automi a stati finiti/Algoritmo di minimizzazione dei DFA.pdf (100%) rename Fond. di programmazione/{PDF => }/Automi a stati finiti/Introduzione agli NFA.pdf (100%) rename Fond. di programmazione/{PDF => }/Automi a stati finiti/Introduzione agli automi e ai sistemi formali.pdf (100%) rename Fond. di programmazione/{PDF => }/Automi a stati finiti/Introduzione agli ε-NFA.pdf (100%) rename Fond. di programmazione/{PDF => }/Automi a stati finiti/Introduzione ai DFA.pdf (100%) rename Fond. di programmazione/{PDF => }/Automi a stati finiti/La funzione di transizione estesa.pdf (100%) rename Fond. di programmazione/{PDF => }/Esercizi svolti/Es. del primo compitino (2012).pdf (100%) rename Fond. di programmazione/{PDF => }/Esercizi svolti/Es. del primo compitino (2019).pdf (100%) rename Fond. di programmazione/{PDF => }/Esercizi svolti/Es. del 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I/{PDF => }/App. lineari/App. bilineari e multilineari.pdf (100%) rename Geometria I/{PDF => }/App. lineari/Endomorfismi e SD-equivalenza.pdf (100%) rename Geometria I/{PDF => }/App. lineari/L(V, W) e matrice di cambiamento.pdf (100%) rename Geometria I/{PDF => }/App. lineari/Multilineari alternanti e determinante.pdf (100%) rename Geometria I/{PDF => }/App. lineari/Rapporto tra matrici e app. lineari.pdf (100%) rename Geometria I/{PDF => }/Esercitazioni/Es. foglio 2 (2022-23).pdf (100%) rename Geometria I/{PDF => }/Esercitazioni/Es. sui sistemi lineari.pdf (100%) rename Geometria I/{PDF => }/Esercitazioni/Es. sulle app. lineari II.pdf (100%) rename Geometria I/{PDF => }/Esercitazioni/Es. sulle app. lineari III.pdf (100%) rename Geometria I/{PDF => }/Esercitazioni/Es. sulle app. lineari IV.pdf (100%) rename Geometria I/{PDF => }/Esercitazioni/Es. sulle app. lineari.pdf (100%) create mode 100644 Geometria I/Matrici/2023-03-06, Teorema degli orlati e calcolo del rango di una matrice/main.pdf create mode 100644 Geometria I/Matrici/2023-03-06, Teorema degli orlati e calcolo del rango di una matrice/main.tex rename Geometria I/{PDF => }/Matrici/Matrici a scala ridotta e proprietà del rango.pdf (100%) rename Geometria I/{PDF => }/Matrici/Metodo di elim. di Gauss.pdf (100%) rename Geometria I/{PDF => }/Somme tra spazi/Somma diretta.pdf (100%) rename Geometria I/{PDF => }/Teoria spettrale degli endomorfismi/2023-03-20, Es. 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\\ + + \leftproof Senza perdità di generalità, supponiamo che $M = A^{1, ..., k}_{1, ..., k}$ (altrimenti è sufficiente + considerare una permutazione delle colonne e delle righe di $A$ per ricadere nel caso studiato in questa dimostrazione). Dal momento che $A^1$, ..., $A^k$ sono per ipotesi colonne linearmente indipendenti (infatti + $\det(M) \neq 0 \implies \rg(A^1 \cdots A^k) = k$), per dimostrare che $\rg(A) = r$ è sufficiente mostrare che + $\forall j > k$, $A^j \in \Span(A^1, ..., A^k)$. \\ + + Si consideri allora la matrice $B = A^{1, ..., k, j}_{1, ..., m}$. Sia $i > k$ e $N_i = A^{1, ..., k, j}_{1, ..., k, i}$. Poiché $N_i$ è un orlato di $M$, $\det(N_i) = 0$, e quindi $\rg(N_i) < k+1$. Tuttavia, poiché le righe $N_{i_1} = B_1$, ..., $N_{i_k} = B_k$ sono linearmente + indipendenti (sono infatti righe di $M$ a cui è stata aggiunta una colonna), $\rg(N_i) \geq k$. Si conclude + allora