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26, Prodotti hermitiani e teorema spettrale/main.tex b/Geometria 1/Prodotto scalare e hermitiano/2023-04-17, 19, 26, Prodotti hermitiani e teorema spettrale/main.tex index f3c8e49..6a2a28d 100644 --- a/Geometria 1/Prodotto scalare e hermitiano/2023-04-17, 19, 26, Prodotti hermitiani e teorema spettrale/main.tex +++ b/Geometria 1/Prodotto scalare e hermitiano/2023-04-17, 19, 26, Prodotti hermitiani e teorema spettrale/main.tex @@ -1532,4 +1532,34 @@ Dacché $\varphi$ è definito positivo, $\varphi(\v, \v) \neq 0 \implies \lambda = -\conj{\lambda}$. Allora $\Re(\lambda) = \frac{\lambda + \conj{\lambda}}{2} = 0$, e quindi $\lambda$ è immaginario, da cui la tesi. \end{proof} + + \begin{exercise} + Sia $V$ uno spazio vettoriale dotato del prodotto $\varphi$. Siano $U$, $W \subseteq V$ due sottospazi + di $V$. Si dimostrino allora le due seguenti + identità. + + \begin{enumerate}[(i)] + \item $(U + W)^\perp = U^\perp \cap W^\perp$, + \item $(U \cap W)^\perp \supseteq U^\perp + W^\perp$, dove vale l'uguaglianza insiemistica se $\varphi$ + è non degenere. + \end{enumerate} + \end{exercise} + + \begin{proof}[Soluzione] + Sia $\v \in (U + W)^\perp$ e siano $\U \in U \subseteq U + W$, $\w \in W \subseteq U + W$. Allora + $\varphi(\v, \U) = 0 \implies \v \in U^\perp$ e $\varphi(\v, \w) = 0 \implies \v \in W^\perp$, + da cui si conclude che $(U + W)^\perp \subseteq U^\perp \cap W^\perp$. Sia adesso + $\v \in U^\perp \cap W^\perp$ e $\v' = \U + \w \in U + W$ con $\U \in V$ e $\w \in W$. Allora + $\varphi(\v, \v') = \varphi(\v, \U) + \varphi(\v, \w) = 0 \implies \v \in (U + W)^\perp$, da cui + si deduca che vale la doppia inclusione, e quindi che $(U + W)^\perp = U^\perp \cap W^\perp$, + dimostrando (i). \\ + + Sia ora $\v' = \U' + \w' \in U^\perp + W^\perp$ con $\U' \in U^\perp$ e $\w' \in W^\perp$. Sia + $\v \in U \cap W$. Allora $\varphi(\v, \v') = \varphi(\v, \U') + \varphi(\v, \w') = 0 \implies + \v' \in (U \cap W)^\perp$, da cui si deduce che $(U \cap W)^\perp \supseteq U^\perp + W^\perp$. + Se $\varphi$ è non degenere, $\dim (U^\perp + W^\perp) = \dim U^\perp + \dim W^\perp - \dim (U^\perp \cap W^\perp) = 2 \dim V - \dim U - \dim W - \dim (U+W)^\perp = \dim V - \dim U - \dim W + \dim (U + W) = + \dim V - \dim (U + W) = \dim (U + W)^\perp$. Valendo pertanto l'uguaglianza dimensionale, si + conclude che in questo caso $(U \cap W)^\perp = U^\perp + W^\perp$, dimostrando (ii). + + \end{proof} \end{document} \ No newline at end of file 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 ecd7add1750fa74576dd3953a5131a93d8912c80..e146778159ae03a15b4293ec4e22f355966bf7e8 100644 GIT binary patch delta 140718 zcma&NbC4&?`mfuzZQHhOYudK$Z*#h*ZBN^_ZQHgvZJ)LF+3W6oe)rsn8&MJU#v2(` zS^rezlbI$;av5aLKdSy?