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1/Spazi affini/2023-05-05, Affinità e spazio proiettivo/main.tex index da13867..fd79b95 100644 --- a/Geometria 1/Spazi affini/2023-05-05, Affinità e spazio proiettivo/main.tex +++ b/Geometria 1/Spazi affini/2023-05-05, Affinità e spazio proiettivo/main.tex @@ -290,7 +290,7 @@ \item Come prima, si dimostra l'identità mostrando che vale la doppia inclusione dei due spazi vettoriali. Sia $\vec u \in D_0 \cap D_0'$. Sia $P \in D \cap D'$. Allora esiste $P_1 \in D$ tale che $\vec u = P - P_1$. Analogamente esiste $P_2 \in D'$ tale che $\vec u = P - P_2$. Poiché - $E$ è $V$-omogeneo, esiste un solo punto $P'$ tale che $P = P' + \vec u$. Si conclude dunque + $V$ agisce liberamente su $E$, esiste un solo punto $P'$ tale che $P = P' + \vec u$. Si conclude dunque che $P_1 = P_2$, e dunque che $P_1$ appartiene anche a $D'$. Pertanto $\vec u \in (D \cap D')_0 \implies D_0 \cap D_0' \subseteq (D \cap D')_0$. @@ -300,15 +300,105 @@ (D \cap D')_0 \subseteq D_0 \cap D_0'$, da cui si conclude che $(D \cap D')_0 = D_0 \cap D_0'$. \qedhere \end{enumerate} \end{proof} + + \begin{definition} [somma affine] + Siano $D$ e $D'$ due sottospazi affini di $E$. Si definisce + allora la \textbf{somma affine} $D + D'$ come $\Aff(D \cup D')$. + \end{definition} \begin{proposition} [formula di Grassmann per i sottospazi affini] Siano $D$ e $D'$ due sottospazi affini di $E$ con $D \cap D' \neq \emptyset$. Allora - $\dim \Aff(D \cup D') = \dim D + \dim D' - \dim (D \cap D')$. + $\dim (D + D') = \dim D + \dim D' - \dim (D \cap D')$. \end{proposition} \begin{proof} Per la proposizione precedente, $\dim \Aff(D \cup D') = \dim (D_0 + D_0')$. Allora, applicando la formula di Grassmann per i sottospazi vettoriali, $\dim (D_0 + D_0') = \dim D_0 + \dim D_0' - \dim (D_0 \cap D_0') = \dim D + \dim D' - \dim (D_0 \cap D_0')$. Sempre per la proposizione precedente, - $D_0 \cap D_0' = (D \cap D')_0$, da cui si deduce che $\dim (D_0 \cap D_0') = \dim (D \cap D')_0 = \dim D \cap D'$. Pertanto $\dim \Aff(D \cup D') = \dim D + \dim D' - \dim (D \cap D')$. + $D_0 \cap D_0' = (D \cap D')_0$, da cui si deduce che $\dim (D_0 \cap D_0') = \dim (D \cap D')_0 = \dim D \cap D'$. Pertanto $\dim (D + D') = \dim \Aff(D \cup D') = \dim D + \dim D' - \dim (D \cap D')$. \end{proof} + + \begin{remark} + Si definisce $\ell_{P,Q} = \{ \lambda P + (1-\lambda) Q \mid \lambda \in \KK \}$ con $P$, $Q \in E$ come la retta passante per due + punti. Allora, in generale, se $D$ e $D'$ sono due sottospazi + affini di $E$, $D + D' = \bigcup_{\substack{P \in D \\ Q \in D'}} \ell_{P,Q}$. \\ + + Infatti ogni elemento di $\ell_{P, Q}$ è una combinazione affine + di due elementi di $D + D'$, e quindi appartiene a $D + D' \implies + D + D' \supseteq \bigcup_{\substack{P \in D \\ Q \in D'}} \ell_{P,Q}$. \\ + + Infine, se $T \in D + D'$, esistono $\lambda_1$, ..., $\lambda_k$, + $\mu_1$, ..., $\mu_{k'} \in \KK$, $P_1$, ..., $P_k \in D$ e + $Q_1$, ..., $Q_{k'} \in D'$ tali che $T = \sum_{i=1}^k \lambda_i P_i + + \sum_{j=1}^{k'} \mu_j Q_j$\footnote{Al più $T$ è un elemento di solo $D$ o $D'$. In tal caso $T$ appartiene già a una qualsiasi retta passante per $T$. Pertanto si può anche assumere successivamente + che $\alpha$, $\beta \neq 0$ -- se infatti uno dei due parametri + fosse nullo, $T$ apparterrebbe a $D$ o $D'$.