From 71b13cecc72167de7bd0f659f45e69ec1a5a48c6 Mon Sep 17 00:00:00 2001 From: Stefano Pigozzi Date: Tue, 4 Jun 2024 08:31:22 +0200 Subject: [PATCH] Importa Algebra lineare da Steffo99/appunti-universitari --- .../1 - Basi di algebra lineare.pdf | Bin 0 -> 71677 bytes .../1 - Basi di algebra lineare.tex | 62 +++++++++ .../2 - Cheatsheet per l'esame.pdf | Bin 0 -> 114557 bytes .../2 - Cheatsheet per l'esame.tex | 121 ++++++++++++++++++ 4 files changed, 183 insertions(+) create mode 100644 1 - Algebra lineare/1 - Basi di algebra lineare/1 - Basi di algebra lineare.pdf create mode 100644 1 - Algebra lineare/1 - Basi di algebra lineare/1 - Basi di algebra lineare.tex create mode 100644 1 - Algebra lineare/2 - Cheatsheet per l'esame/2 - Cheatsheet per l'esame.pdf create mode 100644 1 - Algebra lineare/2 - Cheatsheet per l'esame/2 - Cheatsheet per l'esame.tex diff --git a/1 - Algebra lineare/1 - Basi di algebra lineare/1 - Basi di algebra lineare.pdf b/1 - Algebra lineare/1 - Basi di algebra 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+\usepackage{centernot} +\usepackage{bm} +\usepackage{fullpage} + +% Iniziate a scrivere da qua in poi +\begin{document} + +\section{Moltiplicazioni tra matrici} + +\[ + \begin{bmatrix} + a & b \\ + c & d \\ + \end{bmatrix} + * + \begin{bmatrix} + e & f \\ + g & h \\ + \end{bmatrix} + = + \begin{bmatrix} + ae + cf & be + df \\ + ag + ch & bg + dh \\ + \end{bmatrix} +\] + +\section{Invertibilità di una matrice} + +Si può verificare se una matrice \( A \) quadrata di ordine \( n \) è invertibile verificando una di queste definizioni equivalenti: + +\begin{itemize} +\item Il determinante non è nullo: \( \det A\neq 0 \). +\item Il rango di \( A \) è \( n \). +\item La trasposta \( A^{T} \) è una matrice invertibile. +\item Tutte le righe/colonne di \( A \) sono linearmente indipendenti. +\item Tutte le righe/colonne di \( A \) formano una base di \( \mathbb{K} ^{n} \). +\item Il numero 0 non è un autovalore di \( A \). +\item \( A \) è trasformabile mediante algoritmo di Gauss-Jordan in una matrice con \( n \) pivot. +\end{itemize} + +\section{Stabilire esistenza di funzione lineare} + +Per controllare se esiste o no una funzione lineare è sufficiente verificare che sia valida la proprietà di linearità:\\ +\begin{itemize} +\item Se due vettori sono linearmente indipendenti, anche i risultati della funzione devono essere linearmente indipendenti. +\end{itemize} +Può essere controllata velocemente vedendo se si verificano le seguenti condizioni: +\begin{itemize} +\item Se due vettori di ingresso sono uno multiplo dell'altro, allora anche i vettori di uscita devono essere uno multiplo dell'altro per la stessa costante. +\item Se un vettore di ingresso è dato dalla somma di (multipli di) altri, allora anche il vettore di uscita deve essere dato dalla somma di (multipli degli) stessi. +\end{itemize} + +\section{Determinazione di matrice associata} + +Vogliamo trovare la matrice associata (\(A\)) di una funzione rispetto a delle nuove basi, ad esempio \(< (1, 2, 3), (4, 5, 6), (7, 8, 9)\).\\ + +Procediamo disponendo in verticale gli elementi delle basi, in questo modo: +\[ + M = + \begin{matrix} + 1 & 4 & 7 \\ + 2 & 5 & 8 \\ + 3 & 6 & 9 \\ + \end{matrix} +\] + +Troviamo la matrice inversa con il metodo di Gauss-Jordan: +\[ +... +\] + +Calcoliamo il risultato di: +\[ + B = M^{-1} * A * M +\] + +Il risultato \(B\) sarà la nostra nuova matrice associata. + +\section{Diagonalizzabilità} +Una matrice è \textsc{diagonalizzabile} se ha \textbf{tanti autovalori quanto il suo rango}.\\ +Per trovare gli autovalori trovare dove il polinomio caratteristico (determinante della matrice fatta come quella qui sotto) è uguale a 0: +\[ + \begin{vmatrix} + 1 - x & 2 & 3 \\ + 4 & 5 - x & 6 \\ + 7 & 8 & 9 - x \\ + \end{vmatrix} + = 0 +\] + +\section{Stabilire se una funzione è lineare} + +Se tutti i termini della funzione sono \textbf{polinomi omogenei} di primo grado (non ci sono potenze superiori a 1), allora è automaticamente \textsc{lineare}. + +\section{Immagine} + +Le \textsc{basi dell'immagine} di una funzione sono i \textbf{vettori linearmente indipendenti} che la generano. + +\section{Iniettività e suriettività} + +Una funzione lineare è \textsc{iniettiva} se \textbf{il nucleo è di dimensione 0}, ovvero se l'unico valore che fa risultare 0 alla funzione è il vettore nullo.\\ +\\ +Una funzione lineare è \textsc{suriettiva} se la dimensione dell'immagine è minore o uguale al rango della funzione (degli input, il rango della matrice associata): \(dim(Im(F)) = rk(M_F)\).\\ + +\subsection{Matrici quadrate} + +Se la funzione è un \textbf{automorfismo} (campo input = campo output), allora \(iniettivita' \Leftrightarrow suriettivita'\). + +\section{Somma diretta} + +Un sottospazio è \textsc{somma diretta} se i due sottospazi di cui viene fatta la somma \textbf{non hanno basi in comune}, e quindi \(dim(\pmb{U} \cap \pmb{W}) = 0\). + +\subsection{Trovare basi che diano una somma diretta} + +Per trovare basi che diano una somma diretta, è sufficiente \textbf{trovare basi linearmente indipendenti} con quelle che già abbiamo: solitamente parti della base canonica funzionano alla perfezione. + +\end{document} \ No newline at end of file