2 files changed, 1285 insertions, 88 deletions

diff --git a/šola/la/dn8/dokument.lyx b/šola/la/dn8/dokument.lyx
index 7edbce2..c603fcc 100644
--- a/šola/la/dn8/dokument.lyx
+++ b/šola/la/dn8/dokument.lyx
@@ -1,5 +1,5 @@
-#LyX 2.4 created this file. For more info see https://www.lyx.org/
-\lyxformat 620
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
 \begin_document
 \begin_header
 \save_transient_properties true
@@ -21,17 +21,18 @@
 }%
 \DeclareMathOperator{\Lin}{Lin}
 \DeclareMathOperator{\rang}{rang}
+\DeclareMathOperator{\sled}{sled}
 \end_preamble
 \use_default_options true
 \begin_modules
 enumitem
 theorems-ams
 \end_modules
-\maintain_unincluded_children no
+\maintain_unincluded_children false
 \language slovene
 \language_package default
-\inputencoding auto-legacy
-\fontencoding auto
+\inputencoding auto
+\fontencoding global
 \font_roman "default" "default"
 \font_sans "default" "default"
 \font_typewriter "default" "default"
@@ -39,9 +40,7 @@ theorems-ams
 \font_default_family default
 \use_non_tex_fonts false
 \font_sc false
-\font_roman_osf false
-\font_sans_osf false
-\font_typewriter_osf false
+\font_osf false
 \font_sf_scale 100 100
 \font_tt_scale 100 100
 \use_microtype false
@@ -75,9 +74,7 @@ theorems-ams
 \suppress_date false
 \justification false
 \use_refstyle 1
-\use_formatted_ref 0
 \use_minted 0
-\use_lineno 0
 \index Index
 \shortcut idx
 \color #008000
@@ -100,16 +97,11 @@ theorems-ams
 \papercolumns 1
 \papersides 1
 \paperpagestyle default
-\tablestyle default
 \tracking_changes false
 \output_changes false
-\change_bars false
-\postpone_fragile_content false
 \html_math_output 0
 \html_css_as_file 0
 \html_be_strict false
-\docbook_table_output 0
-\docbook_mathml_prefix 1
 \end_header
 
 \begin_body
@@ -165,13 +157,11 @@ euler{e}
 \end_layout
 
 \begin_layout Enumerate
-Dokaži,
- da je 
+Dokaži, da je 
 \begin_inset Formula $\left[\left(x,y,z\right),\left(u,v,w\right)\right]=2xu-yu-xv+2yv-zv-yw+zw$
 \end_inset
 
- skalarni produkt in ugotovi,
- ali je
+ skalarni produkt in ugotovi, ali je
 \begin_inset Formula 
 \[
 A=\left[\begin{array}{ccc}
@@ -204,8 +194,8 @@ Predpostavljam polje
 \begin_inset Formula $V=\mathbb{R}^{3}$
 \end_inset
 
-,
- saj v kompleksnem to ni skalarni produkt (protiprimer pozitivne definitnosti je 
+, saj v kompleksnem to ni skalarni produkt (protiprimer pozitivne definitnosti
+ je 
 \begin_inset Formula $\left[\left(1,1,1+i\right),\left(1,1,1+i\right)\right]=2$
 \end_inset
 
@@ -214,8 +204,7 @@ Predpostavljam polje
 \begin_inset Formula $\langle\cdot,\cdot\rangle:V\times V\to\mathbb{R}$
 \end_inset
 
- je skalarni produkt,
- če zadošča naslednjim lastnostim.
+ je skalarni produkt, če zadošča naslednjim lastnostim.
  Dokažimo jih za 
 \begin_inset Formula $\left[\cdot,\cdot\right]$
 \end_inset
@@ -256,8 +245,7 @@ Sedaj poiščimo ničle.
 \begin_inset Formula $y$
 \end_inset
 
-,
- 
+, 
 \begin_inset Formula $z$
 \end_inset
 
@@ -300,8 +288,7 @@ Diskriminanta je nenegativna
 \begin_inset Formula $z=0$
 \end_inset
 
-,
- zato 
+, zato 
 \begin_inset Formula $y=0$
 \end_inset
 
@@ -351,7 +338,15 @@ Skalarni produkt je res simetričen.
 
 \begin_inset Formula 
 \[
-\left[\alpha\left(\left(x_{1},y_{1},z_{1}\right)+\left(x_{2},y_{2},z_{2}\right)\right),\left(u,v,w\right)\right]=
+\left[\alpha_{1}\left(x_{1},y_{1},z_{1}\right)+\alpha_{2}\left(x_{2},y_{2},z_{2}\right),\left(u,v,w\right)\right]=
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+=\left[\left(\alpha_{1}x_{1}+\alpha_{2}x_{2},\alpha_{1}y_{1}+\alpha_{2}y_{2},\alpha_{1}z_{1}+\alpha_{2}z_{2}\right),\left(u,v,w\right)\right]=
 \]
 
 \end_inset
@@ -359,7 +354,7 @@ Skalarni produkt je res simetričen.
 