$!fx$tj0884^u?NHG-7__DilV>;&Q}v)E{=vB%H@V#q+O|)_WV?g z^I#}~fhzX#(MT~Be)28%P}0y=Aa@K_s{b?o9vrk02QLFs9$Dq>Kq^RcFe5o^r&Px8 zid)~EtXRob@*nsRwykJO7{dzkfU6z+Y5QLV3#vY#1g8;g$sv&}0n+XIF~r@d zp1ck`9EXLFAig}n8VBTGa(y#BMA-RGhiY>%J!8Cq^;V2>`NM3|fFdRp@1p6i3cxGo z4b`P%cJ-fKB{ax#YC)E1-l^G|U{<+qC_Z?}5rP}fEOM_DRN}Yn*=r9gaZ7#v#2jed zagb7;J0FY;Z?6?Y>2ek10m5aOlhl7>qm!*)nso04@qM7El 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zCc%ghhF@P)VV3eJHMBA`UvLOzRdqF@5*DJ8DgmpG!=Y4=%4$fI0_^{zBL0VtHB3Q) b#n0c~*FVVD$(aShMMM%=L`Bu~)mi=zeyds} diff --git a/Geometria 1/Spazi affini/2023-04-26, Azioni di un gruppo e introduzione agli spazi affini/main.tex b/Geometria 1/Spazi affini/2023-04-26, Azioni di un gruppo e introduzione agli spazi affini/main.tex index 9623590..044550a 100644 --- a/Geometria 1/Spazi affini/2023-04-26, Azioni di un gruppo e introduzione agli spazi affini/main.tex +++ b/Geometria 1/Spazi affini/2023-04-26, Azioni di un gruppo e introduzione agli spazi affini/main.tex @@ -13,13 +13,12 @@ \begin{center} \Large \textbf{Azioni di un gruppo e introduzione agli spazi affini} \end{center} - - \wip + \begin{note} Nel corso delle lezioni si è impiegata la notazione $g.x$ per indicare l'azione di un gruppo su un dato elemento $x \in X$. Tuttavia si è - preferito indicare $g.x$ con $g \cdot x$ nel corso del documento. \\ + preferito utilizzare la notazione $g \cdot x$ nel corso del documento. \\ Inoltre, con $G$ si indicherà un generico gruppo, e con $X$ un generico insieme, sul quale $G$ agisce, qualora non indicato diversamente. @@ -122,21 +121,44 @@ $\exists g \in G \mid g \cdot x = y \implies \tau(g \Stab(x)) = g \cdot x = y$. Siano ora $g$, $g' \in G$ tali che $\tau(g \Stab(x)) = \tau(g' \Stab(x))$, allora $g \cdot x = g' \cdot x \implies (g' g\inv) \cdot x = x \implies g' g\inv \in \Stab(x)$. Pertanto $g \Stab(x) = g' \Stab(x)$, e $\tau$ è allora iniettiva, da cui la tesi. \end{proof} - \begin{remark} Come conseguenza del teorema di orbita-stabilizzatore, + \begin{remark}\nl + \li Come conseguenza del teorema di orbita-stabilizzatore, si osserva che $\abs{G/\Stab(x)} = \abs{\Orb(x)}$, se $\Orb(x)$ è finito, e quindi si conclude, per il teorema di Lagrange, che - $\abs{G} = \abs{\Stab(x)} \abs{\Orb(x)}$. + $\abs{G} = \abs{\Stab(x)} \abs{\Orb(x)}$. \\ + + \li Il teorema di orbita-stabilizzatore implica il primo teorema di omomorfismo. Siano infatti + $G$, $H$ due gruppi e sia $f$ un omomorfismo da $G$ in $H$. Si può allora costruire un azione + di $G$ in $H$ in modo tale che $g \cdot h = f(g) h$ $\forall g \in G$, $h \in H$. Infatti + $e_G \cdot h = f(e_G) h = e_H h = h$ e $g \cdot (g' \cdot h) = g \cdot (f(g') h) = f(g) f(g') h = + f(g g') h = (g g') \cdot h$, $\forall g$, $g' \in G$, $h \in H$. \\ + + Si osserva che $\Stab(e_H) = \Ker