From 2e76fde9ad33885b5f2ffc0f4855eeaf30fc29cc Mon Sep 17 00:00:00 2001 From: Hearot Date: Thu, 22 Jun 2023 21:26:43 +0200 Subject: [PATCH] fix(geometria): modifica alcuni dettagli sulle isometrie --- .../1. Introduzione al prodotto scalare.tex | 40 +++++++++--------- .../main.pdf | Bin 615279 -> 682899 bytes 2 files changed, 20 insertions(+), 20 deletions(-) diff --git a/Geometria 1/4. Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/1. Introduzione al prodotto scalare.tex b/Geometria 1/4. Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/1. Introduzione al prodotto scalare.tex index 1749f4a..e33cf85 100644 --- a/Geometria 1/4. Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/1. Introduzione al prodotto scalare.tex +++ b/Geometria 1/4. Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/1. Introduzione al prodotto scalare.tex @@ -848,7 +848,7 @@ Si osserva infine che, se $V = \RR^3$ e $\basis$ ne è la base canonica, i coni \begin{proof} Senza perdità di generalità si assuma $\iota_-(\varphi) \leq \iota_+(\varphi)$ (il caso $\iota_-(\varphi) > \iota_+(\varphi)$ è analogo). Sia $W$ un sottospazio con $\dim W > \iota_-(\varphi)$. Sia $W^+$ un sottospazio con $\dim W^+ = \iota_+(\varphi)$ e $\restr{\varphi}{W^+} > 0$. Allora, per la formula - di Grassmann, $n \geq \dim (W + W^+) = \dim W + \dim W^+ - \dim (W \cap W^+) > n - \dim (W \cap W^+) \implies \dim (W \cap W^+) > 0$. Quindi $\exists \w \in W$, $\w \neq \vec 0$ tale che $\varphi(\w, \w) > 0$, da cui + di Grassmann, $n - \dim(W \cap W^+) < \dim (W + W^\perp) \leq n \implies \dim (W \cap W^+) > 0$. Quindi $\exists \w \in W$, $\w \neq \vec 0$ tale che $\varphi(\w, \w) > 0$, da cui si ricava che $W$ non è isotropo. Pertanto $W(\varphi) \leq \iota_-(\varphi)$. \\ Sia $a := \iota_+(\varphi)$ e sia $b := \iota_-(\varphi)$. @@ -874,11 +874,14 @@ Si osserva infine che, se $V = \RR^3$ e $\basis$ ne è la base canonica, i coni \[ \varphi(\vec v, \vec w) = \varphi'(f(\vec v), f(\vec w)). \] \end{definition} -\begin{exercise} Sia $f : V \to V'$ un isomorfismo. Allora $f$ è un'isometria $\iff$ $\forall$ base $\basis = \{ \vv 1, \ldots, \vv n \}$ di $V$, $\basis' = \{ f(\vv 1), \ldots, f(\vv n) \}$ è una base di $V'$ e $\varphi(\vv i, \vv j) = \varphi'(f(\vv i), f(\vv j))$ $\forall 1 \leq i, j \leq n$ $\iff$ $\exists$ base $\basis = \{ \vv 1, \ldots, \vv n \}$ di $V$ tale che $\basis' = \{ f(\vv 1), \ldots, f(\vv n) \}$ è una base di $V'$ e $\varphi(\vv i, \vv j) = \varphi'(f(\vv i), f(\vv j))$ $\forall 1 \leq i, j \leq n$. -\end{exercise} +\vskip 0.01in -\begin{solution} Se $f$ è un'isometria, detta $\basis$ una base di $V$, $\basis' = f(\basis)$ è una base di $V'$ - dal momento che $f$ è anche un isomorfismo. Inoltre, dacché $f$ è un'isometria, vale sicuramente che +\begin{proposition} Sia $f : V \to V'$ un isomorfismo. Allora $f$ è un'isometria $\iff$ $\forall$ base $\basis = \{ \vv 1, \ldots, \vv n \}$ di $V$, $\basis' = \{ f(\vv 1), \ldots, f(\vv n) \}$ è una base di $V'$ e $\varphi(\vv i, \vv j) = \varphi'(f(\vv i), f(\vv j))$ $\forall 1 \leq i, j \leq n$ $\iff$ $\exists$ base $\basis = \{ \vv 1, \ldots, \vv n \}$ di $V$ tale che $\basis' = \{ f(\vv 1), \ldots, f(\vv n) \}$ è una base di $V'$ e $\varphi(\vv i, \vv j) = \varphi'(f(\vv i), f(\vv j))$ $\forall 1 \leq i, j \leq n$. + \label{prop:isometrie_base_sufficiente} +\end{proposition} + +\begin{proof} Se $f$ è un'isometria, detta $\basis$ una base di $V$, $\basis' = f(\basis)$ è una base di $V'$ + dal momento che $f$ è prima di tutto un isomorfismo. Inoltre, dacché $f$ è un'isometria, vale sicuramente che $\varphi(\vv i, \vv j) = \varphi'(f(\vv i), f(\vv j))$ $\forall 1 \leq i, j \leq n$. \\ Sia ora assunto per ipotesi che $\forall$ base $\basis = \{ \vv 1, \ldots, \vv n \}$ di $V$, $\basis' = \{ f(\vv 1), \ldots, f(\vv n) \}$ è una base di $V'$ e $\varphi(\vv i, \vv j) = \varphi'(f(\vv i), f(\vv j))$ $\forall 1 \leq i, j \leq n$. Allora, analogamente a prima, detta $\basis = \{ \vv 1, \ldots, \vv n \}$ una base di $V$, $\basis' = f(\basis)$ è una base di $V'$, e in quanto tale, @@ -892,10 +895,10 @@ Si osserva infine che, se $V = \RR^3$ e $\basis$ ne è la base canonica, i coni \sum_{i=1}^n \sum_{j=1}^n a_i b_j \, \varphi(\vv i, \vv j) = \varphi(\v, \w), \] da cui la tesi. -\end{solution} +\end{proof} \begin{proposition} Sono equivalenti le seguenti affermazioni: - + \label{prop:isometrie_equivalenza_base} \begin{enumerate}[(i)] \item $V$ e $V'$ sono isometrici; \item $\forall$ base $\basis$ di $V$, base $\basis'$ di $V'$, @@ -912,11 +915,11 @@ Si osserva infine che, se $V = \RR^3$ e $\basis$ ne è la base canonica, i coni Si dimostra ora (iii) $\implies$ (i). Per ipotesi $M_\basis(\varphi) \cong M_{\basis'}(\varphi')$, quindi $\exists P \in \GL(n, \KK) \mid M_{\basis'}(\varphi') = P^\top M_\basis(\varphi) P$. Allora $\exists$ $\basis''$ base di $V'$ tale che $P = M_{\basis''}^{\basis'}(\Idv)$, da cui $P\inv = M_{\basis'}^{\basis''}(\varphi)$. Per la formula di cambiamento di base del prodotto - scalare, $M_{\basis''}(\varphi) = (P\inv)^\top M_{\basis'} P\inv = M_\basis(\varphi)$. Detta + scalare, $M_{\basis''}(\varphi') = (P\inv)^\top M_{\basis'} P\inv = M_\basis(\varphi)$. Detta $\basis'' = \{ \ww 1, \ldots, \ww n \}$, si costruisce allora l'isomorfismo $f : V \to V'$ tale che $f(\vv i) = \ww i$ $\forall 1 \leq i \leq n$.. Dal momento che per costruzione $M_\basis(\varphi) = M_{\basis''}(\varphi')$, $\varphi(\vv i, \vv j) = \varphi'(\ww i, \ww j) = \varphi'(f(\vv i), f(\vv j))$ $\forall 1 \leq i, j \leq n$. - Si conclude dunque che $\varphi(\v, \w) = \varphi'(f(\v), f(\w))$ $\forall \v, \w \in V$, e dunque + Si conclude dunque, dalla \textit{Proposizione \ref{prop:isometrie_base_sufficiente}}, che $\varphi(\v, \w) = \varphi'(f(\v), f(\w))$ $\forall \v, \w \in V$, e dunque che $f$ è un'isometria, come desiderato dalla tesi. \end{proof} @@ -925,15 +928,12 @@ Si osserva infine che, se $V = \RR^3$ e $\basis$ ne è la base canonica, i coni isometrici $\iff$ $\varphi$ e $\varphi'$ hanno la stessa segnatura. \end{proposition} -\begin{proof}\nl\nl - \rightproof Per la precedente proposizione, esistono due basi $\basis$ e $\basis'$, una di $V$ e una di $V'$, - tali che $M_\basis(\varphi) \cong M_{\basis'}(\varphi)$. Allora queste due matrici condividono la stessa - segnatura, e così quindi anche $\varphi$ e $\varphi'$. \\ - - \leftproof Se $\varphi$ e $\varphi'$ hanno la stessa segnatura, esistono due basi $\basis = \{ \vv 1, \ldots, \vv n \}$ e $\basis' = \{ \ww 1, \ldots, \ww n \}$, una - di $V$ e una di $V'$, tali che $M = M_\basis(\varphi) = M_{\basis'}(\varphi')$ e che $M$ è una matrice di - Sylvester. Allora si costruisce $f : V \to V'$ tale che $f(\vv i) = \ww i$. Esso è un isomorfismo, e per - costruzione $\varphi(\vv i, \vv j) = \varphi'(\ww i, \ww j) = \varphi'(f(\vv i), f(\vv j))$ $\forall 1 \leq i, j \leq n$, da cui - si conclude che $\varphi(\v, \w) = \varphi'(f(\v), f(\w))$ $\forall \v$, $\w \in V$, e quindi che $V$ e - $V'$ sono isometrici. +\begin{proof}Si dimostrano le due implicazioni separatamente.\nl\nl + \rightproof Per la \textit{Proposizione \ref{prop:isometrie_equivalenza_base}}, esistono due basi $\basis$ e $\basis'$, una di $V$ e una di $V'$, + tali che $M_\basis(\varphi) \cong M_{\basis'}(\varphi')$. Allora, poiché queste due matrici sono congruenti, esse devono condividere anche la stessa + segnatura, che è invariante completo per congruenza, e dunque le segnature di $\varphi$ e di $\varphi'$ coincidono. \\ + + \leftproof Se $\varphi$ e $\varphi'$ hanno la stessa segnatura, allora, detta $\basis$ + una base di $V$ e $\basis'$ una base di $V'$, $M_\basis(\phi) \cong M_{\basis'}(\phi')$. + Allora, per la \textit{Proposizione \ref{prop:isometrie_equivalenza_base}}, $V$ e $V'$ sono isometrici. \end{proof} \ No newline at end of file diff --git a/Geometria 1/4. Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/main.pdf b/Geometria 1/4. 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