che $\rg(N_i) = k$, e che, essendo le righe $N_{i_1}$, ..., $N_{i_k}$ linearmente indipendenti, + $N_{i_j} \in \Span(N_{i_1}, ..., N_{i_k}) = \Span(B_1, ..., B_k)$. Allora ogni $B_i \in \Span(B_1, ..., B_k)$, + e quindi $\rg(B) \leq k$. Dal momento però che, come osservato prima, $B_1$, ..., $B_k$ sono linearmente + indipendenti, si conclude che $\rg(B) = k$. Infine, poiché $B^1 = A^1$, ..., $B^k = A^k$ sono linearmente + indipendenti, deve valere che $B^{k+1} = A^j \in \Span(B^1, ..., B^k) = \Span(A^1, ..., A^k)$, da cui la tesi. + \end{proof} + + \begin{example} + Si può impiegare il teorema degli orlati per calcolare agevolmente il rango di una matrice senza impiegare + il metodo di eliminazione di Gauss. Sia per esempio: + + \[ A = \Matrix{1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9}. \] + + Poiché $B_1 = (A_{11}) = (1) \neq (0)$, $\rg(A) \geq 1$. Si consideri l'orlato $B_2 = \Matrix{1 & 2 \\ 4 & 5}$ di $B_1$: + $\det(B_2) = 1\cdot 5 - 4 \cdot 2 = -3 \neq 0$: allora $\rg(A) \geq 2$. Infine, si consideri l'orlato $B_3 = A$ di + $B_2$: poiché $\det(B_3) = \det(A) = 1\cdot \det\Matrix{5 & 6 \\ 8 & 9} - 2 \cdot \det\Matrix{4 & 6 \\ 7 & 9} + 3 \cdot \det\Matrix{4 & 5 \\ 7 & 8} = 0$ e $B_3$ è l'unico orlato di $B_2$, si conclude che $\rg(A) = 2$. + \end{example} + +\end{document} +content... \ No newline at end of file diff --git a/Geometria I/PDF/Matrici/Matrici a scala ridotta e proprietà del rango.pdf b/Geometria I/Matrici/Matrici a scala ridotta e proprietà del rango.pdf similarity index 100% rename from Geometria I/PDF/Matrici/Matrici a scala ridotta e proprietà del rango.pdf rename to Geometria I/Matrici/Matrici a scala ridotta e proprietà del rango.pdf diff --git a/Geometria I/PDF/Matrici/Metodo di elim. di Gauss.pdf b/Geometria I/Matrici/Metodo di elim. di Gauss.pdf similarity index 100% rename from Geometria I/PDF/Matrici/Metodo di elim. di Gauss.pdf rename to Geometria I/Matrici/Metodo di elim. di Gauss.pdf diff --git a/Geometria I/PDF/Somme tra spazi/Somma diretta.pdf b/Geometria I/Somme tra spazi/Somma diretta.pdf similarity index 100% rename from Geometria I/PDF/Somme tra spazi/Somma diretta.pdf rename to Geometria I/Somme tra spazi/Somma diretta.pdf diff --git a/Geometria I/PDF/Teoria spettrale degli endomorfismi/2023-03-20, Es. Algoritmi per la ricerca del polinomio minimo/main.pdf b/Geometria I/Teoria spettrale degli endomorfismi/2023-03-20, Es. Algoritmi per la ricerca del polinomio minimo/main.pdf similarity index 100% rename from Geometria I/PDF/Teoria spettrale degli endomorfismi/2023-03-20, Es. Algoritmi per la ricerca del polinomio minimo/main.pdf rename to Geometria I/Teoria spettrale degli endomorfismi/2023-03-20, Es. Algoritmi per la ricerca del polinomio minimo/main.pdf diff --git a/Geometria I/PDF/Teoria spettrale degli endomorfismi/2023-03-20, Es. Algoritmi per la ricerca del polinomio minimo/main.tex b/Geometria I/Teoria spettrale degli endomorfismi/2023-03-20, Es. Algoritmi per la ricerca del polinomio minimo/main.tex similarity index 100% rename from Geometria I/PDF/Teoria spettrale degli endomorfismi/2023-03-20, Es. Algoritmi per la ricerca del polinomio minimo/main.tex rename to Geometria I/Teoria spettrale degli endomorfismi/2023-03-20, Es. 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