f$: + infatti $\Stab(e_H) = \{ g \in G \mid g \cdot e_H = f(g) e_H = f(g) = e_H \} = \Ker f$. Inoltre, + $\Orb(e_H) = \Im f$, dal momento che $\Orb(e_H) = \{ h \in H \mid \exists g \in G \tc g \cdot h = f(g) h = e_H \iff f(g) = h\inv \} = \{ h \in H \mid \exists g \in G \text{ t.c. } f(g) = h \} = \Im f$, dove + si è usato che $h\inv \in \Im f \iff h \in \Im f$. \\ + + Dal momento allora che $\Stab(e_H)$ è il kernel di $f$, vale che $\Stab(e_H) \nsg G$, e quindi + che $G/\Stab(e_H)$ è un gruppo. + Si verifica allora che l'applicazione $\tau$ costruita nella dimostrazione del teorema di orbita-stabilizzatore + è un omomorfismo. Siano infatti $g \Stab(e_H)$, $g' \Stab(e_H) \in G/\Stab(e_H)$, allora + $\tau(g \Stab(e_H) \, g' \Stab(e_H)) = \tau((g g') \Stab(e_H)) = (g g') \cdot e_H = f(g g') e_H = + f(g) e_H f(g') e_H = \tau(g \Stab(e_H)) \, \tau(g' \Stab(e_H))$. \\ + + Si conclude dunque, per il teorema di orbita-stabilizzatore, che $\tau$ è bigettiva, e dunque + che $G/\Ker f = G/\Stab(e_H) \cong \Orb(e_H) = \Im f$, ossia si ottiene la tesi del primo + teorema di omomorfismo. \end{remark} \begin{definition} - Si dice che $G$ \textbf{opera liberamente} su $X$ se - $\forall x \in X$, l'applicazione da $G$ in $\Orb(x)$ tale che + Si dice che $G$ \textbf{opera liberamente} su $X$ se, dato $x \in X$, l'applicazione da $G$ in $X$ tale che $g \mapsto g \cdot x$ è iniettiva, ossia se $\Stab(x) = \{e\}$. \end{definition} \begin{definition} - Si dice che $G$ \textbf{opera transitivamente} su $X$ se $x \sim_G y$ $\forall x$, $y \in X$, cioè se c'è un'unica orbita, che coincide con $X$. In - tal caso si dice che $X$ è \textbf{omogeneo} per l'azione di $G$. + Si dice che $G$ \textbf{opera transitivamente} su $X$ se, dato $x \in X$, l'applicazione da $G$ in $X$ tale che $g \mapsto g \cdot x$ è surgettiva, ossia se $x \sim_G y$ $\forall x$, $y \in X$, cioè se c'è un'unica orbita, che coincide con $X$. In + tal caso si dice che $X$ è un insieme \textbf{omogeneo} per l'azione di $G$ (o semplicemente che è + un \textit{$G$-insieme omogeneo}). \end{definition} \begin{example} Si possono fare alcuni esempi classici di insiemi $X$ @@ -152,8 +174,8 @@ ultimo elemento, deve essere nulla. Allora $O$ deve essere della seguente forma: - \[ O = \Matrix{A & \rvline & 0 \\ \hline 0 & \rvline & 1}, \] - + \[ O = \Matrix{& & & & & & \rvline & 0 \, \\ & & & & & & \rvline & \vdots \, \\ & & & \mbox{\normalfont\Large $A$} & & & \rvline & \vdots \, \\ & & & & & & \rvline & \vdots \, \\ & & & & & & \rvline & \vdots \, \\ \hline & 0 & \cdots & \cdots & \cdots & \cdots & \rvline & 1 \,}, \] + \vskip 0.05in dove $A \in M(n-1, \RR)$. Affinché allora $O$ sia ortogonale, @@ -166,75 +188,129 @@ \begin{definition} Si dice che $G$ \textbf{opera in maniera