}, con $\sum_{i=1}^k \lambda_i + \sum_{j=1}^{k'} \mu_j = 1$. Se $\alpha = \sum_{i=1}^k \lambda_i$ e + $\beta = \sum_{j=1}^{k'} \mu_j$, si può riscrivere $T$ nel + seguente modo: + + \[ T = \alpha \underbrace{\sum_{i=1}^k \frac{\lambda_i}{\alpha} P_i}_{=\, P'} + + \beta \underbrace{\sum_{j=1}^{k'} \frac{\mu_j}{\beta} Q_j}_{=\, Q'}, \] + + \vskip 0.05in + + dove si osserva che $P' \in D$, essendo combinazione affine di + elementi di $D$, e che analogamente $Q' \in D'$. Allora + $T$ giace sulla retta passante per $P'$ e per $Q'$, ossia + $T \in \ell_{P',Q'} \implies D + D' \subseteq \bigcup_{\substack{P \in D \\ Q \in D'}} \ell_{P,Q}$. + \end{remark} + + \begin{remark} + Siano fissati $P_0 \in D$ e $P_0' \in D'$, e siano $P \in D$ e + $Q \in D'$. Allora vale la seguente identità: + + \[ P - Q = \underbrace{(P - P_0)}_{\in \, D_0} + \underbrace{(P_0 - P_0')}_{\in \, \Span(P_0 - P_0')} + \underbrace{(P_0' - Q)}_{\in \, D_0'}. \] + + \vskip 0.05in + + Si osserva che in generale vale che $(D + D')_0 = D_0 + D_0' + \Span(P_0 - P_0')$. Chiaramente vale che $(D + D')_0 \supseteq D_0 + D_0' + \Span(P_0 - P_0')$, dal momento che $D_0$, $D_0'$ e $\Span(P_0 - P_0')$ sono tutti sottospazi vettoriali di $(D + D')_0$. \\ + + Sia ora $P' \in D + D'$. Allora esistono $P'' \in D$, $Q'' \in Q$ tali + per cui $P' \in \ell_{P'', Q''}$, e quindi esiste $\lambda \in \KK$ + per cui $P' = P'' + \lambda (Q'' - P'')$. Poiché $P'' \in D$, + esiste $\v \in D_0$ tale per cui $P'' = P_0 + \v$. Allora + $P' - P_0 = \v - \lambda (P'' - Q'') \in D_0 + D_0' + \Span(P_0 - P_0')$, pertanto $(D + D')_0 \subseteq D_0 + D_0' + \Span(P_0 - P_0')$, + da cui si conclude che $(D + D')_0 = D_0 + D_0' + \Span(P_0 - P_0')$. + \end{remark} + + \begin{proposition} [formula di Grassmann modificata] + Se $D \cap D' = \emptyset$, allora $\dim (D + D') = \dim D + \dim D' - + \dim (D_0 \cap D_0') + 1$. + \end{proposition} + + \begin{proof} + Dalla precedente osservazione, vale che $(D + D')_0 = D_0 + D_0' + \Span(P_0 - P_0')$. Si dimostra che $P_0 - P_0' \notin D_0 + D_0'$. + Se infatti $P_0 - P_0'$ appartenesse a $D_0 + D_0'$, esisterebbero + $P \in D$, $Q \in D'$ tali per cui $P_0 - P_0' = (P_0 - P) + (Q - P_0')$. \\ + + Allora, facendo agire questo vettore su $P_0'$, + $P_0 = Q + (P_0 - P)$. Tuttavia, poiché l'azione di $V$ su $E$ + è un'azione di gruppo, esiste un solo punto $P'$ tale per + cui $P_0 = P' + (P_0 - P)$, e in particolare $P' = P$. Pertanto + $P = Q \implies P \in D'$. Tuttavia $D \cap D' = \emptyset$, \Lightning. Pertanto $P_0 - P_0' \notin D_0 + D_0'$. + In particolare questo equivale a constatare che + $(D_0 + D_0') \cap \Span(P_0 - P_0') = \zerovecset$, + ossia ad osservare che: + + \[ (D + D')_0 = D_0 + D_0' \oplus \Span(P_0 - P_0'). \] + + \vskip 0.1in + + Si conclude dunque che $\dim (D + D') = \dim (D_0 + D_0') + \dim \Span(P_0 - P_0') = + \dim D + \dim D' - \dim (D_0 \cap D_0') + 1$, da cui la