From f168912de67f8fe3ffbf36cd373e49117f14d633 Mon Sep 17 00:00:00 2001 From: Hearot Date: Wed, 3 May 2023 23:24:28 +0200 Subject: [PATCH] feat(geometria): corregge gli errori nella prima parte degli appunti del 27/03/2023 --- .../main.pdf | Bin 234566 -> 235176 bytes .../main.tex | 85 ++++++++++-------- .../1. 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&\text{(}\textbf{indice di positività}\text{)} \\ \iota_-(\varphi) &= \max\{ \dim W \mid W \subseteq V \E \restr{\varphi}{W} < 0 \}, &\text{(}\textbf{indice di negatività}\text{)}\\ - \iota_0(\varphi) &= \dim V^\perp &\text{(}\textbf{indice di nullità}\text{)} + \iota_0(\varphi) &= \dim V^\perp. &\text{(}\textbf{indice di nullità}\text{)} \end{align*} Quando il prodotto scalare $\varphi$ è noto dal contesto, si omette @@ -233,7 +248,7 @@ prodotto $\varphi$. \end{definition} - \begin{theorem} (di Sylvester, caso reale) Sia $\KK$ un campo ordinato + \begin{theorem}[di Sylvester, caso reale] Sia $\KK$ un campo ordinato i cui elementi positivi sono tutti quadrati (e.g.~$\RR$). Allora esiste una base ortogonale $\basis$ tale per cui: @@ -309,7 +324,7 @@ analogamente per gli altri indici. \end{remark} - \begin{definition} (isometria) + \begin{definition}[isometria tra due spazi vettoriali] Dati due spazi vettoriali $(V, \varphi)$ e $(V', \varphi')$ dotati di prodotto scalare sullo stesso campo $\KK$, si dice che $V$ e $V'$ sono \textbf{isometrici} se esiste un isomorfismo @@ -382,11 +397,7 @@ $V'$ sono isometrici. \end{proof} - % \begin{example} - % Per $\varphi = x_1 y_1 + x_2 y_2 - x_3 y_3$. %TODO: guarda dalle slide. - % \end{example} - - \begin{definition} (sottospazio isotropo) + \begin{definition}[sottospazio isotropo] Sia $W$ un sottospazio di $V$. Allora $W$ si dice \textbf{sottospazio isotropo} di $V$ se $\restr{\varphi}{W} = 0$. \end{definition} @@ -408,7 +419,7 @@ da cui $\dim W \leq \frac{1}{2} \dim V$. \end{proof} - \begin{definition} + \begin{definition}[indice di Witt] Si definisce \textbf{indice di Witt} $W(\varphi)$ di $(V, \varphi)$ come la massima dimensione di un sottospazio isotropo. \end{definition} 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 c90b893..573ceea 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 @@ -64,12 +64,12 @@ \begin{definition} Sia\footnote{In realtà, la definizione è facilmente estendibile a qualsiasi campo, purché esso - sia ordinato.} $\KK = \RR$. Allora un prodotto scalare $\varphi$ si dice \textbf{definito positivo} se $\v \in V$, $\vec{v} \neq \vec{0} \implies - \varphi(\vec{v}, \vec{v}) > 0$. Analogamente $\varphi$ è \textbf{definito negativo} se $\vec{v} \neq \vec 0 \implies \varphi(\v, \v) < 0$. In generale si dice che $\varphi$ è \textbf{definito} se è definito positivo o + sia ordinato.} $\KK = \RR$. Allora un prodotto scalare $\varphi$ si dice \textbf{definito positivo} ($\varphi > 0$) se $\v \in V$, $\vec{v} \neq \vec{0} \implies + \varphi(\vec{v}, \vec{v}) > 0$. Analogamente $\varphi$ è \textbf{definito negativo} ($\varphi < 0$) se $\vec{v} \neq \vec 0 \implies \varphi(\v, \v) < 0$. In generale si dice che $\varphi$ è \textbf{definito} se è definito positivo o definito negativo. \\ - Infine, $\varphi$ è \textbf{semidefinito positivo} se $\varphi(\v, \v) \geq 0$ $\forall \v \in V$ (o - \textbf{semidefinito negativo} se invece $\varphi(\v, \v) \leq 0$ $\forall \v \in V$). Analogamente ai + Infine, $\varphi$ è \textbf{semidefinito