semplicemente transitiva} su $X$ - se $\exists x \in X$ tale che l'applicazione da $G$ in $X$ $g \mapsto g \cdot x$ è una bigezione, + se, dato $x \in X$, l'applicazione da $G$ in $X$ tale che $g \mapsto g \cdot x$ è una bigezione, ossia se $G$ opera transitivamente e liberamente. \end{definition} \begin{definition} - Un insieme $X$ con un'azione semplicemente transitiva di $G$ è - detto un $G$-insieme omogeneo \textit{principale}. + Un insieme $X$ che subisce un'azione del gruppo $G$ che opera in maniera + semplicemente transitiva è + detto un \textbf{$G$-insieme omogeneo principale}. \end{definition} - + \begin{example} - \begin{enumerate}[(i)] - \item $X = G$. L'azione naturale di $G$ su $X$ per moltiplicazione - è semplicemente transitivo (per $g$, $g' \in G$, esiste un - unico $h \in G$ tale che $g = h.g' = hg'$). Quindi $X$ - è $G$-omogeneo principale. - - \item Se $X$ è $G$-omogeneo principale, l'azione è fedele. - - \item Se $X$ è omogeneo per un gruppo $G$ commutativo, allora - $G$ agisce fedelmente su $X$ $\implies$ $X$ è un $G$-insieme - omogeneo principale. - \end{enumerate} + Se $X = G$ e l'azione considerata è quella naturale dell'operazione di $G$, + tale azione opera in maniera semplicemente transitiva. Dato $x \in X$, si consideri infatti l'applicazione $\tau$ + da $G$ in $G$ tale che + $g \mapsto g \cdot x = gx$. Si osserva che $\tau$ è surgettiva, dacché, dato $h \in G$, + $h = h x\inv x = \tau(h x\inv)$. Inoltre $\tau$ è iniettiva, dal momento che, dati $g$, $g'$ + tali che $\tau(g) = \tau(g')$, allora $gx = g' x \implies g = g'$. Pertanto $\tau$ è bigettiva, e + l'azione opera allora in maniera semplicemente transitiva. \end{example} + + \begin{remark}\nl + \li Se $X$ è $G$-omogeneo principale, l'azione di $G$ su $X$ è fedele. Infatti, $f_g = \Id \implies g \cdot x = x$ $\forall x \in X$. Dal momento però che $X$ è $G$-omogeneo principale, $G$ opera liberamente su $X$, + e quindi $\Stab(x) = \{e\}$ $\forall x \in X \implies g = e$. \\ + + \li Se $X$ è $G$-omogeneo e $G$ è abeliano, allora $G$ agisce fedelmente su $X$ $\iff$ $X$ è $G$-omogeneo principale. \\ + + Se $G$ agisce fedelmente su $X$, dato $x \in X$, si può considerare infatti $g \in \Stab(x) \implies g \cdot x = x$. Si osserva allora + che $f_g = \Id$. Dato infatti $y \in X$, dacché $X$ è $G$-omogeneo, $\exists g' \in G \mid y = g' \cdot x$, + da cui si ricava che $f_g(y) = g \cdot y = g \cdot (g' \cdot x) = (gg') \cdot x = (g'g) \cdot x = g' \cdot (g \cdot x) = g' \cdot x = y$, ossia proprio che $f_g = \Id$. Dal momento però che l'azione di $G$ su $X$ è fedele, + $f_g = \Id \implies g = e$, ossia $\Stab(x) = \{e\}$ $\forall x \in X$, per cui si conclude che l'azione + di $G$ opera in maniera semplicemente transitiva su $X$, e dunque che $X$ è $G$-omogeneo principale. \\ + + Viceversa, se $X$ è $G$-omogeneo principale, $\Stab(x) = \{ e \}$ $\forall x \in X$. Allora, se $f_g = \Id$, + per ogni $x \in X$ deve valere che $g \in \Stab(x) = \{ e \} \implies g = e$. + \end{remark} + + \hr \begin{definition} [spazio affine] Sia $V$ uno spazio vettoriale su un campo $\KK$ qualsiasi. Allora uno spazio affine $E$ associato a $V$ è un qualunque - $V$-insieme omogeneo principale. + $V$-insieme omogeneo principale\footnote{Per gruppo $V$ si intende il gruppo abeliano $(V, +)$.}. + In particolare si indica l'azione di $V$ su $E$ $(\v, P) \mapsto \v \cdot P$ come $P + \v$ (o + analogamente come $\v + P$). Inoltre, gli elementi di $E$ si + diranno \textit{punti di} $E$. \end{definition} - - Pertanto, $\forall P, Q \in E$, esiste un unico vettore $\v \in V$ - tale che $Q = \v . P $, denotato come $Q = P + \v = \v + P$. Si - osserva che $\v + (\w + P) = (\v + \w) + P$. Essendo $\v$ unico, - si scrive $\v = Q - P = \vvec{PQ}$. - - %TODO: aggiunge applicazione bigettiva - Fissato $O \in E$, l'applicazione $\v \mapsto \v + O$, $V \to E$ - è una bigezione. - - \begin{remark}\nl - \li $P-P = \vec 0 \in V$, $P-Q = -(Q-P)$, $(P_3 - P_2) + (P_2 - P_1) = P_3 - P_1$. \\ + \begin{remark} + Dal momento che $E$ è un $V$-insieme omogeneo principale, valgono le seguenti proprietà. - \li $O \in E$ l'applicazione $P \mapsto P-O$ è una bigezione di $E$ - su $V$. + \begin{enumerate}[(i)] + \item Poiché $E$ è omogeneo, per ogni $P \in E$, $Q \in E$ esiste $\v \in V$ tale che $P + \v = Q$. + Inoltre, dal momento che $V$ opera liberamente su $E$, tale $\v$ è unico, e si indica come + $Q - P$ o come $\vvec{PQ}$. + + \item Vale l'identità $P + \vec 0 = P$, dal momento che $\vec 0$ è l'identità del gruppo $(V, +)$ + e l'applicazione $P + \v$ è un azione di $V$. Allo stesso modo, vale che $(P + \v) + \w = P + (\v + \w) = + P + (\w + \v) = (P + \w) + \v$, pertanto si può scrivere, senza alcuna ambiguità, $P + \v + \w$. + + \item Fissato $O \in E$, l'applicazione da $V$ in $E$ tale che $\v \mapsto O + \v$ è una bigezione, + dal momento che $V$ opera su $E$ in maniera semplicemente transitiva. + + \item Analogamente, fissato $O \in E$, l'applicazione $\tau$ da $E$ in $V$ tale che $P \mapsto P - O = \vvec{OP}$ + è una bigezione. Infatti $\tau$ è surgettiva: $\forall \v \in V$, $\tau(O + \v) = (O + \v) - O = \v$, + coerentemente con le operazioni aritmetiche. Infine, $\tau$ è iniettiva: siano $P$, $Q \in E$ tali che + $\tau(P) = \tau(Q)$, allora $P = O + (P - O) = O + \tau(P) = O + \tau(Q) = O + (Q - O) = Q$, per + cui $\tau$ è bigettiva. + + \item Dati $P$, $Q \in E$, vale l'identità $P - Q = -(Q - P)$. Infatti $P = Q + (P-Q) = P + (Q-P) + (P-Q) = + P + ((Q-P) + (P-Q))$. Allora, essendo l'azione di $V$ libera