From 07c1575f6f77ee02b28895c9d05d1a4b75dcf683 Mon Sep 17 00:00:00 2001 From: Hearot Date: Wed, 3 May 2023 18:05:52 +0200 Subject: [PATCH] fix(geometria): corregge gli errori degli appunti del 22/03/2023 --- .../main.pdf | Bin 202575 -> 224862 bytes .../main.tex | 67 +++++++++-------- .../1. 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$\varphi$ di $V$, ad ogni vettore $\vec{v} \in V$ si associa una \textbf{forma quadratica} - $q : V \to \KK$ tale che $q(\vec{v}) = \varphi(\vec{v}, \vec{v})$. + Ad un dato prodotto scalare $\varphi$ di $V$ si associa una mappa + $q : V \to \KK$, detta \textbf{forma quadratica}, tale che $q(\vec{v}) = \varphi(\vec{v}, \vec{v})$. \end{definition} \begin{remark} @@ -100,31 +100,29 @@ forma bilineare simmetrica, ossia un prodotto scalare su $\KK^n$. \begin{example} Rispetto al prodotto scalare $\varphi : \RR^3 \to \RR$ tale che $\varphi((x_1, x_2, x_3), (y_1, y_2, y_3)) = - x_1 y_1 + x_2 y_2 - x_3 y_3$, i vettori isotropi $(x, y, z)$ sono quelli tali che $x^2 + y^2 = z^2$, ossia - i vettori stanti sul cono di eq.~$x^2 + y^2 = z^2$. + x_1 y_1 + x_2 y_2 - x_3 y_3$, i vettori isotropi sono i vettori della forma $(x, y, z)$ tali che $x^2 + y^2 = z^2$, ossia + i vettori stanti sul cono di equazione $x^2 + y^2 = z^2$. \end{example} \begin{remark} - Come già osservato in generale per le app.~multilineari, il prodotto scalare è univocamente determinato + Come già osservato in generale per le applicazioni multilineari, il prodotto scalare è univocamente determinato dai valori che assume nelle coppie $\vv{i}, \vv{j}$ estraibili da una base $\basis$. Infatti, se $\basis = (\vv1, ..., \vv{k})$, $\vec{v} = \sum_{i=1}^k \alpha_i \vv{i}$ e $\vec{w} = \sum_{i=1}^k \beta_i \vv{i}$, allora: - \[ \varphi(\vec{v}, \vec{w}) = \sum_{1 \leq i \leq j \leq k} \alpha_i \beta_j \, \varphi(\vv{i}, \vv{j}). \] + \[ \varphi(\vec{v}, \vec{w}) = \sum_{i=1}^k \sum_{j=1}^k \alpha_i \beta_j \, \varphi(\vv{i}, \vv{j}). \] \end{remark} \begin{definition} - Sia $\varphi$ un prodotto scalare di $V$ e sia $\basis = (\vv1, ..., \vv{n})$ una base ordinata di $V$. Allora si denota con \textbf{matrice associata} - a $\varphi$ la matrice: + Sia $\varphi$ un prodotto scalare di $V$ e sia $\basis = (\vv1, ..., \vv{n})$ una base ordinata di $V$. Allora si definisce la \textbf{matrice associata} + a $\varphi$ come la matrice: \[ M_\basis(\varphi) = (\varphi(\vv{i}, \vv{j}))_{i,\,j = 1\text{---}n} \in M(n, \KK). \] \end{definition} - \begin{remark} - Si possono fare alcune osservazioni riguardo $M_\basis(\varphi)$. \\ - - \li $M_\basis(\varphi)$ è simmetrica, infatti $\varphi(\vv{i}, \vv{j}) = \varphi(\vv{j}, \vv{i})$ per - definizione di prodotto scalare, \\ + \begin{remark}\nl + \li $M_\basis(\varphi)$ è simmetrica, infatti $\varphi(\vv{i}, \vv{j}) = \varphi(\vv{j}, \vv{i})$, + dal momento che il prodotto scalare è simmetrico, \\ \li $\varphi(\vec{v}, \vec{w}) = [\vec{v}]_\basis^\top M_\basis(\varphi) [\vec{w}]_\basis$. \end{remark} @@ -140,10 +138,10 @@ forma bilineare simmetrica, ossia un prodotto scalare su $\KK^n$. \end{proof} \begin{definition} - Si definisce \textbf{congruenza} la relazione di equivalenza $\cong$ definita nel seguente + Si definisce \textbf{congruenza} la relazione di equivalenza $\cong$ (denotata anche come $\equiv$) definita nel seguente modo su $A, B \in M(n, \KK)$: - \[ A \cong B \iff \exists P \in GL(n, \KK) \mid A = P^\top A P. \] + \[ A \cong B \defiff \exists P \in GL(n, \KK) \mid A = P^\top A P. \] \end{definition} \begin{remark} @@ -151,7 +149,7 @@ forma bilineare simmetrica, ossia un prodotto scalare su $\KK^n$. \li $A = I^\top A I \implies A \cong A$ (riflessione), \\ \li $A \cong B \implies A = P^\top B P \implies B = (P^\top)\inv A P\inv = (P\inv)^\top A P\inv \implies B \cong A$ (simmetria), \\ - \li $A \cong B \implies A = P^\top B P$, $B \cong C \implies B = Q^\top C Q$, quindi $A = P^\top Q^\top C Q P = + \li $A \cong B$, $B \cong C$ $\implies A = P^\top B P$, $B = Q^\top C Q$, quindi $A = P^\top Q^\top C Q P = (QP)^\top C (QP) \implies A \cong C$ (transitività). \end{remark} @@ -163,14 +161,14 @@ forma bilineare simmetrica, ossia un prodotto scalare su $\KK^n$. endomorfismo sono sempre simili). \li Se $A$ e $B$ sono congruenti, $A = P^\top B P \implies \rg(A) = \rg(P^\top B P) = \rg(BP) = \rg(B)$, dal momento che $P$ e $P^\top$ sono invertibili; quindi il rango è un invariante per congruenza. Allora - è ben definito il rango $\rg(\varphi)$ di un prodotto scalare come il rango di una sua qualsiasi matrice - associata. + si può ben definire il rango $\rg(\varphi)$ di un prodotto scalare come il rango della matrice + associata di $\varphi$ in una qualsiasi base di $V$. \\ \li Se $A$ e $B$ sono congruenti, $A = P^\top B P \implies \det(A) = \det(P^\top B P) = \det(P^\top) \det(B) \det(P)= - \det(P)^2 \det(B)$. Quindi, per $\KK = \RR$, il segno del determinante è invariante per congruenza. + \det(P)^2 \det(B)$. Quindi, per $\KK = \RR$, il segno del determinante è un altro invariante per congruenza. \end{remark} \begin{definition} - Si dice \textbf{radicale} di un prodotto scalare $\varphi$ lo spazio: + Si definisce il \textbf{radicale} di un prodotto scalare $\varphi$ come lo spazio: \[ V^\perp = \{ \vec{v} \in V \mid \varphi(\vec{v}, \vec{w}) = 0 \, \forall \vec{w} \in V \} \] @@ -178,7 +176,9 @@ forma bilineare simmetrica, ossia un prodotto scalare su $\KK^n$. \end{definition} \begin{remark} - Il radicale