15 files changed, 5391 insertions, 0 deletions

diff --git a/.gitignore b/.gitignore
index 524229a..1e4a91f 100644
--- a/.gitignore
+++ b/.gitignore
@@ -16,3 +16,4 @@ vendor/
 node_modules/
 *.class
 javac.*.args
+*.emergency
diff --git a/prog/sear.php/index.php b/prog/sear.php/index.php
index 1f90c9c..2e695c5 100644
--- a/prog/sear.php/index.php
+++ b/prog/sear.php/index.php
@@ -59,6 +59,8 @@ function query_google ($s, $image = false, $noredirect = false, $bindstring = fa
 	}
 	$results = [];
 	$x = new DOMDocument();
+	// file_put_contents("/tmp/x", $txtdoc);
+	$txtdoc = mb_convert_encoding($txtdoc, "ISO-8859-1");
 	$x->loadHTML($txtdoc);
 	if ($image)
 		foreach ($x->getElementsByTagName("div") as $div) {
diff --git a/skripti/namesti.sh b/skripti/namesti.sh
index 88253bf..e36a894 100755
--- a/skripti/namesti.sh
+++ b/skripti/namesti.sh
@@ -7,4 +7,5 @@ Uporabnik $DOAS_USER na računalniku `hostname` je pravkar zahteval poganjanje p
 EOF
 p=`grep -Eo '[a-zA-Z0-9+][a-zA-Z0-9_+.-]*[a-zA-Z0-9_+./]*' <<<$1 | head -n1`
 echo Začel bom nameščanje $p.
+[ $DOAS_USER = dijak ] && echo "PREKINJAM NAMEŠČANJE. uporabnik dijak ne sme nameščati programov iz varnostnih razlogov."
 echo | emerge $p
diff --git a/skripti/registracija_uporabnika.sh b/skripti/registracija_uporabnika.sh
new file mode 100755
index 0000000..a2419fa
--- /dev/null
+++ b/skripti/registracija_uporabnika.sh
@@ -0,0 +1,12 @@
+#!/bin/bash
+# $1 je username, $2 je fullname, $3 je komentar, $4 je expiredate YYYY-MM-DD, $5 je groups, $6 je crypt password
+# GECOS: full name,room number,office tel,home tel,other contact
+set -euo pipefail
+username=`grep -o "[a-z][-_.a-z0-9]*" <<<"$1" | head -n1 | head -c16`
+if [ ! "$username" = "$1" ]
+then
+	echo uporabniško ime ni pravilne oblike. predlagam takole uporabniško ime: $username -- uporabnika nisem izdelal
+	exit 1
+fi
+useradd --expiredate "$4" --groups "$5" --password "$6" --comment "`sed -e s/,//g<<<$2`,,,,`sed -e s/,//<<<$3`-reg.sh $DOAS_USER `date +%s`" $username
+echo uporabnik uspešno izdelan!
