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proiettivo/main.tex @@ -38,7 +38,7 @@ \[ f(\v) = A \v + \vec b, \] per ogni $\v \in \AnK$. Se $f \in A(E)$, allora vale anche che: - \[ f\inv(O + \w) = f\inv(f(O) + (O - f(O)) + \w) = O - g\inv(f(O) - O) + g(\w), \] + \[ f\inv(O + \w) = f\inv(f(O) + (O - f(O)) + \w) = O - g\inv(f(O) - O) + g\inv(\w), \] dove si è usato che $g$ è invariante per cambiamento del punto d'origine $O$. Pertanto, in questo caso, passando alle coordinate, vale che: @@ -128,7 +128,22 @@ \vskip 0.05in - formano un sottogruppo di $(M(n+1, \KK), \cdot)$ canonicamente isomorfo a $A(\AnK)$. + formano un sottogruppo di $(M(n+1, \KK), \cdot)$ canonicamente isomorfo a $A(\AnK)$. In particolare si osserva che un'affinità dipende + da esattamente $n^2 + n$, dove $n^2$ sono i parametri su cui + si basa $A$, e $n$ sono i parametri su cui si basa $\vec b$. \\ + + Se $D \subseteq E$ è un sottospazio affine di $E$, l'insieme + $T = \{ f \in A(E) \mid f(D) = D \}$ forma un sottogruppo di $(A(E), \circ)$. In particolare, se $\dim D = k$, un'affinità di $T$ + dipende da esattamente $(k+1)k + (n-k)n$ parametri. \\ + + Infatti in tal caso, scegliendo una base opportuna di $D_0$, estesa + poi a base di $E_0$, e riferendosi ad un'origine di $D$, + $A$ conterrà un blocco $k^2$ relativo alle immagini della base + di $D$ ed un blocco $(n-k)n$ relativo alle immagini degli altri + vettori, non appartenenti a $D$. Inoltre dovranno essere scelti + i parametri riguardanti il vettore $\vec b$, che, essendo stato + scelto come riferimento un punto d'origine appartenente a $D$, + richiederà la scelta di $k$ parametri. \end{remark} \hr @@ -164,4 +179,54 @@ esistervi obbligatoriamente un vettore non nullo. In particolare, se esiste un'intersezione tra $T_i$ e un elemento di $\PP^n(\KK)$, questa è unica. \end{remark} + + \hr + + \begin{theorem} + Sia $E$ uno spazio affine sullo spazio $V$ di dimensione $n$. Allora + valgono i seguenti due risultati. + + \begin{enumerate}[(i)] + \item Se $f \in A(E)$ e i punti $P_1$, ..., $P_k$ sono affinemente + indipendenti, allora anche i punti $f(P_1)$, ..., $f(P_k)$ sono + affinemente indipendenti. + + \item Se i punti $P_1$, ..., $P_{n+1}$ sono affinemente indipendenti, e lo sono anche i punti $Q_1$, ..., $Q_{n+1}$, + allora esiste un'unica affinità + $f \in A(E)$ tale che $f(P_i) = Q_i$ $\forall 1 \leq i \leq n+1$. + \end{enumerate} + \end{theorem} + + \begin{proof} + Si dimostrano i due risultati separatamente. + + \begin{enumerate}[(i)] + \item Poiché $f \in A(E)$, allora $g \in \GL(V)$, ed è + dunque invertibile. Si considerino i vettori $f(P_i) - f(P_1) = g(P_i - P_1)$ + con $2 \leq i \leq k$. Dal momento che è invertibile, + $g$ mappa vettori linearmente indipendenti a vettori + ancora linearmente indipendenti. + + Allora, poiché i punti + $P_1$, ..., $P_k$ sono affinemente indipendenti, i + vettori $P_i - P_1$ sono linearmente indipendenti per + $2 \leq i \leq k$. Pertanto anche i vettori $g(P_i - P_1) = f(P_i) - f(P_1)$ con $2 \leq i \leq k$ sono linearmente indipendenti, da cui si conclude che i punti $f(P_1)$, ..., + $f(P_k)$ sono affinemente indipendenti. + + \item Dal momento che i punti $P_1$, ..., $P_{n+1}$ sono + affinemente indipendenti, allora i vettori $P_i - P_1$ con + $2 \leq i \leq n+1$ sono linearmente indipendenti, e formano + dunque una base di $V$, essendo tanti quanti la dimensione + di $V$. Analogamente anche i vettori $Q_i - Q_1$ con $2 \leq i \leq n+1$ formano una base di $V$. + + In particolare esiste una sola applicazione lineare $g$ che + associa a $P_i - P_1$ il vettore $Q_i - Q_1$, con $2 \leq i \leq n+1$. Dacché le immagini formano una base di $V$, $g$ è suriettiva, + e dunque, poiché $g \in \End(V)$, $g$ è anche invertibile. + Un'affinità $f \in A(E)$ tale che $f(P_i) = Q_i$ con $1 \leq i \leq n+1$ è per esempio $f(P) = Q_1 + g(P - P_1)$. \\ + + Si mostra che tale $f$ è anche unica. Se esistesse $f' \in A(E)$ + con le stesse proprietà di $f$, varrebbe che $Q_i - Q_1 = f'(P_i) - f'(P_1) = g'(P_i - P_1)$ $\forall 2 \leq i \leq n+1$. Tuttavia + una $g'$ tale che mappi $P_i - P_1$ a $Q_i - P_1$ $\forall 2 \leq i \leq n+1$ è unica, e quindi $g' = g$. Allora $f'(P) = Q_1 + g(P - P_1) = f(P)$ $\forall P \in E$ $\implies f' = f$. + \end{enumerate} + \end{proof} \end{document}