di $\RR^n$ con il prodotto scalare canonico ha dimensione nulla, dal momento che $\forall \vec{v} \in \RR^n \setminus \{\vec{0}\}$, $q(\vec{v}) = \varphi(\vec{v}, \vec{v}) > 0$. + Il radicale del prodotto scalare canonico su $\RR^n$ ha dimensione nulla, dal momento che $\forall \vec{v} \in \RR^n \setminus \{\vec{0}\}$, $q(\vec{v}) = \varphi(\vec{v}, \vec{v}) > 0 \implies \v \notin V^\perp$. In + generale ogni prodotto scalare definito positivo (o negativo) è non degenere, dal momento che ogni vettore + non nullo non è isotropo, e dunque non può appartenere a $V^\perp$. \end{remark} \begin{definition} @@ -186,14 +186,19 @@ forma bilineare simmetrica, ossia un prodotto scalare su $\KK^n$. dimensione non nulla. \end{definition} - %TODO: spiegare perché \alpha_\varphi è lineare e aggiungere esempi nella parte precedente. - %TODO: aggiungere osservazioni sul radicale (i.e. che è uno spazio, che ogni suo vettore è isotropo, ...). - \begin{remark} - Si definisce l'applicazione lineare $\alpha_\varphi : V \to \dual{V}$ in modo tale che - $\alpha_\varphi(\vec{v}) = p$, dove $p(\vec{w}) = \varphi(\vec{v}, \vec{w})$. \\ - - Allora $V^\perp$ altro non è che $\Ker \alpha_\varphi$. Se $V$ ha dimensione finita, $\dim V = \dim \dual{V}$, + Sia $\alpha_\varphi : V \to \dual{V}$ la mappa\footnote{In letteratura questa mappa, se invertibile, è nota come \textit{isomorfismo musicale}, ed è in realtà indicata come $\flat$.} tale che + $\alpha_\varphi(\vec{v}) = p$, dove $p(\vec{w}) = \varphi(\vec{v}, \vec{w})$ $\forall \v$, $\w \in V$. \\ + + Si osserva che $\alpha_\varphi$ è un'applicazione lineare. Infatti, $\forall \v$, $\w$, $\U \in V$, + $\alpha_\varphi(\v + \w)(\U) = \varphi(\v + \w, \U) = \varphi(\v, \U) + \varphi(\w, \U) = + \alpha_\varphi(\v)(\U) + \alpha_\varphi(\w)(\U) \implies \alpha_\varphi(\v + \w) = \alpha_\varphi(\v) + \alpha_\varphi(\w)$. Inoltre $\forall \v$, $\w \in V$, $\lambda \in \KK$, $\alpha_\varphi(\lambda \v)(\w) = + \varphi(\lambda \v, \w) = \lambda \varphi(\v, \w) = \lambda \alpha_\varphi(\v)(\w) \implies + \alpha_\varphi(\lambda \v) = \lambda \alpha_\varphi(\v)$. + + Si osserva inoltre che $\Ker \alpha_\varphi$ raccoglie tutti + i vettori $\v \in V$ tali che $\varphi(\v, \w) = 0$ $\forall \w \in W$, ossia esattamente i vettori di $V^\perp$, per cui si conclude che $V^\perp = \Ker \alpha_\varphi$ (per cui $V^\perp$ è effettivamente uno + spazio vettoriale). Se $V$ ha dimensione finita, $\dim V = \dim \dual{V}$, e si può allora concludere che $\dim V^\perp > 0 \iff \Ker \alpha_\varphi \neq \{\vec{0}\} \iff \alpha_\varphi$ non è invertibile (infatti lo spazio di partenza e di arrivo di $\alpha_\varphi$ hanno la stessa dimensione). In particolare, $\alpha_\varphi$ non è invertibile se e solo se $\det(\alpha_\varphi) = 0$. \\ @@ -205,6 +210,6 @@ forma bilineare simmetrica, ossia un prodotto scalare su $\KK^n$. $M_{\basisdual}^\basis(\alpha_\varphi) = M_\basis(\varphi)$. \\ Si conclude allora che $\varphi$ è degenere se e solo se $\det (M_\basis(\varphi)) = 0$ e che - $V^\perp \cong \Ker M_\basis(\varphi)$ con l'isomorfismo è il passaggio alle coordinate. + $V^\perp \cong \Ker M_\basis(\varphi)$ mediante l'isomorfismo del passaggio alle coordinate. \end{remark} \end{document} diff --git a/Geometria 1/Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/1. Introduzione al prodotto scalare.tex b/Geometria 1/Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/1. Introduzione al prodotto scalare.tex index 62c9782..a87ce1d 100644 --- a/Geometria 1/Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/1. Introduzione al prodotto scalare.tex +++ b/Geometria 1/Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/1. Introduzione al prodotto scalare.tex @@ -1,4 +1,4 @@ -\chapter{Introduzione al prodotto scalare} +\chapter{Il prodotto scalare} \begin{note} Nel corso del documento, per $V$, qualora non specificato, si intenderà uno spazio vettoriale di dimensione @@ -14,7 +14,7 @@ \li $\varphi(A + A', B) = \tr((A + A')B) = \tr(AB + A'B) = \tr(AB) + \tr(A'B) = \varphi(A, B) + \varphi(A', B)$ (linearità nel primo argomento), \\ - \li $\varphi(\alpha A, B) = \tr(\alpha A B) = \alpha \tr(AB) = \alpha \varphi(A, B)$ (omogeneità nel secondo argomento), \\ + \li $\varphi(\alpha A, B) = \tr(\alpha A B) = \alpha \tr(AB) = \alpha \varphi(A, B)$ (omogeneità nel primo argomento), \\ \li $\varphi(A, B) = \tr(AB) = \tr(BA) = \varphi(B, A)$ (simmetria), \\ \li poiché $\varphi$ è simmetrica, $\varphi$ è lineare e omogenea anche nel secondo argomento, e quindi è una forma bilineare simmetrica, ossia un prodotto scalare su $M(n, \KK)$. @@ -63,15 +63,15 @@ \begin{example} Il prodotto scalare canonico di $\RR^n$ è definito positivo: infatti $\varphi((x_1, ..., x_n), (x_1, ..., x_n)) = - \sum_{i=1}^n x_i^2 = 0 \iff x_i = 0$, $\forall 1 \leq i \leq n$ $\iff (x_1, ..., x_n) = \vec{0}$. \\ + \sum_{i=1}^n x_i^2 > 0$, se $(x_1, ..., x_n) \neq \vec 0$. \\ Al contrario, il prodotto scalare $\varphi : \RR^2 \to \RR$ tale che $\varphi((x_1, x_2), (y_1, y_2)) = x_1 y_1 - x_2 y_2$ non è definito positivo: $\varphi((x, y), (x, y)) = 0$, $\forall$ $(x, y) \mid x^2 = y^2$, ossia se $y = x$ o $y = -x$. \end{example} \begin{definition} - Dato un prodotto scalare $\varphi$ di $V$, ad ogni vettore $\vec{v} \in V$ si associa una \textbf{forma quadratica} - $q : V \to \KK$ tale che $q(\vec{v}) = \varphi(\vec{v}, \vec{v})$. + Ad un dato prodotto scalare $\varphi$ di $V$ si