su $E$ (ovvero, come osservato prima, + essendo $\vvec{PP}$ unicamente zero), $(Q-P) + (P-Q) = \vec 0 \implies P-Q = -(Q-P)$. + + \item Dati $P_1$, $P_2$, $P_3 \in E$, vale l'identità $(P_3 - P_2) + (P_2 - P_1) = P_3 - P_1$. Infatti + $P_1 + (P_2 - P_1) + (P_3 - P_2) = P_2 + (P_3 - P_2) = P_3 \implies (P_2 - P_1) + (P_3 - P_2) = P_3 - P_1$. + \end{enumerate} \end{remark} - Siano $P_1$, ..., $P_n \in E$. $\forall \lambda_1$, ..., $\lambda_k \in \KK$. $\forall O \in E$ possiamo individuare il punto $P = O + \sum_{i=1}^n \lambda_i (P_i - O)$. + Siano adesso $P_1$, ..., $P_n$ punti di $E$. Dati $\lambda_1$, ..., $\lambda_n \in \KK$ e $O \in E$ si può allora individuare il punto $P = O + \sum_{i=1}^n \lambda_i (P_1 - O) \in E$. - $P = P' = \iff O + \sum_{i=1}^n \lambda_i (P_i - O) = O' + \sum_{i=1}^n \lambda_i (P_i - O') \iff O + \sum_{i=1}^n \lambda_i (O' - O) = O' \iff - (\sum \lambda_i) (O' - O) = O' - O \iff \sum \lambda_i = 1$. + \begin{proposition} + Dati $P_1$, ..., $P_n$ punti di $E$ e $\lambda_1$, ..., $\lambda_n \in \KK$, il punto + $P(O) = O + \sum_{i=1}^n \lambda_i (P_i - O)$ rappresenta lo stesso identico punto al + variare del punto $O$ se e solo se $\sum_{i=1}^n \lambda_i = 1$. + \end{proposition} - \begin{definition} + \begin{proof} + Siano $O$, $O'$ due punti distinti di $E$. Allora $P(O) = P(O') \iff O + \sum_{i=1}^n \lambda_i (P_i - O) = + O' + \sum_{i=1}^n \lambda_i (P_i - O') = O + (O' - O) + \sum_{i=1}^n \lambda_i (P_i - O') \iff \sum_{i=1}^n \lambda_i (P_i - O) = (O' - O) + \sum_{i=1}^n \lambda_i ((P_i - O) + (O - O'))$. Distribuendo + la somma e utilizzando l'identità dell'\textit{Osservazione} (v), si ottiene allora che $P(O) = P(O') \iff + \sum_{i=1}^n \lambda_i = 1$. + \end{proof} + + \begin{definition} [combinazione affine di punti] Un punto $P \in E$ è \textbf{combinazione affine} dei punti - $P_1$, ..., $P_k$ se $P = O + \sum \lambda_i (P_i - O)$ se - $\sum \lambda_i = 1$. Si scriverà, in particolare, che - $P = \sum \lambda_i P_i$. + $P_1$, ..., $P_n$ se $\exists \lambda_1$, ..., $\lambda_n \in \KK$, $O \in E$ tali che $P = O + \sum_{i=1}^n \lambda_i (P_i - O)$ e che + $\sum_{i=1}^n \lambda_i = 1$. Dal momento che per la precedente proposizione $P$ è invariante al variare di $O \in E$, si scriverà, senza alcuna ambiguità, che + $P = \sum_{i=1}^n \lambda_i P_i$. \end{definition} - - Si chiama retta affine l'insieme dei punti che sono combinazione affine di - due punti. Analogamente si fa per un piano e uno spazio. - \begin{definition} - Un sottoinsieme $D \subseteq E$ si dirà \textbf{sottospazio affine} - se è chiuso per combinazioni affini (finite). + \begin{definition} [sottospazio affine] + Un sottoinsieme $D \subseteq E$ si dice \textbf{sottospazio affine} di $E$ se ogni combinazione + affine