 \begin_inset Formula 
 \[
-=2\alpha\left(x_{1}+x_{2}\right)u-\alpha\left(y_{1}+y_{2}\right)u-\alpha\left(x_{1}+x_{2}\right)v+2\alpha\left(y_{1}+y_{2}\right)v-\alpha\left(z_{1}+z_{2}\right)v-\alpha\left(y_{1}+y_{2}\right)w+\alpha\left(z_{1}+z_{2}\right)w=
+=2\left(\alpha_{1}x_{1}+\alpha_{2}x_{2}\right)u-\left(\alpha_{1}y_{1}+\alpha_{2}y_{2}\right)u-\left(\alpha_{1}x_{1}+\alpha_{2}x_{2}\right)v+
 \]
 
 \end_inset
@@ -367,7 +362,7 @@ Skalarni produkt je res simetričen.
 
 \begin_inset Formula 
 \[
-=\alpha\left(2\left(x_{1}+x_{2}\right)u-\left(y_{1}+y_{2}\right)u-\left(x_{1}+x_{2}\right)v+2\left(y_{1}+y_{2}\right)v-\left(z_{1}+z_{2}\right)v-\left(y_{1}+y_{2}\right)w+\left(z_{1}+z_{2}\right)w\right)=
++2\left(\alpha_{1}y_{1}+\alpha_{2}y_{2}\right)v-\left(\alpha_{1}z_{1}+\alpha_{2}z_{2}\right)v-\left(\alpha_{1}y_{1}+\alpha_{2}y_{2}\right)w+\left(\alpha_{1}z_{1}+\alpha_{2}z_{2}\right)w=
 \]
 
 \end_inset
@@ -375,7 +370,7 @@ Skalarni produkt je res simetričen.
 
 \begin_inset Formula 
 \[
-=\alpha\left(2x_{1}u+2x_{2}u-y_{1}u-y_{2}u-x_{1}v-x_{2}v+2y_{1}v+2y_{2}v-z_{1}v-z_{2}v-y_{1}w-y_{2}w+z_{1}w+z_{2}w\right)=
+=2\alpha_{1}x_{1}u+2\alpha_{2}x_{2}u-\alpha_{1}y_{1}u-\alpha_{2}y_{2}u-\alpha_{1}x_{1}v-\alpha_{2}x_{2}v+
 \]
 
 \end_inset
@@ -383,7 +378,7 @@ Skalarni produkt je res simetričen.
 
 \begin_inset Formula 
 \[
-=\alpha\left(2x_{1}u-y_{1}u-x_{1}v+2y_{1}v-z_{1}v-y_{1}w+z_{1}w\right)+\alpha\left(2x_{2}u-y_{2}u-x_{2}v+2y_{2}v-z_{2}v-y_{2}w+z_{2}w\right)=
++2\alpha_{1}y_{1}v+2\alpha_{2}y_{2}v-\alpha_{1}z_{1}v-\alpha_{2}z_{2}v-\alpha_{1}y_{1}w-\alpha_{2}y_{2}w+\alpha_{1}z_{1}w+\alpha_{2}z_{2}w=
 \]
 
 \end_inset
@@ -391,7 +386,15 @@ Skalarni produkt je res simetričen.
 
 \begin_inset Formula 
 \[
-=\alpha\left[\left(x_{1},y_{1},z_{1}\right),\left(u,v,w\right)\right]+\alpha\left[\left(x_{2},y_{2},z_{2}\right),\left(u,v,w\right)\right]
+=\alpha_{1}\left(2x_{1}u-y_{1}u-x_{1}v+2y_{1}v-z_{1}v-y_{1}w+z_{1}w\right)+\alpha_{2}\left(2x_{2}u-y_{2}u-x_{2}v+2y_{2}v-z_{2}v-y_{2}w+z_{2}w\right)=
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+=\alpha_{1}\left[\left(x_{1},y_{1},z_{1}\right),\left(u,v,w\right)\right]+\alpha_{2}\left[\left(x_{2},y_{2},z_{2}\right),\left(u,v,w\right)\right]
 \]
 
 \end_inset
@@ -417,22 +420,23 @@ Po definiciji
 \end_layout
 
 \begin_layout Itemize
-Na predavanjih 2024-05-08 smo dokazali,
- da za vsak skalarni produkt 
+Na predavanjih 2024-05-08 smo dokazali, da za vsak skalarni produkt 
 \begin_inset Formula $\left[u,v\right]$
 \end_inset
 
- obstaja taka pozitivno definitna matrika 
+ obstaja taka ortogonalna (
+\begin_inset Formula $M^{*}=M^{-1}$
+\end_inset
+
+) pozitivno definitna matrika 
 \begin_inset Formula $M$
 \end_inset
 
-,
- da velja 
-\begin_inset Formula $\left[u,v\right]=\langle u,Mv\rangle=u^{*}v$
+, da velja 
+\begin_inset Formula $\left[u,v\right]=\langle u,Mv\rangle$
 \end_inset
 
-,
- kjer je 
+, kjer je 
 \begin_inset Formula $\langle\cdot,\cdot\rangle$
 \end_inset
 
@@ -440,20 +444,6 @@ Na predavanjih 2024-05-08 smo dokazali,
 \end_layout
 
 \begin_layout Itemize
-Na predavanjih 2024-04-17 smo dokazali,
- da 
-\begin_inset Formula $\left[L^{*}\right]_{C\leftarrow B}=\left(\left[L\right]_{B\leftarrow C}\right)^{*}$
-\end_inset
-
-,
- torej 
-\begin_inset Formula $PLP^{-1}=\left(P^{-1}L^{*}P\right)^{*}$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Itemize
 Izpeljimo predpis za 
 \begin_inset Formula $A^{*}$
 \end_inset
@@ -632,11 +622,9 @@ Da preverimo pravilnost matrike
 \begin_inset Formula $A^{*}$
 \end_inset
 
-,
- lahko napravimo preizkus:
+, lahko napravimo preizkus:
 \begin_inset Float figure
 placement H
-alignment document
 wide false
 sideways false
 status open
@@ -665,6 +653,98 @@ Preizkus s programom SageMath.
 