associa una mappa + $q : V \to \KK$, detta \textbf{forma quadratica}, tale che $q(\vec{v}) = \varphi(\vec{v}, \vec{v})$. \end{definition} \begin{remark} @@ -86,31 +86,29 @@ \begin{example} Rispetto al prodotto scalare $\varphi : \RR^3 \to \RR$ tale che $\varphi((x_1, x_2, x_3), (y_1, y_2, y_3)) = - x_1 y_1 + x_2 y_2 - x_3 y_3$, i vettori isotropi $(x, y, z)$ sono quelli tali che $x^2 + y^2 = z^2$, ossia - i vettori stanti sul cono di eq.~$x^2 + y^2 = z^2$. + x_1 y_1 + x_2 y_2 - x_3 y_3$, i vettori isotropi sono i vettori della forma $(x, y, z)$ tali che $x^2 + y^2 = z^2$, ossia + i vettori stanti sul cono di equazione $x^2 + y^2 = z^2$. \end{example} \begin{remark} - Come già osservato in generale per le app.~multilineari, il prodotto scalare è univocamente determinato + Come già osservato in generale per le applicazioni multilineari, il prodotto scalare è univocamente determinato dai valori che assume nelle coppie $\vv{i}, \vv{j}$ estraibili da una base $\basis$. Infatti, se $\basis = (\vv1, ..., \vv{k})$, $\vec{v} = \sum_{i=1}^k \alpha_i \vv{i}$ e $\vec{w} = \sum_{i=1}^k \beta_i \vv{i}$, allora: - \[ \varphi(\vec{v}, \vec{w}) = \sum_{1 \leq i \leq j \leq k} \alpha_i \beta_j \, \varphi(\vv{i}, \vv{j}). \] + \[ \varphi(\vec{v}, \vec{w}) = \sum_{i=1}^k \sum_{j=1}^k \alpha_i \beta_j \, \varphi(\vv{i}, \vv{j}). \] \end{remark} \begin{definition} - Sia $\varphi$ un prodotto scalare di $V$ e sia $\basis = (\vv1, ..., \vv{n})$ una base ordinata di $V$. Allora si denota con \textbf{matrice associata} - a $\varphi$ la matrice: + Sia $\varphi$ un prodotto scalare di $V$ e sia $\basis = (\vv1, ..., \vv{n})$ una base ordinata di $V$. Allora si definisce la \textbf{matrice associata} + a $\varphi$ come la matrice: \[ M_\basis(\varphi) = (\varphi(\vv{i}, \vv{j}))_{i,\,j = 1\text{---}n} \in M(n, \KK). \] \end{definition} -\begin{remark} - Si possono fare alcune osservazioni riguardo $M_\basis(\varphi)$. \\ - - \li $M_\basis(\varphi)$ è simmetrica, infatti $\varphi(\vv{i}, \vv{j}) = \varphi(\vv{j}, \vv{i})$ per - definizione di prodotto scalare, \\ +\begin{remark}\nl + \li $M_\basis(\varphi)$ è simmetrica, infatti $\varphi(\vv{i}, \vv{j}) = \varphi(\vv{j}, \vv{i})$, + dal momento che il prodotto scalare è simmetrico, \\ \li $\varphi(\vec{v}, \vec{w}) = [\vec{v}]_\basis^\top M_\basis(\varphi) [\vec{w}]_\basis$. \end{remark} @@ -126,10 +124,10 @@ \end{proof} \begin{definition} - Si definisce \textbf{congruenza} la relazione di equivalenza $\cong$ definita nel seguente + Si definisce \textbf{congruenza} la relazione di equivalenza $\cong$ (denotata anche come $\equiv$) definita nel seguente modo su $A, B \in M(n, \KK)$: - \[ A \cong B \iff \exists P \in GL(n, \KK) \mid A = P^\top A P. \] + \[ A \cong B \defiff \exists P \in GL(n, \KK) \mid A = P^\top A P. \] \end{definition} \begin{remark} @@ -137,7 +135,7 @@ \li $A = I^\top A I \implies A \cong A$ (riflessione), \\ \li $A \cong B \implies A = P^\top B P \implies B = (P^\top)\inv A P\inv = (P\inv)^\top A P\inv \implies B \cong A$ (simmetria), \\ - \li $A \cong B \implies A = P^\top B P$, $B \cong C \implies B = Q^\top C Q$, quindi $A = P^\top Q^\top C Q P = + \li $A \cong B$, $B \cong C$ $\implies A = P^\top B P$, $B = Q^\top C Q$, quindi $A = P^\top Q^\top C Q P = (QP)^\top C (QP) \implies A \cong C$ (transitività). \end{remark} @@ -149,14 +147,14 @@ endomorfismo sono sempre simili). \li Se $A$ e $B$ sono congruenti, $A = P^\top B P \implies \rg(A) = \rg(P^\top B P) = \rg(BP) = \rg(B)$, dal momento che $P$ e $P^\top$ sono invertibili; quindi il rango è un invariante per congruenza. Allora - è ben definito il rango $\rg(\varphi)$ di un prodotto scalare come il rango di una sua qualsiasi matrice - associata. + si può ben definire il rango $\rg(\varphi)$ di un prodotto scalare come il rango della matrice + associata di $\varphi$ in una qualsiasi base di $V$. \\ \li Se $A$ e $B$ sono congruenti, $A = P^\top B P \implies \det(A) = \det(P^\top B P) = \det(P^\top) \det(B) \det(P)= - \det(P)^2 \det(B)$. Quindi, per $\KK = \RR$, il segno del determinante è invariante per congruenza. + \det(P)^2 \det(B)$. Quindi, per $\KK = \RR$, il segno del determinante è un altro invariante per congruenza. \end{remark} \begin{definition} - Si dice \textbf{radicale} di un prodotto scalare $\varphi$ lo spazio: + Si definisce il \textbf{radicale} di un prodotto scalare $\varphi$ come lo spazio: \[ V^\perp = \{ \vec{v} \in V \mid \varphi(\vec{v}, \vec{w}) = 0 \, \forall \vec{w} \in V \} \] @@ -164,7 +162,9 @@ \end{definition} \begin{remark} - Il radicale di $\RR^n$ con il prodotto scalare canonico ha dimensione nulla, dal momento che $\forall \vec{v} \in \RR^n \setminus \{\vec{0}\}$, $q(\vec{v}) = \varphi(\vec{v}, \vec{v}) > 0$. + Il radicale del prodotto scalare canonico su $\RR^n$ ha dimensione nulla, dal momento che $\forall \vec{v} \in \RR^n \setminus \{\vec{0}\}$, $q(\vec{v}) = \varphi(\vec{v}, \vec{v}) > 0 \implies \v \notin V^\perp$. In + generale ogni prodotto scalare definito positivo (o negativo) è non degenere, dal momento che ogni vettore + non nullo non è isotropo, e dunque non può appartenere a $V^\perp$. \end{remark} \begin{definition} @@ -172,14 +172,19 @@ dimensione non nulla. \end{definition} -%TODO: spiegare perché \alpha_\varphi è lineare e aggiungere esempi nella parte precedente. -%TODO: aggiungere osservazioni sul radicale (i.e. che è uno spazio, che ogni suo vettore è isotropo, ...). - \begin{remark} - Si