positivo} ($\varphi \geq 0$) se $\varphi(\v, \v) \geq 0$ $\forall \v \in V$ (o + \textbf{semidefinito negativo}, e quindi $\varphi \leq 0$, se invece $\varphi(\v, \v) \leq 0$ $\forall \v \in V$). Analogamente ai prodotti definiti, si dice che $\varphi$ è \textbf{semidefinito} se è semidefinito positivo o semidefinito negativo. \end{definition} @@ -96,18 +96,28 @@ $\RR^n$. \end{remark} -\begin{definition} +\begin{definition}[vettore isotropo] Un vettore $\vec{v} \in V$ si dice \textbf{isotropo} rispetto al prodotto scalare $\varphi$ se $q(\vec{v}) = \varphi(\vec{v}, \vec{v}) = 0$. \end{definition} +\begin{definition}[cono isotropo] + Si definisce \textbf{cono isotropo} di $V$ rispetto al prodotto scalare $\varphi$ il seguente insieme: + + \[ \CI(\varphi) = \{ \v \in V \mid \varphi(\v, \v) = 0 \}, \] + + \vskip 0.05in + + ossia l'insieme dei vettori isotropi di $V$. +\end{definition} + \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 sono i vettori della forma $(x, y, z)$ tali che $x^2 + y^2 = z^2$, ossia - i vettori stanti sul cono di equazione $x^2 + y^2 = z^2$. + 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 $x^2 + y^2 = z^2$, e quindi $\CI(\varphi)$ è l'insieme dei + vettori stanti sul cono di equazione $x^2 + y^2 = z^2$. \end{example} -\subsection{Matrice associata a \texorpdfstring{$\varphi$}{φ} e congruenza} +\subsection{Matrice associata a \texorpdfstring{$\varphi$}{φ} e relazione di congruenza} \begin{remark} Come già osservato in generale per le applicazioni multilineari, il prodotto scalare è univocamente determinato 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 845ef29592466aed0bd7b6d396e9a7537c77cbd9..99be17190b22548c4fab9c0a947ecc3cccd6ec3a 100644 GIT binary patch delta 31611 zcmV)PK()WxxD?>L6o9k=a5w`nH#wKlg99jk?Oa)t8@CaD-(SHzX;=|q+{fB-m0Yo_ zl%wL1F*=?3P)5Oa{rmE)?sEV;uO9DRKFy3v>~Uf+E36{CE{ zC~?%W+3UA6&J>Z7&5W^xay5Isnth+K`9e{ez1q&X&5rxsd;y=Uhu@0bc0FGRA+mgb z6F%>&A9u&&E?hdy7c5)mn|wcK*?Rut>%V^TmDcS(7mRDd)NCOb;Z}q`Uxhn`Xy3^s zQBu!nxaHn=wJVl5te^1D50s0!G`#9ZnfB8{5Xo5e^s`h4jyPH00j=?}HRwoXhvM#T zecTtz{CTQ*t%+g!^5!?icAf9bf69-4-QDip8ZL+b@-2uEW^1ya%HOx^O*yvpa<^T1 z;ma)FzuVodQSL!wyXSD-$*R%!55;y>;P^oFLK3SiegVHPLD6PXqTBD2zEi0-0trWeT8R87}w^$;7t(pnq zR5^idO8AOsp~1+S+eo<0*vJ|MxB#_9EPdlY-TxXJ$en~Z>2^TWb-&ixH)xJ;ajuOVsnp^LGpdz)j?LTEv zf>3cCY1ISE)6E!drbw%9=6$*uuGv|->m7|A^S#(78s8iDhwqc{0+MFaE$YW|QG+!$ zsW?ZpGo49l8V|`dckra2Q*cf(eO_rF=Rl>5(;!p8t!ah`=zIFyH~aqW9O;S~Qg)p9S;O zt3B$tGeHChq;=QF1f@rgo1KjNsWe!sQ%Q`7v{?2_2uTJ^z^^5j z=c!?v4y(8k?T%N);aHTMxd$gL=ni7B1(1DV-T#5|e-tQ-yb3$J|2sa55#poFvxOV)!QQlh(L~D?wD> z4}h95Ev7T(boPgT$VTRG(y&fNZAis7HL-=HhG@#;5N*o*QWzI*`F)t6x0ahPU*EjH zK|KK%23iBg!t_it?)`Fkck}&^bhd&Yf!YuWJ^bu%y{pZ0GyC@DUpHT1p0YD$&WKdd z6a@B;M=17dqG!-BcMfg`jxFU-&!A`-cib^}!9k5K?s+7Cu44hl1|*{>vEWCSNhZpQ zFl7ML7&Haf6gw+3@-ihl!-?SbDx$ND7%F4Y4bdd#Jd9r@@n@o$ju0YEbQ6)jOoekm z(K+Ltc&?-MD4y%j5s9<>Y9L2s&dU*X^R`uLQ)6pRDIRzdb-?N*d~trpf&CGf{(PYn z&A!HTreK(V&ICaB8BOrP)%pj@*GaI9e9fvU7l)4AoGQlB73>n?DqM5 zAwgmfb4=d*uOVY>s*EvSsxrn&G>4|`@Gb|y4CnQK_znd&s(~~w*D0L{1kakLw^gfz zMh6x=+Z~Ey@nMeNQXRrA;8PgW#@h@N=*u%09LVPLWOiBk{i5 zSxppw*!ERs=3wFAE}oiBjhi+;;#Ejb2gd!XqNWW@DmApcaIglcc|5QeDey*kA5DKE z3>h}cip*K)ozM-}l2CK0Zn_H>fHf?TkqAm!l!HjF0 z|GWoeMYESm0U(X!*~@Ld0bJa@%kOY=v)mPbOZvKnDVbpqWI&5ko3RN9grz~-6aq?T zvhDg^J_3HQ9>9bz0zY9y_E{4A1jHn~1AfA>KJYU{3}S?op7ki9KrW9(KS?CRwvc3+ zi5P)1O<)A>S~7+9$SO-2K|93z0BQuPra0=T@^@

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