 \end_layout
 
+\begin_layout Standard
+Dokazati, da 
+\begin_inset Formula $A$
+\end_inset
+
+ ni normalna, je moč še lažje.
+ Dokažemo lahko namreč, da eden izmed potrebnih pogojev za normalnost matrike
+ ni izpolnjen.
+ Na primer: 
+\begin_inset Formula $AA^{*}=A^{*}A\rightarrow A=PDP^{-1}$
+\end_inset
+
+, kjer je 
+\begin_inset Formula $P$
+\end_inset
+
+ ortogonalna in 
+\begin_inset Formula $D$
+\end_inset
+
+ diagonalna 
+\begin_inset Formula $\Rightarrow$
+\end_inset
+
+ lastni vektorji 
+\begin_inset Formula $A$
+\end_inset
+
+ tvorijo ortogonalno množico.
+\end_layout
+
+\begin_layout Standard
+Lastne vrednosti 
+\begin_inset Formula $A$
+\end_inset
+
+ so (s kalkulatorjem) 
+\begin_inset Formula $\left\{ -2,1\right\} $
+\end_inset
+
+, kjer ima 1 algebrajsko večkratnost 2.
+ Lastni vektorji:
+\begin_inset Formula 
+\[
+A-\left(-2\right)I=\left[\begin{array}{ccc}
+2 & 2 & -2\\
+0 & 3 & 0\\
+-1 & 2 & 1
+\end{array}\right]\sim\left[\begin{array}{ccc}
+2 & 2 & -2\\
+0 & 3 & 0\\
+0 & 3 & 0
+\end{array}\right]\sim\left[\begin{array}{ccc}
+2 & 2 & -2\\
+0 & 3 & 0\\
+0 & 0 & 0
+\end{array}\right]\sim\left[\begin{array}{ccc}
+2 & 0 & -2\\
+0 & 3 & 0\\
+0 & 0 & 0
+\end{array}\right]\Rightarrow x=z,y=0\Rightarrow v_{1}=\left(1,0,1\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+A-1I=\left[\begin{array}{ccc}
+-1 & 2 & -2\\
+0 & 0 & 0\\
+-1 & 2 & -2
+\end{array}\right]\sim\left[\begin{array}{ccc}
+-1 & 2 & -2\\
+0 & 0 & 0\\
+0 & 0 & 0
+\end{array}\right]\Rightarrow x=2y-2z\Rightarrow v_{2}=\left(2,1,0\right),\quad v_{3}=\left(-2,0,1\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\left[v_{1},v_{2}\right]=\left[\left(1,0,1\right),\left(2,1,0\right)\right]=4-0-1+0-1-0+0=2\not=0\Rightarrow v_{1}\not\perp v_{2}\Rightarrow A\text{ ni normalna}
+\]
+
+\end_inset
+
+
+\end_layout
+
 \end_deeper
 \begin_layout Enumerate
 Pokaži 
@@ -693,7 +773,7 @@ Definiciji:
 \end_inset
 
  je normalna 
-\begin_inset Formula $\Leftrightarrow A^{*}A=A^{*}$
+\begin_inset Formula $\Leftrightarrow A^{*}A=AA^{*}$
 \end_inset
 
 
@@ -746,9 +826,63 @@ Po predpostavki velja
 \begin_inset Formula $\left(AA^{*}-A^{*}A\right)^{*}=AA^{*}-A^{*}A$
 \end_inset
 
+ in 
+\begin_inset Formula $\forall v\in V:\left\langle \left(AA^{*}-A^{*}A\right)v,v\right\rangle \geq0$
+\end_inset
+
+.
+\begin_inset Formula 
+\[
+\sled\left(AA^{*}-A^{*}A\right)=\sled\left(AA^{*}\right)-\sled\left(A^{*}A\right)\overset{\text{lastnost sledi}}{=}\sled\left(AA^{*}\right)-\sled\left(A^{*}A\right)=0
+\]
+
+\end_inset
+
+Sled 
+\begin_inset Formula $M$
+\end_inset
+
+ je vsota lastnih vrednosti 
+\begin_inset Formula $M$
+\end_inset
+
+, torej je vsota lastnih vrednosti 
+\begin_inset Formula $\left(AA^{*}-A^{*}A\right)=0$
+\end_inset
+
+.
  