definisce l'applicazione lineare $\alpha_\varphi : V \to \dual{V}$ in modo tale che - $\alpha_\varphi(\vec{v}) = p$, dove $p(\vec{w}) = \varphi(\vec{v}, \vec{w})$. \\ - - Allora $V^\perp$ altro non è che $\Ker \alpha_\varphi$. Se $V$ ha dimensione finita, $\dim V = \dim \dual{V}$, + Sia $\alpha_\varphi : V \to \dual{V}$ la mappa\footnote{In letteratura questa mappa, se invertibile, è nota come \textit{isomorfismo musicale}, ed è in realtà indicata come $\flat$.} tale che + $\alpha_\varphi(\vec{v}) = p$, dove $p(\vec{w}) = \varphi(\vec{v}, \vec{w})$ $\forall \v$, $\w \in V$. \\ + + Si osserva che $\alpha_\varphi$ è un'applicazione lineare. Infatti, $\forall \v$, $\w$, $\U \in V$, + $\alpha_\varphi(\v + \w)(\U) = \varphi(\v + \w, \U) = \varphi(\v, \U) + \varphi(\w, \U) = + \alpha_\varphi(\v)(\U) + \alpha_\varphi(\w)(\U) \implies \alpha_\varphi(\v + \w) = \alpha_\varphi(\v) + \alpha_\varphi(\w)$. Inoltre $\forall \v$, $\w \in V$, $\lambda \in \KK$, $\alpha_\varphi(\lambda \v)(\w) = + \varphi(\lambda \v, \w) = \lambda \varphi(\v, \w) = \lambda \alpha_\varphi(\v)(\w) \implies + \alpha_\varphi(\lambda \v) = \lambda \alpha_\varphi(\v)$. + + Si osserva inoltre che $\Ker \alpha_\varphi$ raccoglie tutti + i vettori $\v \in V$ tali che $\varphi(\v, \w) = 0$ $\forall \w \in W$, ossia esattamente i vettori di $V^\perp$, per cui si conclude che $V^\perp = \Ker \alpha_\varphi$ (per cui $V^\perp$ è effettivamente uno + spazio vettoriale). Se $V$ ha dimensione finita, $\dim V = \dim \dual{V}$, e si può allora concludere che $\dim V^\perp > 0 \iff \Ker \alpha_\varphi \neq \{\vec{0}\} \iff \alpha_\varphi$ non è invertibile (infatti lo spazio di partenza e di arrivo di $\alpha_\varphi$ hanno la stessa dimensione). In particolare, $\alpha_\varphi$ non è invertibile se e solo se $\det(\alpha_\varphi) = 0$. \\ @@ -191,5 +196,5 @@ $M_{\basisdual}^\basis(\alpha_\varphi) = M_\basis(\varphi)$. \\ Si conclude allora che $\varphi$ è degenere se e solo se $\det (M_\basis(\varphi)) = 0$ e che - $V^\perp \cong \Ker M_\basis(\varphi)$ con l'isomorfismo è il passaggio alle coordinate. + $V^\perp \cong \Ker M_\basis(\varphi)$ mediante l'isomorfismo del passaggio alle coordinate. \end{remark} \ No newline at end of file diff --git a/Geometria 1/Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/main.pdf b/Geometria 1/Prodotto scalare e hermitiano/I prodotti di uno spazio vettoriale/main.pdf index bc53281ce51132694a9a634374dee3f3391c404f..7073f907a2e70dc6554fa96e3bafc9b1318d4b89 100644 GIT binary patch delta 307404 zcmZs?Q;;r95GB~QZQHi(zHQsK@wI*1wr|_EZQHhu`DbEwXEtIVD(jrAe9MZ=tgNb= z3&y($rhq130b&qlt~8|xU}`{fV!gs36Oz!Y|M1@yv8_jLg)8((bu$qv9iuuF(9{JA 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Introduzione