tesi. + \end{proof} + + \begin{remark} + In generale vale che $\Span(P_0 - P_0') \subseteq D_0 + D_0' \iff + D \cap D' \neq \emptyset$. Infatti $\Span(P_0 - P_0') \subseteq D_0 + D_0' \iff D \cap D' \neq \emptyset$, come appena dimostrato. + Inoltre, se $D \cap D' \neq \emptyset$, esiste un punto + $P \in D \cap D'$. Allora $P_0 - P_0' = \underbrace{(P_0 - P)}_{\in \, D_0} + \underbrace{(P - P_0')}_{\in \, D_0'} \implies \Span(P_0 - P_0') \subseteq D_0 + D_0'$. \\ + + Si poteva dunque dimostrare la \textit{formula di Grassmann} (non + modificata, per $D \cap D' \neq \emptyset$) utilizzando questa osservazione, così come si sarebbe + potuto dimostrare che $(D + D')_0 = D_0 + D_0'$. + \end{remark} \end{document} diff --git a/Geometria 1/Teoria spettrale degli endomorfismi/2023-03-29, Es. Computo delle basi di Jordan/main.pdf b/Geometria 1/Teoria spettrale degli endomorfismi/2023-03-29, Es. 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Computo delle basi di Jordan/main.tex b/Geometria 1/Teoria spettrale degli endomorfismi/2023-03-29, Es. Computo delle basi di Jordan/main.tex index 63bb7f3..ea6ce6b 100644 --- a/Geometria 1/Teoria spettrale degli endomorfismi/2023-03-29, Es. Computo delle basi di Jordan/main.tex +++ b/Geometria 1/Teoria spettrale degli endomorfismi/2023-03-29, Es. Computo delle basi di Jordan/main.tex @@ -29,7 +29,7 @@ devono esservi obbligatoriamente $2$ blocchi di ordine $1$. Pertanto la sua forma canonica è la seguente: - \[ J=\Matrix{0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0}. \] + \[ J=\Matrix{0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0}. \] \vskip 0.05in @@ -95,10 +95,16 @@ \end{proposition} \begin{proof} - Sia $B \in C(J_{0, m})$. Si osserva che $J_{0,m} B = \Matrix{B_2 \\ \hline B_3 \\ \hline \vdots \\ \hline B_{m} \\ \hline 0}$, mentre $B J_{0,m} = \Matrix{0 & \rvline & B^1 & \rvline & B^2 & \rvline & \cdots & \rvline & B^{m-1} }$. Per ipotesi deve valere che $J_{0,m} B = B J_{0,m}$, e quindi, uguagliando le matrici colonna a colonna, si osserva + Sia $B \in C(J_{0, m})$. Si osserva che vale la seguente identità: + + \[ J_{0,m} B = \Matrix{B_2 \\ \hline B_3 \\ \hline \vdots \\ \hline B_{m} \\ \hline 0}, \] + + \vskip 0.05in + + mentre $B J_{0,m} = \Matrix{0 & \rvline & B^1 & \rvline & B^2 & \rvline & \cdots & \rvline & B^{m-1} }$. Per ipotesi deve valere che $J_{0,m} B = B J_{0,m}$, e quindi, uguagliando le matrici colonna a colonna, si osserva la colonna $B^1$ è tutta nulla eccetto per il primo elemento; si osserva poi che la colonna $B^2$ è composta da elementi di $B^1$ traslata in basso di una posizione; e così via ciclando sulle colonne, ottenendo che, - data $B^m = \Vector{ a_0 \\ a_1 \\ \vdots \\ a_{m-1} }$, $B = a_0 I + a_1 J_{0,m} + \ldots + a_{m-1} J_{0, m}^{m-1}$, + data $B^m = \Vector{ a_0 & a_1 & \cdots & a_{m-1} }^\top$, $B = a_0 I + a_1 J_{0,m} + \ldots + a_{m-1} J_{0, m}^{m-1}$, quindi $B \in \Span(I, J_{0, m}, J_{0, m}^2, ..., J_{0, m}^{m-1})$. Dal momento che ogni elemento generatore di $\Span(I, J_{0, m}, J_{0, m}^2, ..., J_{0, m}^{m-1})$ commuta con $J_{0,m}$, vale la doppia inclusione, da cui la tesi.