-\series bold
-TODO TODO TODO XXX XXX XXX XXX XXX XXX TODO TODO TODO
+\begin_inset Formula $AA^{*}-A^{*}A\geq0\Rightarrow$
+\end_inset
+
+ vse lastne vrednosti so nenegativne.
+ Iz teh dveh trditev sledi, da je vsaka lastna vrednost 
+\begin_inset Formula $AA^{*}-A^{*}A=0$
+\end_inset
+
+.
+ 
+\begin_inset Formula $AA^{*}-A^{*}A\geq0\Rightarrow AA^{*}-A^{*}A$
+\end_inset
+
+ normalna.
+ Normalne matrike je moč diagonalizirati v ortonormirani bazi:
+\begin_inset Formula 
+\[
+AA^{*}-A^{*}A=PDP^{-1}\overset{\text{diagonalci so lastne vrednosti}}{=}P0P^{-1}=0
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+AA^{*}=A^{*}A\Rightarrow A\text{ je normalna}
+\]
+
+\end_inset
+
+
 \end_layout
 
 \end_deeper
@@ -757,8 +891,7 @@ Naj bo
 \begin_inset Formula $w_{1}=\left(1,1,1,1\right)$
 \end_inset
 
-,
- 
+, 
 \begin_inset Formula $w_{2}=\left(3,3,-1,-1\right)$
 \end_inset
 
@@ -852,10 +985,8 @@ Dopolnimo
 \begin_inset Formula $W^{\perp}$
 \end_inset
 
-,
- nato uporabimo Fourierov razvoj po dopolnjeni bazi.
- Bazo podprostora dopolnimo tako,
- da rešimo sistem enačb.
+, 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
@@ -943,8 +1074,7 @@ Iščemo
 \begin_inset Formula $U$
 \end_inset
 
-,
- 
+, 
 \begin_inset Formula $\Sigma$
 \end_inset
 
@@ -952,8 +1082,7 @@ Iščemo
 \begin_inset Formula $V$
 \end_inset
 
-,
- da velja 
+, da velja 
 \begin_inset Formula $A=U\Sigma V^{*}$
 \end_inset
 
@@ -978,18 +1107,15 @@ Diagonalci
 \begin_inset Formula $A^{*}A$
 \end_inset
 
-,
- torej 
+, torej 
 \begin_inset Formula $\sigma_{1}=2$
 \end_inset
 
-,
- 
+, 
 \begin_inset Formula $\sigma_{2}=1$
 \end_inset
 
-,
- 
+, 
 \begin_inset Formula $\sigma_{3}=0$
 \end_inset
 
@@ -1054,8 +1180,8 @@ Stolpci
 A^{*}A-4I=\left[\begin{array}{ccc}
 -3 & 0 & 0\\
 0 & 0 & 0\\
-0 & 0 & 0
-\end{array}\right]\Rightarrow x=0\Rightarrow v_{1}=\left(0,1,0\right)
+0 & 0 & -4
+\end{array}\right]\Rightarrow x=z=0\Rightarrow v_{1}=\left(0,1,0\right)
 \]
 
 \end_inset
@@ -1066,8 +1192,8 @@ A^{*}A-4I=\left[\begin{array}{ccc}
 A^{*}A-1I=\left[\begin{array}{ccc}
 0 & 0 & 0\\
 0 & 3 & 0\\
-0 & 0 & 0
-\end{array}\right]\Rightarrow y=0\Rightarrow v_{2}=\left(1,0,0\right)
+0 & 0 & -1
+\end{array}\right]\Rightarrow y=z=0\Rightarrow v_{2}=\left(1,0,0\right)
 \]
 
 \end_inset
@@ -1114,8 +1240,7 @@ Stolpci
 \begin_inset Formula $v_{\rang A+1},\dots,v_{m}$
 \end_inset
 
- najdemo tako,
- da dopolnimo 
+ najdemo tako, da dopolnimo 
 \begin_inset Formula $v_{1},\dots,v_{\rang A}$
 \end_inset
 
@@ -1136,8 +1261,7 @@ U=\left[\begin{array}{cccc}
 \end_layout
 
 \begin_layout Itemize
-Dobljene matrike zmnožimo,
- s čimer potrdimo veljavnost singularnega razcepa:
+Dobljene matrike zmnožimo, s čimer potrdimo veljavnost singularnega razcepa:
 \begin_inset Formula 
 \[
 U\Sigma V^{*}=\left[\begin{array}{cccc}
@@ -1169,9 +1293,7 @@ U\Sigma V^{*}=\left[\begin{array}{cccc}
 
 \end_deeper
 \begin_layout Standard
-Rokopisi,
- ki sledijo,
- naj služijo le kot dokaz samostojnega reševanja.
+Rokopisi, ki sledijo, naj služijo le kot dokaz samostojnega reševanja.
  Zavedam se namreč njihovega neličnega izgleda.
 \end_layout
 
@@ -1185,6 +1307,13 @@ Rokopisi,
 
 \begin_inset External
 	template PDFPages
+	filename /mnt/slu/shramba/upload/www/d/1ladn8aq.jpg
+
+\end_inset
+
+
+\begin_inset External
+	template PDFPages
 	filename /mnt/slu/shramba/upload/www/d/1ladn8b.jpg
 