al prodotto scalare} - + \newpage \thispagestyle{empty} ~\newpage @@ -35,5 +35,4 @@ ~\newpage \include{3. Prodotti hermitiani, spazi euclidei e teorema spettrale} - \end{document} diff --git a/Geometria 1/Spazi affini/2023-04-26, Azioni di un gruppo e introduzione agli spazi affini/main.pdf b/Geometria 1/Spazi affini/2023-04-26, Azioni di un gruppo e introduzione agli spazi affini/main.pdf index ab3b5241606ea952842255652efbb40d00e00507..273d33f13d024e855535fe41dcae4deff7f17333 100644 GIT binary patch delta 5616 zcmai02{e^k8y1o&9g#>GqiFhubI$jjqD&bgxs)l&Tx7~T<&z=vaGgR_#>k~ehG;+{ zm5XvS>nb6+R7eT=x9;u!`TT4BXD#cT_1OD;_xrrh`|O?3!*{iZueg>w`I#=O9P=GF zgH5@=jw#E76MausRajN!Nd{l-nOeZAK&aHIyo#Q-`i2+7p*vey1 z-}rDlU1$IGfO&5kSC(2uR$*WB>`>tEZ8kl(d+0@Tr|6$QeCd#yx^b8yZ(Px~ty)bt zDM81kj&U-fz(Hq1+|tXnujmWDRYYCT05#1`d;2k}Ky6Y$!$)-IA8ORI?xKmQ-+Jxe z?q{xJzUPr;=m^xV+tzBVx30|8T{qNHIyx?un%5_EoIN&7Aa#28%O3F){Ikp`UfI+2 zagWMzG0_TZ|HjX@qZLeDUJ;@q*EepB?emTCwqgo08}5zIRPu}#{^HhUXp;JS`HDWc ze|`S(3?=zs8+rHWlM!4|Ic6cZ&)FIof8yiI57`ywso6erG;a(0r;N~F?wg{jf(MIe zE#_Z19rUV)GZm@Ny0?EkutPuguD-xAb4ytHg?1!jX9shzp5qdUxs_6uMNns`F7Bn*NLw}gg15a}??mX1YfJJw06X)HisRJg%VRhQ3h>3MJ}=I~?s+rpG55n@^GYicPfk2{T4?qIFaD9`ZC+^n7*1RsYVaZ%ZKVGry_ja6E; zO3j7~J6?r=IQi;wkP{i}ns zoK=&4SKEhG0<^1ZiUfY|IdJaF=ZVO?LG@b13H9Uf=pIZLN~wEsRl82*8_Tv=Ol*xh z5$gOjcFVkqt+|NiNXgY4bd$Yg&iL~WHvI=6Zl3r=hz{a z@^f?|U%mE<)v15jlv=k-rahBgTUx4N{pA(?qZGmTcfM-7R*G*>KJ&Y$X_h9}Cb#S9 z*FMl?MjBV-s0H_37kOsyr3L5j_%kl)$Q1#h@T_;Eb-W%6CLd?>1I`p3PBhv--%!L{ zOM&sZEGPcI(h9Fe%V%F`&$L`o-$9MlEs>BvSLEPzQ!ttJSG#=AOpYxR`}Jt8EmPC; z>8>AB@~qCTW%^SXL|@iH156UN72&kLCSbZinUV<(QTvi(81I+oxWd8gFZpt%I|=3I zwuBuJbN{QPbdcj|#_B6l`vk|OH@ECq$?EQHt~eSXc;mwwY)dslgpPHWT;uU3CU$jo zSLq3`_lj6-_3aYkSK%$2ELIO5SjY29O~v&ulT!QX{T+Hdo1KY2HwAJDE2!GrG{0`+ zx!YI71HCOYFd6nZn_S{B}G*jCU=4jf{hOYaJ4_m!< zx)x{W)_AV+AD>>BKi|=l@mpg|gyYSpi8m=WHJ3+ooz!=!Xj2ZS@8}V7N(-l`HbicG zQhBD(@R@OJtiG4F2ti}t!5$ir)Z99L;*r&YF}=3t`sPfw_u`I|s#`ku(>fK)yyA95 zH=GG-pRek@QrI;RVlh;KX7B!LYNNsEu)Ly< z$Jp*aUU~R+TUyjlzY_~qHiu^Guih^m%KpQkc>B!jw3M}Vald&Ja-TE;BQK7m2{+i@ z=j%z-c@fYYrdy_Kt7XDC(!J?xTt&zAtdbCU*UFY2)k?X()wnC=?Wq6Lx$?QIg>$PE zbJO*?yS2lIx;ps3#qc_gem*s6zRtN))hk0YmAR4%c%W?0o{0S`W^Pm|(kPN=mzrG6`}REriS&FY(IdP+UzV*7kjp4x+^ zU};`9RDnJR$rq?8?u<~MJayRQ&mp&T4Tzw}oIE3{(5)7vQ1bNcNrGqFm5rP9*Ihp{ zAtQC#VQq8^cJPAtxRrJ-X7_T`Vzud{OTRx!2-LoQ9UmLML^^@$u; zUiz?l)^_$#}D`*H5XQh0+#tD6FH>ZtAI6 z3rB7K3c_Dto=FjuCf?k#WIr9V={A)4QPL;1qpp;)XVL4S{q^RsO9qmFPHT9pZlkKa`47poVY3(9+lOR>7o zC3Hty3sij-&_M873tF1X*LU1*qrsnVw)dbT z`*Vt*PVYxT4+j!@3q0f-Pa|0=b-q!rY!q1Rj|!C758QN(=Z_y?4htS7j_S>XsRnXB z4IBeG;-=}0lCb-Crz5AkKUM8&saIF1l)Wg3`A*c^*%q4Bn^itB(?l-J2b4bH3*3-q z;GUh!pzZw-6_u>mVrHN1NjhL+|KehXfnoP*L3AJDNJYXbqJN! 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