From 3dd41c4f87c6f72d7e7df105ce37a8ea60eaed57 Mon Sep 17 00:00:00 2001 From: Hearot Date: Wed, 28 Jun 2023 20:23:09 +0200 Subject: [PATCH] =?UTF-8?q?feat(geometria/schede):=20aggiunge=20propriet?= =?UTF-8?q?=C3=A0=20principali=20dei=20prodotti=20hermitiani?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- Geometria 1/Scheda riassuntiva/main.pdf | 1762 ++++++++++++----------- Geometria 1/Scheda riassuntiva/main.tex | 31 +- 2 files changed, 980 insertions(+), 813 deletions(-) diff --git a/Geometria 1/Scheda riassuntiva/main.pdf b/Geometria 1/Scheda riassuntiva/main.pdf index 37ea689..65bd127 100644 --- a/Geometria 1/Scheda riassuntiva/main.pdf +++ b/Geometria 1/Scheda riassuntiva/main.pdf @@ -191,57 +191,61 @@ endobj endobj 128 0 obj << -/Length 9538 +/Length 9352 /Filter /FlateDecode >> stream -x]Y䶑~ׯ7wi ]!=$+쐼U=uȲ~f 2yԡ/S=l $2<ޱ_}F>?⣟R; -͝Vic -&Ow_zQZ}o}| xjֻ0= -Y/LlV-RzZo} D287HKQ?")*+K|2w0\d R¯*˸#[QhʰH_3a_w;aUo}{|?_7fOCmU#1@cU(Y탨Hqn1 TL3,7<" Q:< 4<`%,=_|8%HЬӼ,W._܃01gÌYa`Kf Swm\^JIJ{eipkхQ&.?^M\xlLG.yxxTVPpzz,xX=5=/d)<fB\(*}ς:P]$^666hs*=w20 eI:pjŭ'o-%;آKE-ԢWi]0 -}N5ؖNPYY߀pP^e"^q!?znR:͸eV, -A*Xt fA88z[)0^I"]ګ=Ja왮tF($Ά)7`BsGyzm#o*pN'pal^5h:ocS@FC#VEi-]J{P/.lޢcME'~Ǣ#',A=+_j6=Ap:٦ J~S -J1N~n)`(A> '^zv3Tx$t/v9txܹ?DO' u1:Ud~pAk*<82*] %tR#pe:/^RCi/#_D#rdOGjj8R[,pRJxs6{Eg7`e ۓl&2/ 7^MVMW.{/aNwM;㘩n!xYu #*n/Z73S*X+7YD6u}dr3T)`iKe 60Ox.|&6:=R mW]NOy'eoKĢwM{C- O򥭂j(Um Jkʘ5QPȘ-(qdJXs 6)JXm{D=oB/da=b4ۂPf3, LDX+kjTbɰA3i@\'[=4Р͙u9]`K~Fꮸ7 -x5M"wt<5mcC<,f*sS94u,^6ɬi!nU"΃ǡ낗@rrЬu yϞ_xj=o9M]~yr7|>a0I3>*ܼo߶˪ am$fxW]Z8-1YE>nv;ؘ C3E\-N$$bO.AOB}_hz =ǼkojWK'wCS*3bb/*jޣ˽"¢H0* -/M8H:z5\EQd8glYY1w3i3 ~1S7&X,﫠X],ߡ>+YJjcM+Is "䔻mbf(YRU6ZulYfJjCj-]J%gngtE[Ե"@ -Lh{qŶfiK iFY攰"9gy{ڽNh( aHYhv;;|E6kvjL?>2S &ߪ뉻Ipw~qa`]ET`H/<2S$nigcKNoGkw%g_GGKcuA8=D܏у ^YJR׀uMM.)8fjq7V.@e?ĤG.\vx+\*XΨL _|Q}K=ǡ'.s]#^I -1'2Easߎx[5J)3 -mS@qan*hp[I@sScQ7d`LtuGk Ϋȷ Ξ,hggi677u& 3ayhsATdB:Kݒ Tu6zn8 )!XIic.M!M֯Mc{XXïى/v|y3rr LئY_[+1<)ſH`H8h׉o/~KL/OQTӮ3it{2atDZƁSP J[_cݮGbLHxc%6R63ȧ-KMZ@KR$C[;ln]~I:yD<57rnܚaN62xfѤne0wIvD UOo]2I^Rvҕ%sg@M/r.&HB/iX -tD Oc)]b郤*=qzr6<]HXb}Ev_j3i{#reu-sT,t@ċPvsS0[@ܹ8%r( X]O]A|9lV`}- 1,vȇr-kOΣ~ -) .eQk ^܄}=&ƒ]p}tQDtKf9HGc?wH'yk{1 H7&sqJu} -dW]<3ϊg$:Qq -VmJ"&'^R^xm]u0-2\Z 1ҫ/m)C9ÎAx ]/KTpS9[O |.. -Bi"O~C'-"sKqN S[cM0ҥ. 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+0000410993 00000 n +0000411031 00000 n +0000411159 00000 n trailer -<< /Size 366 -/Root 364 0 R -/Info 365 0 R -/ID [<3C8CAB4B5DF1DD09132A005CCB557A93> <3C8CAB4B5DF1DD09132A005CCB557A93>] >> +<< /Size 377 +/Root 375 0 R +/Info 376 0 R +/ID [ ] >> startxref -399264 +411433 %%EOF diff --git a/Geometria 1/Scheda riassuntiva/main.tex b/Geometria 1/Scheda riassuntiva/main.tex index 073fcaf..1db6140 100644 --- a/Geometria 1/Scheda riassuntiva/main.tex +++ b/Geometria 1/Scheda riassuntiva/main.tex @@ -179,6 +179,8 @@ di $V$ contenenti tali vettori, \item $\Span(\basis) = V$, \item $\Span(\emptyset) = \zerovecset$, + \item $U \cap W$ è sempre un sottospazio se $U$ e $W$ sono due sottospazi di $V$, + \item $U \cup W$ è un sottospazio se e