 \end_inset
diff --git a/šola/la/kolokvij4.lyx b/šola/la/kolokvij4.lyx
new file mode 100644
index 0000000..3e8a3e8
--- /dev/null
+++ b/šola/la/kolokvij4.lyx
@@ -0,0 +1,1068 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass article
+\begin_preamble
+\usepackage{siunitx}
+\usepackage{pgfplots}
+\usepackage{listings}
+\usepackage{multicol}
+\sisetup{output-decimal-marker = {,}, quotient-mode=fraction, output-exponent-marker=\ensuremath{\mathrm{3}}}
+\end_preamble
+\use_default_options true
+\begin_modules
+enumitem
+\end_modules
+\maintain_unincluded_children false
+\language slovene
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry true
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification false
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\leftmargin 1cm
+\topmargin 2cm
+\rightmargin 1cm
+\bottommargin 2cm
+\headheight 1cm
+\headsep 1cm
+\footskip 1cm
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style german
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+newcommand
+\backslash
+euler{e}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{multicols}{2}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Drobnarije od prej
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\det A=\det A^{T}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Vsota je direktna 
+\begin_inset Formula $\Leftrightarrow V\cap U=\left\{ 0\right\} $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Skalarni produkt
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\left\langle v,v\right\rangle >0$
+\end_inset
+
+, 
+\begin_inset Formula $\left\langle v,u\right\rangle =\overline{\left\langle u,v\right\rangle }$
+\end_inset
+
+, 
+\begin_inset Formula $\left\langle \alpha_{2}u_{1}+\alpha_{2}u_{2},v\right\rangle =\alpha_{1}\left\langle u_{1},v\right\rangle +\alpha_{2}\left\langle u_{2},v\right\rangle $
+\end_inset
+
+, 
+\begin_inset Formula $\left\langle u,\alpha_{1}v_{1}+\alpha_{2}v_{2}\right\rangle =\overline{\alpha_{1}}\left\langle u,v_{1}\right\rangle +\overline{\alpha_{2}}\left\langle u,v_{2}\right\rangle $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Standardni: 
+\begin_inset Formula $\left\langle \left(\alpha_{1},\dots,\alpha_{n}\right),\left(\beta_{1},\dots,\beta_{n}\right)\right\rangle =\alpha_{1}\overline{\beta_{1}}+\cdots\alpha_{n}\overline{\beta_{n}}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Norma: 
+\begin_inset Formula $\left|\left|v\right|\right|^{2}=\left\langle v,v\right\rangle $
+\end_inset
+
+: 
+\begin_inset Formula $\left|\left|v\right|\right|>0\Leftrightarrow v\not=0$
+\end_inset
+
+, 
+\begin_inset Formula $\left|\left|\alpha v\right|\right|=\left|\alpha\right|\left|\left|v\right|\right|$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Trikotniška neenakost: 
+\begin_inset Formula $\left|\left|u+v\right|\right|\leq\left|\left|u\right|\right|+\left|\left|v\right|\right|$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Cauchy-Schwarz: 
+\begin_inset Formula $\left|\left\langle u,v\right\rangle \right|\leq\left|\left|v\right|\right|\left|\left|u\right|\right|$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $v\perp u\Leftrightarrow\left\langle u,v\right\rangle =0$
+\end_inset
+
+.
+ 
+\begin_inset Formula $M$
+\end_inset
+
+ ortog.
+ 
+\begin_inset Formula $\Leftrightarrow\forall u,v\in M:v\perp u\wedge v\not=0$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $M$
+\end_inset
+
+ normirana 
+\begin_inset Formula $\Leftrightarrow\forall u\in M:\left|\left|u\right|\right|=1$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $M$
+\end_inset
+
+ ortog.
+ 
+\begin_inset Formula $\Rightarrow M$
+\end_inset
+
+ lin.
+ neod., Ortog.
+ baza 
+\begin_inset Formula $\sim$
+\end_inset
+
+ ortog.
+ ogrodje
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $v\perp M\Leftrightarrow\forall u\in M:v\perp u$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Fourierov razvoj
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $v_{i}$
+\end_inset
+
+ ortog.
+ baza za 
+\begin_inset Formula $V$
+\end_inset
+
+, 
+\begin_inset Formula $v\in V$
+\end_inset
+
+ poljuben.
+ 
+\begin_inset Formula $v=\sum_{i=1}^{n}\frac{\left\langle v,v_{i}\right\rangle }{\left\langle v_{i},v_{i}\right\rangle }v_{i}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Parsevalova identiteta: 
+\begin_inset Formula $\left|\left|v\right|\right|^{2}=\sum_{i=1}^{n}\frac{\left|\left\langle v,v_{i}\right\rangle \right|^{2}}{\left\langle v_{i},v_{i}\right\rangle }$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Projekcija na podprostor
