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z(#X{WM;_^*G7tc=V^AruEI{t*Q^-?s7SaD-6=N20aeQ|Vb2pDyZdTU#5HJV=hTwB> J$g0cX|3B{NrrrPm diff --git a/Geometria 1/Prodotto scalare e hermitiano/Appunti originali/2023-04-17, 19, 26, Prodotti hermitiani e teorema spettrale/main.tex b/Geometria 1/Prodotto scalare e hermitiano/Appunti originali/2023-04-17, 19, 26, Prodotti hermitiani e teorema spettrale/main.tex index 007794f..27d8c67 100644 --- a/Geometria 1/Prodotto scalare e hermitiano/Appunti originali/2023-04-17, 19, 26, Prodotti hermitiani e teorema spettrale/main.tex +++ b/Geometria 1/Prodotto scalare e hermitiano/Appunti originali/2023-04-17, 19, 26, Prodotti hermitiani e teorema spettrale/main.tex @@ -1429,11 +1429,7 @@ \dim V - \rg A_{1, \ldots, k}$, dimostrando (iii). \\ Se $\varphi$ è non degenere, $A$ è invertibile, dacché $\dim V^\perp = \dim \Ker A = 0$. Allora - ogni minore di taglia $k$ di $A$ ha determinante diverso da zero. Dacché ogni minore di taglia $k$ - di $A_{1,\ldots,k}$ è anche un minore di taglia $k$ di $A$, si ricava che anche ogni minore di taglia - $k$ di $A_{1, \ldots, k}$ ha determinante diverso da zero, e quindi che $\rg(A_{1,\ldots,k}) \geq k$. - Dacché deve anche valere $\rg(A_{1,\ldots,k}) \leq \min\{k,n\} = k$, si conclude che $\rg(A_{1,\ldots,k})$ - vale esattamente $k = \dim W$. Allora, dal punto (iii), vale che $\dim W^\perp + \dim W = \dim W^\perp + \rg(A_{1,\ldots,k}) = \dim V$, dimostrando il punto (iv). + $\rg(A_{1,\ldots,k}) = k = \dim W$, dal momento che le prime $k$ righe di $A$ devono essere linearmente indipendenti. Allora, dal punto (iii), vale che $\dim W^\perp + \dim W = \dim W^\perp + \rg(A_{1,\ldots,k}) = \dim V$, dimostrando il punto (iv). \end{proof} \begin{exercise} diff --git a/Geometria 1/Scheda riassuntiva/main.pdf b/Geometria 1/Scheda riassuntiva/main.pdf index 0ba68a77007ddcc29b9c676b7f0ff0f09f7a3d41..881a6f2c4643588d1c0b58d8f84cb912f5192b6c 100644 GIT binary patch delta 68908 zcmZ^qb9iOjvbST~?%1|%b=a|O8!NVLJL%X~$96{@+eRn(y7xKz+}`(m_m7M@#+Yl> znDy4LYCg|;Dn|#XwLej7VS&N4nON8xAb~kh6B$uq5SbaN2>JP8Ol*zq3@u?e6A{7D zA*8e!85y18zY$}Z{GMquvi|yGWOV*LIQ||C5;uU!65~*z5)nb@;4G;A^L3`gZ7>W_ zEl%b{Av6q3RzgO?-&T2fVHjjhY|Wg_3AtF9IZ|hXz$gLdeBTrHn&U5DB7B|P%rl1I z=T@!IQqb$WHbo>>h%87r7qc*i7ghOOqR4#SOap+R<3hX!*b;=1m@_zO`E1_=7GRTvv$q+!RfHM%@erwF@_qkW)?2IUC|cVM#aYl zFY#XVC=890_n>JfBL_2!HV2{&btE-x@L0=62CnwR$}Jqst5Q$-8TV@A;G7;;E;g@~orZ&`X=lL*g0ev-bvc)@eycA3oCUTW(=g;Klz)@UEXeeDo6amM0CM= z<@Jmk&|L%~Pw%;xESMU%>?YfQjAE_vAOTnx0PZBudYc{`BjiilnVs{tJ^n2DRK*2= 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