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applicazioni affini/main.tex b/Geometria 1/Spazi affini/2023-04-28, Indipendenza e applicazioni affini/main.tex index 1da3aa8..24e7a45 100644 --- a/Geometria 1/Spazi affini/2023-04-28, Indipendenza e applicazioni affini/main.tex +++ b/Geometria 1/Spazi affini/2023-04-28, Indipendenza e applicazioni affini/main.tex @@ -14,6 +14,14 @@ \Large \textbf{Indipendenza e applicazioni affini} \end{center} + \wip + + \begin{note} Qualora non specificato diversamente, si intenderà per + $E$ uno spazio affine sullo spazio vettoriale $V$ e + per $E'$ uno spazio affine sullo spazio vettoriale $V'$, dove sia $V$ che $V'$ sono costruiti + sul campo $\KK$. + \end{note} + Fissato un origine $O$ dello spazio affine, si possono sempre considerare due bigezioni: @@ -180,6 +188,26 @@ sé stesso. \end{remark} + \begin{definition} [riferimento affine] Sia $D \subseteq E$ un sottospazio affine di $E$ di dimensione $k-1$. + Siano i punti $P_1$, ..., $P_k$ dei punti affinemente indipendenti. Allora + si dice che tali punti formano un \textbf{riferimento affine} di $D$. + \end{definition} + + \begin{definition} [coordinate affini] Sia $D \subseteq E$ un sottospazio affine di $E$ di dimensione $k-1$ + e siano i punti $P_1$, ..., $P_k$ un riferimento affine $R$ di $D$. Allora, se $P = \lambda_1 P_1 + \ldots + \lambda_k P_k \in D$ con $\lambda_1 + \ldots + \lambda_k = 1$, si dice che le \textbf{coordinate affine} + di $P$ sono rappresentate dal punto $[P]_\basis$, dove: + + \[ [P]_R = \Vector{\lambda_1 \\ \vdots \\ \lambda_k} \in \AnK. \] + \end{definition} + + \begin{remark}\nl + \li Esiste sempre un riferimento affine di un sottospazio affine $D$ di $E$. Infatti, dato un punto $P_1$ + di $E$, e una base $\basis = \{\vv 1, \ldots, \vv k\}$ della direzione $D_0$, i punti $P_1$, $P_1 + \vv 1$, ..., + $P_1 + \vv k$ formano un riferimento affine. \\ + \li Dalla definizione sopra si deduce che, scelto un riferimento affine $R$, esiste una mappa iniettiva $[\cdot]_R : D \to \AnK$, dove l'immagine di $P$ mediante $[\cdot]_R$ è esattamente il vettore + contenente le coordinate affini di $P$. + \end{remark} + \begin{proposition} Sia $E = \AnK$. Allora i punti $P_1$, ..., $P_k$ sono affinemente indipendenti se e solo se i vettori $\hat P_1 = \Vector{P_1 \\ \hline 1}$, ..., $\hat P_k = \Vector{P_k \\ \hline 1}$ sono @@ -198,6 +226,8 @@ \[ \lambda_2 (P_2 - P_1) + \ldots + \lambda_k (P_k - P_1) = 0. \] + \vskip 0.05in + Poiché i punti $P_1$, ..., $P_k$ sono affinemente indipendenti, per la proposizione precedente, allora i vettori $P_2 - P_1$, ..., $P_k - P_1$ sono linearmente indipendenti, per cui $\lambda_2 = \cdots = \lambda_k = 0$. Pertanto anche $\lambda_1 = 0$, e quindi i vettori $\hat P_1$, ..., $\hat P_k$ sono linearmente indipendenti. \\ @@ -208,34 +238,90 @@ che $\lambda_1 + \ldots + \lambda_k = 0$ e che $\lambda_1 P_1 + \ldots + \lambda_k P_k = 0$, da cui si deduce che $\lambda_1 \hat P_1 + \ldots + \lambda_k \hat P_k = 0$. Dal momento però che $\hat P_1$, ..., $\hat P_k$ sono linearmente indipendenti, $\lambda_2 = \cdots = \lambda_k = 0$, - da cui la tesi. + da