+\end_layout
+
+\begin_layout Standard
+let 
+\begin_inset Formula $V$
+\end_inset
+
+ podprostor 
+\begin_inset Formula $W$
+\end_inset
+
+.
+ 
+\begin_inset Formula $v'$
+\end_inset
+
+ je ortog.
+ proj vektorja 
+\begin_inset Formula $v$
+\end_inset
+
+ 
+\begin_inset Formula $\Leftrightarrow\forall w\in W:\left|\left|v-v'\right|\right|\leq\left|\left|v-w\right|\right|\sim\text{v'}$
+\end_inset
+
+ je najbližje 
+\begin_inset Formula $V$
+\end_inset
+
+ izmed elementov 
+\begin_inset Formula $W$
+\end_inset
+
+.
+ 
+\begin_inset Formula $\sun$
+\end_inset
+
+ Pitagora:
+\end_layout
+
+\begin_layout Standard
+Zadošča preveriti ortogonalnost 
+\begin_inset Formula $v-v'$
+\end_inset
+
+ na vse elemente 
+\begin_inset Formula $W$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Formula za ort.
+ proj.: 
+\begin_inset Formula $v'=\sum_{i=0}^{n}\frac{\left\langle v,w_{i}\right\rangle }{\left\langle w_{i},w_{i}\right\rangle }$
+\end_inset
+
+, kjer je 
+\begin_inset Formula $w_{i}$
+\end_inset
+
+ OB 
+\begin_inset Formula $W$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Paragraph
+Obstoj ortogonalne baze (Gram-Schmidt)
+\end_layout
+
+\begin_layout Standard
+let 
+\begin_inset Formula $\left\{ u_{1},\dots,u_{n}\right\} $
+\end_inset
+
+ baza 
+\begin_inset Formula $V$
+\end_inset
+
+.
+ Zanj konstruiramo OB 
+\begin_inset Formula $\left\{ v_{1},\dots,v_{n}\right\} $
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $v_{1}=u_{1}$
+\end_inset
+
+, 
+\begin_inset Formula $v_{2}=u_{2}-\frac{\left\langle u_{2},v_{1}\right\rangle }{\left\langle v_{1},v_{1}\right\rangle }v_{1}$
+\end_inset
+
+, 
+\begin_inset Formula $v_{3}=u_{3}-\frac{\left\langle u_{3},v_{2}\right\rangle }{\left\langle v_{2},v_{2}\right\rangle }v_{2}-\frac{\left\langle u_{3},v_{1}\right\rangle }{\left\langle v_{1},v_{1}\right\rangle }v_{1}$
+\end_inset
+
+...
+ 
+\begin_inset Formula $v_{k}=u_{k}-\sum_{i=1}^{k-1}\frac{\text{\left\langle u_{k},v_{i}\right\rangle }}{\left\langle v_{i},v_{i}\right\rangle }v_{i}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Ortogonalni komplement
+\end_layout
+
+\begin_layout Standard
+let 
+\begin_inset Formula $S\subseteq V$
+\end_inset
+
+.
+ 
+\begin_inset Formula $S^{\perp}=\left\{ v\in V;v\perp S\right\} $
+\end_inset
+
+.
+ Velja: 
+\begin_inset Formula $S^{\perp}$
+\end_inset
+
+ podprostor 
+\begin_inset Formula $V$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $W$
+\end_inset
+
+ podprostor 
+\begin_inset Formula $V$
+\end_inset
+
+.
+ Velja: 
+\begin_inset Formula $W\oplus W^{\perp}=V$
+\end_inset
+
+ in 
+\begin_inset Formula $\left(W^{\perp}\right)^{\perp}=W$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Če je 
+\begin_inset Formula $\left\{ u_{1},\dots,u_{k}\right\} $
+\end_inset
+
+ OB podprostora 
+\begin_inset Formula $V$
+\end_inset
+
+, je dopolnitev do baze vsega 
+\begin_inset Formula $V^{\perp}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Za vektorske podprostore 
+\begin_inset Formula $V_{i}$
+\end_inset
+
+ VPSSP 
+\begin_inset Formula $W$
+\end_inset
+
+ velja:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $S\subseteq W\Rightarrow\left(S^{\perp}\right)^{\perp}=\mathcal{L}in\left\{ S\right\} $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $V_{1}\subseteq V_{2}\Rightarrow V_{2}^{\perp}\subseteq V_{1}^{\perp}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\left(V_{1}+v_{2}\right)^{\perp}=V_{1}^{\perp}\cup V_{2}^{\perp}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\left(V_{1}\cap V_{2}\right)^{\perp}=V_{1}^{\perp}+V_{2}^{\perp}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Linearni funkcional
+\end_layout
+
+\begin_layout Standard
+je linearna preslikava 
+\begin_inset Formula $V\to F$
+\end_inset
+
+, če je 
+\begin_inset Formula $V$
+\end_inset
+
+ nad poljem 
+\begin_inset Formula $F$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Rieszov izrek o reprezentaciji linearnih funkcionalov: 
+\begin_inset Formula $\forall\text{l.f.}\varphi:V\to F\exists!w\in V\ni:\forall v\in V:\varphi v=\left\langle v,w\right\rangle $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Za 
+\begin_inset Formula $L:U\to V$
+\end_inset
+
+ je 
+\begin_inset Formula $L^{*}:V\to U$
+\end_inset
+
+ adjungirana linearna preslika 
