From 818a6da7dfbbc49eb2f1464550588a8c0e8d1bee Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Anton=20Luka=20=C5=A0ijanec?= <anton@sijanec.eu>
Date: Mon, 20 May 2024 20:05:39 +0200
Subject: ladn8

---
 "\305\241ola/la/dn8/dokument.lyx" | 107 ++++++++++++++++++++++----------------
 1 file changed, 61 insertions(+), 46 deletions(-)

(limited to 'šola/la')

diff --git "a/\305\241ola/la/dn8/dokument.lyx" "b/\305\241ola/la/dn8/dokument.lyx"
index c01e7bc..6d8a154 100644
--- "a/\305\241ola/la/dn8/dokument.lyx"
+++ "b/\305\241ola/la/dn8/dokument.lyx"
@@ -725,7 +725,9 @@ Po predpostavki velja
 \begin_inset Formula $\left(AA^{*}-A^{*}A\right)^{*}=AA^{*}-A^{*}A$
 \end_inset
 
-
+ 
+\series bold
+TODO TODO TODO XXX XXX XXX XXX XXX XXX TODO TODO TODO
 \end_layout
 
 \end_deeper
@@ -804,25 +806,76 @@ Baza za
 \end_inset
 
  je 
-\begin_inset Formula $B=\left\{ v_{1},v_{2}\right\} =\left\{ \left(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right),\left(\frac{1}{2},\frac{-1}{2},\frac{-1}{2},\frac{-1}{2}\right)\right\} $
+\begin_inset Formula $B=\left\{ v_{1},v_{2}\right\} =\left\{ \left(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right),\left(\frac{1}{2},\frac{1}{2},\frac{-1}{2},\frac{-1}{2}\right)\right\} $
 \end_inset
 
 .
 \end_layout
 
 \begin_layout Enumerate
-Uporabimo fourierov razvoj 
-\begin_inset Formula $y$
+Dopolnimo 
+\begin_inset Formula $B$
 \end_inset
 
- po 
-\begin_inset Formula $B$
+ do baze 
+\begin_inset Formula $\mathbb{R}^{4}$
 \end_inset
 
 .
+ Dopolnitev 
+\begin_inset Formula $\left\{ u_{1},u_{2}\right\} $
+\end_inset
+
+ bo ortonormirane baze za 
+\begin_inset Formula $W^{\perp}$
+\end_inset
+
+, nato uporabimo Fourierov razvoj po dopolnjeni bazi.
+ Bazo podprostora dopolnimo tako, da rešimo sistem enačb.
+\begin_inset Formula 
+\[
+\left\langle \left(x_{1},y_{1},z_{1},w_{1}\right),\left(3,3,-1,-1\right)\right\rangle =0\quad\quad\quad\left\langle \left(x_{2},y_{2},z_{2},w_{2}\right),\left(1,1,1,1\right)\right\rangle =0
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\left[\begin{array}{cccc}
+1 & 1 & 1 & 1\\
+3 & 3 & -1 & -1
+\end{array}\right]\sim\left[\begin{array}{cccc}
+1 & 1 & 0 & 0\\
+0 & 0 & 1 & 1
+\end{array}\right]\Rightarrow x=-y,\quad z=-w\Rightarrow\tilde{u_{1}}=\left(1,-1,0,0\right),\quad\tilde{u_{2}}=\left(0,0,1,-1\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+u_{1}=\tilde{u_{1}}/\left|\left|\tilde{u_{1}}\right|\right|=\left(\frac{1}{\sqrt{2}},\frac{-1}{\sqrt{2}},0,0\right)\quad\quad\quad u_{2}=\tilde{u_{2}}/\left|\left|\tilde{u}_{2}\right|\right|=\left(0,0,\frac{1}{\sqrt{2}},\frac{-1}{\sqrt{2}}\right)
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+y=\sum_{i=1}^{n}\frac{\left\langle y,v_{i}\right\rangle }{\left\langle v_{i},v_{i}\right\rangle }v_{i}=\left\langle \left(6,0,2,0\right),\left(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)\right\rangle v_{1}+\left\langle \left(6,0,2,0\right),\left(\frac{1}{2},\frac{1}{2},\frac{-1}{2},\frac{-1}{2}\right)\right\rangle v_{2}+
+\]
+
+\end_inset
+
+
 \begin_inset Formula 
 \[
-y=\sum_{i=1}^{n}\frac{\left\langle y,v_{i}\right\rangle }{\left\langle v_{i},v_{i}\right\rangle }v_{i}=\left\langle \left(-6,0,2,0\right),\left(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)\right\rangle v_{i}+\left\langle \left(-6,0,2,0\right),\left(\frac{1}{2},\frac{1}{2},\frac{-1}{2},\frac{-1}{2}\right)\right\rangle \left(\frac{1}{2},\frac{1}{2},\frac{-1}{2},\frac{-1}{2}\right)=
++\left\langle \left(6,0,2,0\right),\left(\frac{1}{\sqrt{2}},\frac{-1}{\sqrt{2}},0,0\right)\right\rangle u_{1}+\left\langle \left(6,0,2,0\right),\left(0,0,\frac{1}{\sqrt{2}},\frac{-1}{\sqrt{2}},\right)\right\rangle u_{2}=4v_{1}+2v_{2}+\frac{6}{\sqrt{2}}u_{1}+\frac{2}{\sqrt{2}}u_{2}=
 \]
 
 \end_inset
@@ -830,7 +883,7 @@ y=\sum_{i=1}^{n}\frac{\left\langle y,v_{i}\right\rangle }{\left\langle v_{i},v_{
 
 \begin_inset Formula 
 \[
-=4\left(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)+2\left(\frac{1}{2},\frac{1}{2},\frac{-1}{2},\frac{-1}{2}\right)=\left(2,2,2,2\right)+\left(1,1,-1,-1\right)
+=\left(2,2,2,2\right)+\left(1,1,-1,-1\right)+\left(3,-3,0,0\right)+\left(0,0,1,-1\right)=\left(3,3,1,1\right)\in W+\left(3,-3,1,-1\right)\in W^{\perp}
 \]
 
 \end_inset
@@ -1092,43 +1145,5 @@ Rokopisi, ki sledijo, naj služijo le kot dokaz samostojnega reševanja.
  Zavedam se namreč njihovega neličnega izgleda.
 \end_layout
 
-\begin_layout Standard
-\begin_inset External
-	template PDFPages
-	filename /mnt/slu/shramba/upload/www/d/1ladn8a.jpg
-
-\end_inset
-
-
-\begin_inset External
-	template PDFPages
-	filename /mnt/slu/shramba/upload/www/d/1ladn8b.jpg
-
-\end_inset
-
-
-\begin_inset External
-	template PDFPages
-	filename /mnt/slu/shramba/upload/www/d/1ladn8c.jpg
-
-\end_inset
-
-
-\begin_inset External
-	template PDFPages
-	filename /mnt/slu/shramba/upload/www/d/3ladn8.jpg
-
-\end_inset
-
-
-\begin_inset External
-	template PDFPages
-	filename /mnt/slu/shramba/upload/www/d/4ladn8.jpg
-
-\end_inset
-
-
-\end_layout
-
 \end_body
 \end_document
-- 
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