cui la tesi, per la proposizione precedente. \end{proof} + + \begin{definition} [combinazione convessa] + Si dice che una combinazione affine $\sum_{i=1}^k \lambda_i P_i$ nei punti $P_1$, ..., $P_k$ con + $\sum_{i=1}^k \lambda_i = 1$ è + una \textbf{combinazione convessa} se $\lambda_i \geq 0$ $\forall 1 \leq i \leq k$. + \end{definition} + + \begin{definition} [baricentro] + Si definisce \textbf{baricentro} dei punti $P_1$, ..., $P_k$ la combinazione convessa + $\sum_{i=1}^k \frac{1}{k} P_i$. + \end{definition} - Se si impone $\lambda_i \geq 0$, si definisce che la - combinazione è una combinazione convessa. Si definisce - baricentro il punto con $\lambda_i = \frac{1}{n}$. - - \begin{definition} [inviluppo convesso] Si dice $\IC(S)$ di un insieme - $S \subseteq E$ l'insieme delle combinazioni convesse di $S$ (finite). - %TODO: dimostrare che è un insieme convesso + \begin{definition} [inviluppo convesso] Si definisce l'\textbf{inviluppo complesso} $\IC(S)$ di un insieme + $S \subseteq E$ l'insieme delle combinazioni convesse finite di $S$. \end{definition} - % TODO: aggiungere baricentro + \begin{remark}\nl + \li L'insieme $\IC(S)$ è, effettivamente, un insieme convesso, se $S \subseteq E$. Se infatti $P$, $Q \in \IC(S)$, + allora $\lambda_1 P + \lambda_2 Q \in \IC(S)$, con $\lambda_1$, $\lambda_2 \geq 0$, e quindi + $[P, Q] \subseteq \IC(S)$. \\ + \li Se $E = \Aa_2(\RR)$, e $P_1$, $P_2$, $P_3$ sono tre punti di $E$, l'inviluppo convesso dei + tre punti è esattamente il triangolo costruito sui tre punti. Analogamente, presi quattro + punti di $\Aa_3(\RR)$, l'inviluppo convesso dei quattro punti è un tetraedro. + \end{remark} - \begin{definition} Sia $E$ uno spazio affine su $V$, $E'$ spazio - affine su $V'$ (sullo stesso $\KK$) un'applicazione $f : E \to E'$ - si dice app. affine se conserva le combinazioni affini - ($f(\sum \lambda_i P_i) = \sum \lambda_i f(P_i)$, $\sum \lambda_i = 1$). + \begin{definition} [applicazione affine] Si definisce \textbf{applicazione affine} da $E$ a $E'$ un'applicazione $\varphi : E \to E'$ + che conservi le combinazioni affini, ossia tale che: + + \[ \varphi\left( \sum_{i=1}^k \lambda_i P_i \right) = \sum_{i=1}^k \lambda_i \varphi(P_i), \quad \se \sum_{i=1}^k \lambda_i = 0. \] \end{definition} - - \begin{theorem} Sia $f : E \to E'$ affine. Allora $\exists$ unica - app. lineare $g : V \to V'$ lineare tale che valga - $f(O + \v) = f(O) + g(\v)$, per ogni scelta di $O \in E$. + + \begin{remark}\nl + \li Come per le applicazioni lineari, la somma e la composizione di più applicazioni affini è + ancora una applicazione affine. \\ + \li Se si sceglie un riferimento affine di $E$, $\varphi$ è univocamente determinata + da come agisce su tale riferimento. + \end{remark} + + \begin{theorem} + Sia $\varphi : E \to E'$ un'applicazione affine. Allora esiste un'unica applicazione lineare $g : V \to V'$ + tale per cui $\varphi(P) = \varphi(O) + g(P-O)$ $\forall P \in E$, invariante per la scelta di $O \in E$. \end{theorem} - + \begin{proof} - Sia $O \in E$. L'applicazione $g_O : V \to V'$ data da - $g_O(\v) = f(O + \v) - f(O)$. Si dimostra che $g_O$ è - lineare. + Sia $O \in E$. Si consideri l'applicazione $g : V \to V'$ tale per cui $g(\v) = \varphi(O + \v) - \varphi(O)$. + Si verifica che $g$ è lineare: + + \begin{itemize} + \item $g(\v + \w) = \varphi(O + \v + \w) - \varphi(O) = \varphi((O + \v) + (O + \w) - O) - \varphi(O) = + \varphi(O + \v) - \varphi(O) + \varphi(O + \w) - \varphi(O) = g(\v) + g(\w)$ (additività), + \item $g(a\v) = \varphi(O + a\v) - \varphi(O) = \varphi(a(O + \v) + (1-a)O) - \varphi(O) = + a\varphi(O + \v) + (1-a)\varphi(O) - \varphi(O) = ag(\v)$ (omogeneità). + \end{itemize} + + Inoltre, $\varphi(P) = \varphi(O + P - O) = \varphi(O) + \varphi(P) - \varphi(O) = \varphi(O) + g(P-O)$. Si + osserva infine che $g$ è unica per costruzione. Si + verifica allora che scegliendo $O' \in E$ al posto di $O$, la costruzione di $g$ è invariante, ossia + che $\varphi(O' + \v) - \varphi(O') = \varphi(O + \v) - \varphi(O)$ $\forall \v \in V$. Infatti + $\varphi(O' + \v) - \varphi(O') = \varphi(O' - O + (O + \v)) - \varphi(O') = + \varphi(O') - \varphi(O) + \varphi(O + \v) - \varphi(O') = \varphi(O + \v) - \varphi(O)$, da cui + la tesi. \end{proof} + + \begin{remark} + Data un'applicazione lineare $g$ da $V$ in $V'$ e dati $O \in E$, $O' \in E$, si può sempre costruire un'applicazione affine $\varphi$ tale che $\varphi(P) = O' + g(P - O)$. Infatti, se $\sum_{i=1}^n \lambda_i = 1$, + $\varphi\left( \sum_{i=1}^n \lambda_i P_i \right) = O' + g\left( \sum_{i=1}^n \lambda_i (P_i - O) ) \right) = + O' + \sum_{i=1}^n \lambda_i \, g(P_i - O) = O' + \sum_{i=1}^n \lambda_i \, (\varphi(P_i) - O') = \sum_{i=1}^n \lambda_i \, \varphi(P_i)$. + \end{remark} + + \begin{definition} [applicazione lineare associata ad un'applicazione affine] + Data un'applicazione affine $\varphi : E \to E'$ e dato $O \in E$, si definisce $g : V \to V'$ tale che + $g(\v) = \varphi(O + \v) - \varphi(O)$ come l'\textbf{applicazione lineare associata a $\varphi$}. + \end{definition} + + \begin{remark}\nl + \li Siano $E = \AnK$ ed $E' = \Aa_m(\KK)$. Allora, se $\varphi$ è un'applicazione affine da $E$ a $E'$, + $\varphi(\vec x) = \varphi(\vec 0) + g(\vec x - \vec 0) = A \vec x + \vec b$ $\forall \vec x \in E$, dove $A$ è la matrice associata + di $g$ nelle basi canoniche di $\KK^n$ e $\KK^m$ e $\vec b = \varphi(\vec 0)$. \\ + + \li Se $g$ e $g'$ sono le applicazioni lineari associate alle applicazioni affini $\varphi : E \to E'$ e $\varphi' : E' \to E''$, + allora $g \circ g'$ è l'applicazione lineare associata a $\varphi \circ \varphi'$ e + $\varphi + \varphi'$. Infatti, se $O \in E$, $\varphi(\varphi'(P)) = \varphi(\varphi'(O) + g'(P-O)) = + \varphi(\varphi'(O)) + g(g'(P-O))$. + \end{remark} \end{document} diff --git a/tex/latex/style/personal_commands.sty b/tex/latex/style/personal_commands.sty index d19f7f7..a5a10c8 100644 --- a/tex/latex/style/personal_commands.sty +++ b/tex/latex/style/personal_commands.sty @@ -95,6 +95,7 @@ \newcommand{\RRbar}{\overline{\RR}} % Spesso utilizzati al corso di Geometria 1. +\newcommand{\Aa}{\mathcal{A}} \DeclareMathOperator{\An}{\mathcal{A}_n} \DeclareMathOperator{\AnK}{\mathcal{A}_n(\KK)} \DeclareMathOperator{\Giac}{Giac}