+\begin_inset Formula $\Leftrightarrow\forall u\in U,v\in V:\left\langle Lu,v\right\rangle =\left\langle v,L^{*}u\right\rangle $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Za std.
+ skal.
+ prod.
+ velja: 
+\begin_inset Formula $A^{*}=\overline{A}^{T}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\left(AB\right)^{*}=B^{*}A^{*}$
+\end_inset
+
+, 
+\begin_inset Formula $\left(L^{*}\right)_{B\leftarrow C}=\left(L_{C\leftarrow B}\right)^{*}$
+\end_inset
+
+, 
+\begin_inset Formula $\left(\alpha A+\beta B\right)^{*}=\overline{\alpha}A^{*}+\overline{\beta}B^{*}$
+\end_inset
+
+, 
+\begin_inset Formula $\left(A^{*}\right)^{*}=A$
+\end_inset
+
+, 
+\begin_inset Formula $\text{Ker}L^{*}=\left(\text{Im}L\right)^{\perp}$
+\end_inset
+
+, 
+\begin_inset Formula $\left(\text{Ker}L^{*}\right)^{\perp}=\text{Im}L$
+\end_inset
+
+, 
+\begin_inset Formula $\text{Ker}\left(L^{*}L\right)=\text{Ker}L$
+\end_inset
+
+, 
+\begin_inset Formula $\text{Im}\left(L^{*}L\right)=\text{Im}L$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Lastne vrednosti 
+\begin_inset Formula $A^{*}$
+\end_inset
+
+ so konjugirane lastne vrednosti 
+\begin_inset Formula $A$
+\end_inset
+
+.
+ Dokaz: 
+\begin_inset Formula $B=A-\lambda I$
+\end_inset
+
+.
+ 
+\begin_inset Formula $B^{*}=A^{*}-\overline{\lambda}I$
+\end_inset
+
+.
+ 
+\begin_inset Formula $\det B^{*}=\det\overline{B}^{T}=\det B=\overline{\det B}$
+\end_inset
+
+, torej 
+\begin_inset Formula $\det B=0\Leftrightarrow\det B^{*}=0$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\Delta_{A^{*}}$
+\end_inset
+
+ ima konjugirane koeficiente 
+\begin_inset Formula $\Delta_{A}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Paragraph
+Normalne matrike 
+\begin_inset Formula $A^{*}A=AA^{*}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Velja: 
+\begin_inset Formula $A$
+\end_inset
+
+ kvadratna, 
+\begin_inset Formula $Av=\lambda v\Leftrightarrow A^{*}v=\overline{\lambda}v$
+\end_inset
+
+ (isti lastni vektorji)
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $Au=\lambda u\wedge Av=\mu v\wedge\mu\not=\lambda\Rightarrow v\perp u$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Je podobna diagonalni: 
+\begin_inset Formula $\forall m:\text{Ker}\left(A-\lambda I\right)^{m}=\text{Ker}\left(A-\lambda I\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $A=PDP^{-1}\Leftrightarrow$
+\end_inset
+
+ stolpci 
+\begin_inset Formula $P$
+\end_inset
+
+ so ONB, diagonalci 
+\begin_inset Formula $D$
+\end_inset
+
+ lavr, zdb 
+\begin_inset Formula $P$
+\end_inset
+
+ je unitarna/ortogonalna.
+\end_layout
+
+\begin_layout Paragraph
+Unitarne 
+\begin_inset Formula $\mathbb{C}$
+\end_inset
+
+/ortogonalne 
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+ matrike 
+\begin_inset Formula $AA^{*}=A^{*}A=I$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $A$
+\end_inset
+
+ kvadratna z ON stolpci.
+ 
+\begin_inset Formula $A$
+\end_inset
+
+ ortog.
+ 
+\begin_inset Formula $\Rightarrow A$
+\end_inset
+
+ normalna
+\end_layout
+
+\begin_layout Standard
+Lavr: let 
+\begin_inset Formula $Av=\lambda v\Rightarrow\left\langle Av,Av\right\rangle =\left\langle \lambda v,\lambda v\right\rangle =\left\langle v,v\right\rangle =\lambda\overline{\lambda}\left\langle v,v\right\rangle \Rightarrow\left|\lambda\right|=1\Rightarrow\lambda=e^{i\varphi}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $A=PDP^{-1},A^{*}=A^{-1}$
+\end_inset
+
+ 
+\end_layout
+
+\begin_layout Paragraph
+Simetrične 
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+/hermitske 
+\begin_inset Formula $\mathbb{C}$
+\end_inset
+
+ matrike 
+\begin_inset Formula $A=A^{*}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Sebiadjungirane linearne preslikave.
+\end_layout
+
+\begin_layout Standard
+Hermitska 
+\begin_inset Formula $\Rightarrow$
+\end_inset
+
+ Normalna
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $Av=\lambda v=A^{*}v=\overline{\lambda}v\Rightarrow\lambda\in\mathbb{R}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $A=A^{*}\Leftrightarrow\forall v:\left\langle Av,v\right\rangle \in\mathbb{R}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Pozitivno (semi)definitne 
+\begin_inset Formula $A\geq0$
+\end_inset
+
+ (
+\begin_inset Formula $>$
+\end_inset
+
+ za PD)
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $A$
+\end_inset
+
+ P(S)D 
+\begin_inset Formula $\Rightarrow$
+\end_inset
+
+ 
+\begin_inset Formula $A$
+\end_inset
+
+ sim./ortog.
+ 
+\begin_inset Formula $\Rightarrow A$
+\end_inset
+
+ normalna
+\end_layout
