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aggiunta di grafici capitolo 4
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12 changes: 8 additions & 4 deletions Primo anno/algebra e geometria/algebraegeometria.aux
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]
Capitolo 4.
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89 changes: 87 additions & 2 deletions Primo anno/algebra e geometria/algebraegeometria.tex
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\input{pag/numeri_complessi.tex}
\input{pag/determinante.tex}
\chapter{Applicazione lineari e prodotto di matrici}
\section{Applicazioni lineari: definizione e esempi}
\section{Applicazioni lineari: definizione e esempi\label{applin}}
Inizieremo ora a parlare di funzioni tra spazi vettoriali. Ricordiamo che una funzione $f:X\to Y$ tra due insiemi
$X$ (detto \textit{dominio}) e $Y$ (detto \textit{codominio}) è una legge che associa a ogni $x\in X$ un ben
preciso elemento di $Y$, detto \textit{immagine di $x$} e denotato $f(x)$.\\
Expand All @@ -155,7 +155,92 @@ \section{Applicazioni lineari: definizione e esempi}
\begin{esempio}
si può vedere che se\footnote{Sappiamo che $\mathds{R}^n$ è una spazio vettoriale di dimensione $n$, in
particolare per $n=1$ si ottiene $\mathds{R}^1=\mathds{R}$ (che risulta quindi spazio vettoriale come da
dimostrazione 1).}
dimostrazione 1).} $V=W=\mathds{R}$, le uniche funzioni lineari $f:\mathds{R}\to\mathds{R}$ sono quelle
del tipo $f(x)=ax$, con $a\in \mathds{R}$ fissato.
\end{esempio}
Tuttavia, vediamo subito che tra le applicazioni lineari vi sono funzioni di grante impotanza e utilità in
geometria e nello sue applicazioni:
\begin{esempio}
Dato lo spazio $V_O^2$ dei vettori geometrici applicati nel piano, consideriamo la funzione
$f:V_O^2\to V_O^2$ che associa a ogni vettore $\vec{OP}$ il vettore che si ottiene ruotando $\vec{OP}$ di
un angolo $\theta$ fissato in senso antiorario attorno all'origine $O$,come nel disegno sequente
\begin{figure}[th]
\centering
\includegraphics[width=5cm]{img/finiti/imgex4-2-1.eps}
\caption{$f:V_O^2\to V_O^2$}
\end{figure}
Ora, come si vede nel disegno sequente, dati due vettori $\vec{OP}$ e $\vec{OP}^\prime$, sommarli e poi
ruotare il vettore risultante oppure prima ruotarli e poi sommare i vettori ruotati è equivalente, ovvero
\clearpage
\begin{figure}[th]
\centering
\includegraphics[width=6cm]{img/finiti/imgex4-2-2.eps}
\caption{$f(\vec{OP})+f(\vec{OP}^\prime)=f(\vec{OP}+\vec{OP})$}
\end{figure}
Quindi vale la
\begin{equation}
f(\vec{OP})+f(\vec{OP}^\prime)=f(\vec{OP}+\vec{OP})
\end{equation}
che ci dice che questa funzione soddisfala proprietà (\ref{applin}) della Definizione \ref{4.1-4.2}\\
Analogamente, dato un vettore $\vec{OP}$ e un numero reale $c$, moltiplicare il vettore per $c$ e poi
ruotarlo e poi moltiplicarlo per $c$ è equivalente:
\begin{figure}[th]
\centering
\includegraphics[width=6cm]{img/finiti/imgex4-2-3.eps}
\caption{$f(c\vec{OP})=cf(\vec{OP})$}
\end{figure}
Quindi si ha
\begin{equation}
f(c\vec{OP})=cf(\vec{OP})
\end{equation}
che ci dice che questa funzione soddisfa anche la proprietà (\ref{matasapplin}), della definizione
\ref{4.1-4.2} -- Concludiamo quindi che le rotazioni attorno a $O$ sono applicazioni lineari dallo
spazio vettoriale $V_O^2$ in se stessso.\\
Possiamo arrivare alla stessa conclusione anche per altre importanti trasformazioni geometriche: ad
esempio, si consideri la riflessione rispetto a una retta $r$ passante per $O$, che manda ogni vettore
$\vec{OP}\in V_O^2$ nel vettore simmetrico rispetto alla retta, come nel seguente disegno
\begin{figure}[th]
\centering
\includegraphics[width=6cm]{img/finiti/imgex4-2-4.eps}
\end{figure}
Allora, come già fatto per le rotazioni, notiamo che, dati due vettori $\vec{OP}$ e $\vec{OP}^\prime$,
sommarli e poi riflettere il vettore risultante oppure prima rifletterli e poi sommare i vettori riflessi
è equivalente
\clearpage
\begin{figure}[th]
\centering
\includegraphics[width=6cm]{img/finiti/imgex4-2-5.eps}
\end{figure}
$f\left(\vec{OP}+\vec{OP}^\prime\right)=f\left(\vec{OP}\right)+f\left(\vec{OP}^\prime\right)$ e
dato uin vettore $\vec{OP}$ e un numero
reale $c$, moltiplicatore il vettore per $c$ e poi rifletterlo oppure prima rifletterlo e poi moltiplicarlo
per $c$ è equivalente
\begin{figure}[th]
\centering
\includegraphics[width=6cm]{img/finiti/imgex4-2-6.eps}
\end{figure}

$f(c\vec{OP})=cf(\vec{OP})$: quindi concludiamo che anche la riflessione rispetto a una retta ce passa per
$O$, avendo le proprietà entrambe le proprietà (\ref{4.1-4.2}) richieste nella Definizione (\ref{applin}),
è un'applicazione lineare $f:V_O^2\to V_O^2$.\\
Come terzo esempio esempio di applicazione lineare $V_O^2\to V_O^2$ citiamo la propiezione ortogonale,
che proieta ortogonalmente i vettori su una retta fissata passante per $O$.
\begin{figure}[th]
\centering
\includegraphics[width=8cm]{img/finiti/imgex4-2-7.eps}
\end{figure}

per la quale è difficile vedere che valgono ancora le proprieta citate in (\ref{4.1-4.2}).\\
Analogamente a quanto visto per rotazioni, riflessioni e proiezioni nel piano, anche le corrispondenti
trasformazioni $V_O^3\to V_O^3$ dello spazio tridimensionale come da proprietà (\ref{4.1-4.2}) della
Definizione \ref{applin} la rotazione di un angolo fissato $\Theta$ attorno a una retta data passante per $O$
(detta ase della rotazione)
\clearpage
\begin{figure}[th]
\centering
\includegraphics[width=3cm]{img/finiti/imgex4-2-8.eps}
\end{figure}
la riflessione rispetto a un piano passante per $O$
\end{esempio}
\end{definizione}
\end{document}
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