di finiti termini di $D$ appartiene a $D$. \end{definition} - - \begin{definition} - Il sottospazio affine $D \subseteq E$ generato da un sottoinsieme $S \subseteq E$ è l'insieme delle combinazioni affini (finite) di punti - di $S$, detto $D = \Aff(S)$. %TODO: mostrare che è chiuso per combinazioni affini. + + \begin{definition} [sottospazio affine generato da alcuni punti] + Dati $S \subseteq E$, si dice \textbf{sottospazio affine generato da $S$} + l'insieme delle combinazioni affini di finiti termini dei punti di $S$, denotato con $\Aff(S)$. \end{definition} + + \begin{remark}\nl + \li Come avviene per $\Span$ nel caso degli spazi vettoriali, dati $P_1$, ..., $P_n \in E$, si usa scrivere $\Aff(P_1, \ldots, P_n)$ + per indicare $\Aff(\{P_1, \ldots, P_n\})$. \\ + + \li Si osserva che in effetti, dato $S \subseteq E$, $\Aff(S)$ è un sottospazio affine, ossia è + chiuso per combinazioni affini dei propri punti. Siano infatti $P_1$, ..., $P_n$ punti di $\Aff(S)$ + e siano $\lambda_1$, ..., $\lambda_n \in \KK$ tali che $\sum_{i=1}^n \lambda_i = 1$. Si deve + mostrare dunque che $\sum_{i=1}^n \lambda_i P_i \in \Aff(S)$. Dal momento che $P_i \in \Aff(S)$ esiste $k_i \in \NN^+$ tale per cui esistano + $S_{i,1}$, ..., $S_{i,k_i} \in S$ e $\lambda_{i,1}$, ..., $\lambda_{i,k_i} \in \KK$ tali per cui + $P_i = \sum_{j=1}^{k_i} \lambda_{i,j} S_{i,j}$ e $\sum_{j=1}^{k_i} \lambda_{i,j} = 1$. Allora + $\sum_{i=1}^n \lambda_i P_i = \sum_{i=1}^n \lambda_i (\sum_{j=1}^{k_i} \lambda_{i,j} S_{i,j}) = + \sum_{i=1}^n \sum_{j=1}^{k_i} \lambda_i \lambda_{i,j} S_{i,j}$. Inoltre $\sum_{i=1}^n \sum_{j=1}^{k_i} \lambda_i \lambda_{i,j} = \sum_{i=1}^n \lambda_i (\sum_{j=1}^{k_i} \lambda_{i,j}) = \sum_{i=1}^n \lambda_i = 1$. + Pertanto $\sum_{i=1}^n \lambda_i P_i$ è combinazione affine di elementi di $S$, e quindi $\sum_{i=1}^n \lambda_i P_i \in \Aff(S)$. \\ + + \li Siano $P_1$, $P_2 \in E$. Allora il sottospazio affine $\Aff(P_1, P_2) = \{ \lambda_1 P_1 + \lambda_2 P_2 \mid \lambda_1 + \lambda_2 = 1, \lambda_1, \lambda_2 \in \KK \} = \{ (1-\lambda) P_1 + \lambda P_2 \mid \lambda \in \KK \} = \{ P_1 + \lambda (P_2 - P_1) \mid \lambda \in \KK \}$ è detto \textit{retta affine passante per $P_1$ e $P_2$}. Analogamente il sottospazio affine generato da tre elementi è detto \textit{piano affine}. + \end{remark} \end{document} diff --git a/tex/latex/style/personal_commands.sty b/tex/latex/style/personal_commands.sty index 8b1be40..b761072 100644 --- a/tex/latex/style/personal_commands.sty +++ b/tex/latex/style/personal_commands.sty @@ -32,6 +32,7 @@ \newcommand{\E}{\text{ e }} \newcommand{\altrimenti}{\text{altrimenti}} \newcommand{\se}{\text{se }} +\newcommand{\tc}{\text{ t.c. }\!} \newcommand{\epari}{\text{ è pari}} \newcommand{\edispari}{\text{ è dispari}}