+
+\begin_layout Standard
+Def.: 
+\begin_inset Formula $A=A^{*}\wedge\forall v:\left\langle Av,v\right\rangle \geq0$
+\end_inset
+
+ (
+\begin_inset Formula $>$
+\end_inset
+
+ za PD)
+\end_layout
+
+\begin_layout Standard
+Za poljubno 
+\begin_inset Formula $B$
+\end_inset
+
+ je 
+\begin_inset Formula $B^{*}B$
+\end_inset
+
+ PSD.
+ Če ima 
+\begin_inset Formula $B$
+\end_inset
+
+ LN stolpce, je 
+\begin_inset Formula $B^{*}B$
+\end_inset
+
+ PD.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\forall\text{lavr}\lambda_{i}:A>0\Rightarrow\lambda_{i}>0$
+\end_inset
+
+, 
+\begin_inset Formula $A\geq0\Rightarrow\lambda_{i}\geq0$
+\end_inset
+
+.
+ Dokaz: let 
+\begin_inset Formula $A\geq0,v\not=0,Av=\lambda v\Rightarrow\left\langle Av,v\right\rangle =\left\langle \lambda v,v\right\rangle =\lambda\left\langle v,v\right\rangle \geq0\wedge\left\langle v,v\right\rangle >0\Rightarrow\lambda\geq0$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Lavr isto kot hermitska, lave isto kot normalna, diag.
+ isto kot normalna.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\text{A\ensuremath{\geq0}}\Rightarrow\exists B=B^{*},B\geq0\ni:B^{2}=A$
+\end_inset
+
+.
+ Dokaz: let 
+\begin_inset Formula $E\text{diag s koreni lavr}\geq0,A=PDP^{-1},P^{*}=P^{-1},D=\text{\text{diag z lavr}}\geq0,B=PEP^{-1}=PEP^{*}\Rightarrow B=B^{*}\Rightarrow B^{2}=PEP^{-1}PEP^{-1}=PE^{2}P^{-1}=PDP=A$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+NTSE: 
+\begin_inset Formula $A\geq0\Leftrightarrow A=A^{*}\wedge\forall\lambda\text{lavr}A:\lambda\geq0\Leftrightarrow A=PDP^{-1}\wedge P\text{ unit.}\wedge\text{diag.}D\geq0\Leftrightarrow A=A^{*}\wedge\exists\sqrt{A}\ni:\sqrt{A}^{2}=A\Leftrightarrow A=B^{*}B$
+\end_inset
+
+ (oz.
+ 
+\begin_inset Formula $>$
+\end_inset
+
+ za PD)
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\forall\left[\cdot,\cdot\right]:V^{2}\to F\exists M>0\ni:\forall v,u\in V:\left[v,u\right]=\left\langle Au,v\right\rangle $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\forall A>0:\left\langle A\cdot,\cdot\right\rangle $
+\end_inset
+
+ je skalarni produkt.
+\end_layout
+
+\begin_layout Paragraph
+Singularni razcep (SVD)
+\end_layout
+
+\begin_layout Standard
+Singularne vrednosti 
+\begin_inset Formula $A$
+\end_inset
+
+ so kvadratni koreni lastnih vrednosti 
+\begin_inset Formula $A^{*}A$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Št.
+ ničelnih singvr 
+\begin_inset Formula $=\dim\text{Ker}\left(A^{*}A\right)=\dim\text{Ker}A$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Št.
+ nenič.
+ singvr 
+\begin_inset Formula $n\times n$
+\end_inset
+
+ matrike 
+\begin_inset Formula $=n-\dim\text{Ker}A=\text{rang}A$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Za posplošeno diagonalno matriko 
+\begin_inset Formula $D$
+\end_inset
+
+ velja 
+\begin_inset Formula $\forall i,j:i\not=j\Rightarrow D_{ij}=0$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Izred o SVD: 
+\begin_inset Formula $\forall A\in M_{m\times n}\left(\mathbb{C}\right)\exists\text{unit. }Q_{1},\text{unit. }Q_{2},\text{diag. }D\ni:A=Q_{1}DQ_{2}^{-1}=Q_{1}DQ_{2}^{*}$
+\end_inset
+
+.
+ Diagonalci 
+\begin_inset Formula $D$
+\end_inset
+
+ so singvr 
+\begin_inset Formula $A$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $A^{*}=Q_{2}D^{*}Q_{1}^{*}$
+\end_inset
+
+, 
+\begin_inset Formula $A^{*}A=Q_{2}D^{*}DQ_{1}^{*}\sim D^{*}D$
+\end_inset
+
+.
+ Diagonalci 
+\begin_inset Formula $D^{*}D$
+\end_inset
+
+ so lavr 
+\begin_inset Formula $A^{*}A$
+\end_inset
+
+ in stolpci 
+\begin_inset Formula $Q_{2}$
+\end_inset
+
+ so ONB lave 
+\begin_inset Formula $A^{*}A$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Konstrukcija 
+\begin_inset Formula $Q_{2}$
+\end_inset
+
+: ONB iz pripadajočih ONB 
+\begin_inset Formula $A^{*}A$
+\end_inset
+
+.
+ 
+\begin_inset Formula $r=\text{rang}A$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Konstrukcija 
+\begin_inset Formula $Q_{1}$
+\end_inset
+
+: 
+\begin_inset Formula $\forall i\in\left\{ 1..r\right\} :u_{i}=\frac{1}{\sigma_{i}}Av_{i}$
+\end_inset
+
+.
+ 
+\begin_inset Formula $\left\{ u_{1},\dots,u_{r}\right\} $
+\end_inset
+
+ dopolnimo do ONB, 
+\begin_inset Formula $Q_{1}=\left[\begin{array}{ccccc}
+u_{1} & \cdots & u_{r} & \cdots & u_{m}\end{array}\right]$
+\end_inset
+
+ unitarna (ONB stolpci)
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{multicols}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document