solo se $U \subseteq W$ o $U \supseteq W$ (e quindi $U \cup W = U$ o $U \cup W = W$), \item dato $X$ generatore di $V$, $X \setminus \{\vec{x_0}\}$ genera $V \iff \vec{x_0} \in \Span(X \setminus \{\vec{x_0}\})$, \item $X \subseteq Y$ è un sottospazio di $Y \iff \Span(X) = X$, @@ -1863,12 +1865,37 @@ \end{enumerate} Un prodotto hermitiano $\varphi$ si comporta pressoché come un prodotto - scalare su $\RR$. Infatti tale prodotto soddisfa le seguenti proprietà: + scalare su $\RR$ (le definizioni principali sono infatti le medesime). Se $\basis$ è una base di $V$, la matrice associata $M_\basis(\varphi)$ è definita in + modo tale che $M_\basis(\varphi)_{ij} = \varphi(\vv i, \vv j)$. Infatti tale prodotto soddisfa le seguenti proprietà: \begin{itemize} - \item + \item $\varphi(\lambda \v + \w, \U) = \conj{\lambda} \varphi(\v, \U) + \varphi(\w, \U)$, $\forall \v$, $\w$, $\U \in V$, $\lambda \in \CC$, + \item $V^\perp = []_\basis \inv (\Ker M_\basis(\varphi))$, + \item $\varphi$ è non degenere se e solo se $\Ker M_\basis(\varphi) = \zerovecset$, + \item $\dim W + \dim W^\perp = \dim V + \dim (W \cap V^\perp)$ (formula delle dimensioni), + \item $\varphi(\v, \w) = [\v]_\basis^* M_\basis(\varphi) [\w]_\basis$, + \item $M_{\basis'}(\varphi) = \left(M_\basis^{\basis'}(\Idv)\right)^* M_\basis(\varphi) \, M_\basis^{\basis'}(\Idv)$ (formula del cambiamento di base), + \item si può definire una relazione di equivalenza analoga alla congruenza: $A \sim_* B \defiff \exists M \in \GL(n, \CC) \mid A = M^* B M$, + \item $\varphi$ è completamente determinato dalla sua forma quadratica $q$ secondo le seguenti due formule di polarizzazione: + \begin{itemize} + \item $q(\v + \w) - q(\v) - q(\w) = 2 \Re(\varphi(\v, \w))$, + \item $q(\v + i \w) - q(\v) - q(\w) = 2 i \imm(\varphi(\v, \w))$, + \end{itemize} + \item esiste sempre una base ortogonale per $\varphi$ (\textit{teorema di Lagrange}), + \item vale il teorema di Sylvester reale e la segnatura in senso hermitiano è un invariante per la relazione $\sim_*$, + \item $\varphi > 0 \iff \sigma(\varphi) = (n, 0, 0)$, + \item $\varphi < 0 \iff \sigma(\varphi) = (0, n, 0)$, + \item $\varphi \geq 0 \iff \sigma(\varphi) = (n-k, 0, k)$, dove $k = \dim V^\perp = \dim \Ker M_\basis(\varphi)$, + \item $\varphi \leq 0 \iff \sigma(\varphi) = (0, n-k, k)$, dove $k = \dim V^\perp = \dim \Ker M_\basis(\varphi)$. \end{itemize} + Esiste un unico modo per complessificare un prodotto scalare $\varphi$, ossia + esiste un unico prodotto hermitiano $\varphi_\CC$ tale per cui $\varphi_\CC(\v, \w) = \varphi(\v, \w)$ se $\v$, $\w$ sono vettori della parte reale dello spazio complessificato. In particolare $\varphi_\CC$ è determinato dalla seguente + formula: + \begin{multline*} + \varphi_\CC(\vv 1 + i \vv 2, \ww 1 + i \ww 2) = \varphi(\vv 1, \ww 1) + \varphi(\vv 2, \ww 2) \\ + i (\varphi(\vv 1, \ww 2) - \varphi(\vv 2, \ww 1)). + \end{multline*} + \subsubsection{Funzionali rappresentabili} Un funzionale $f \in \dual V$ si dice rappresentabile tramite $\varphi$ se