diff --git a/skripti/transcribe_large_file.sh b/skripti/transcribe_large_file.sh
new file mode 100755
index 0000000..e9bcc27
--- /dev/null
+++ b/skripti/transcribe_large_file.sh
@@ -0,0 +1,14 @@
+#!/bin/bash
+set -xeuo pipefail
+p=`rev <<<$0 | cut -d/ -f1 | rev`
+d=`mktemp -p "" -d $p.XXX`
+trap "rm -fr $d" EXIT
+ffmpeg -i $1 -f segment -segment_time 240 -acodec pcm_s16le -ac 1 -ar 16000 $d/%03d.wav
+for i in $d/*.wav
+do
+	curl -X 'POST' \
+		'http://splet.4a.si.:8000/api/transcribe' \
+		-H 'accept: application/json' \
+		-H 'Content-Type: multipart/form-data' \
+		-F "audio_file=@$i;type=audio/wav" | jq --raw-output .result > $1.`rev <<<$i | cut -d. -f2 | cut -d/ -f1 | rev`.txt
+done
diff --git a/šola/ars/kol1.txt b/šola/ars/kol1.txt
new file mode 100644
index 0000000..8d10834
--- /dev/null
+++ b/šola/ars/kol1.txt
@@ -0,0 +1,51 @@
+registri:	x0	zero	0
+		x1	ra	return address
+		x2	sp	stack pointer
+		x5-7	t0-2	temporary
+		x8-9	fp/s0-1	saved
+		x10-17	a0-7	argument
+		x18-27	s2-11	saved
+		x28-31	t3-6	temporary
+
+psevdoinštrukcije:
+
+	delta = symbol - pc
+	la rd, sym		auipc rd, delta[31:12]+delta[11]; addi rd, rd, delta[11:0]
+	lla rd, sym		TODO
+	l[bhwd] rd, sym		auipc rd, delta[31:12]+delta[11]; l[bhwd] rd, delta[11:0](rd)
+	s[bhwd] rd, sym, rt	auipc rt, delta[31:12]+delta[11]; s[bhws] rd, delta[11:0](rt)
+
+	nop		addi x0, x0, 0
+	mv rd, rs	addi rd, rs, 0
+	not rd, rs	xori rd, rs, -1
+	neg rd, rs	sub rd, x0, rs
+
+	bgt[u] rs, rt, off	blt[u] rt, rs, off
+	ble[u] rs, rt, off	bge[u] rt, rs, off
+	
+	j off	jal x0, off
+	jal off	jal x1, off
+	jr rs	jalr x0, 0(rs)
+	jalr rs	jalr x1, 0(rs)
+	ret	jalr x0, 0(x1)
+	call of	auipc x1, of[31:12]+of[11]; jalr x1, of[11:0](x1)
+	tail of auipc x6, of[31:12]+of[11]; jalr x0, of[11:0](x6)
+	
+float:	(-1)^s*2^(E-e)*1,M_1 M_2 M_3 M_4...M_n
+		exp	e	man
+	single	32	8	127	23
+	double	64	11	1023	52
+
+	inf: E=255/2047, m=0
+	nan: E=255/2047, m!=0
+	računamo z več biti: varovalni, zaokroževalni, lepljivi
+
+.data, .text, .byte, .half, .word, .dword, .align
+
+a*10 = a*(8+2) = a*8+a*2
+
+rabimo ceil(log_base(1/error)) decimalk
+
+Dostop do visokih naslovov: %hi(A) -- top 20b, %lo(A) low 12b
+
+TODO: slide 3b množenje floatov, chapter 2.4 intager computation
diff --git a/šola/ds2/kolokvij1.lyx b/šola/ds2/kolokvij1.lyx
new file mode 100644
index 0000000..2227d5e
--- /dev/null
+++ b/šola/ds2/kolokvij1.lyx
@@ -0,0 +1,1779 @@
+#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}}}
+\DeclareMathOperator{\g}{g}
+\DeclareMathOperator{\sled}{sled}
+\DeclareMathOperator{\Aut}{Aut}
+\DeclareMathOperator{\Cir}{Cir}
+\DeclareMathOperator{\ecc}{ecc}
+\DeclareMathOperator{\rad}{rad}
+\DeclareMathOperator{\diam}{diam}
+\newcommand\euler{e}
+\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 xetex
+\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 1cm
+\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
+setlength{
+\backslash
+columnseprule}{0.2pt}
+\backslash
+begin{multicols}{2}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Največja stopnja 
+\begin_inset Formula $\Delta G$
+\end_inset
+
+, najmanjša 
+\begin_inset Formula $\delta G$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Rokovanje: 
+\begin_inset Formula $\sum_{v\in VG}\deg_{G}v=2\left|EG\right|$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Vsak graf ima sodo mnogo vozlišč lihe stopnje.
+\end_layout
+
+\begin_layout Standard
+Presek/unija grafov je presek/unija njunih 
+\begin_inset Formula $V$
+\end_inset
+
+ in 
+\begin_inset Formula $E$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G\cup H$
+\end_inset
+
+ je disjunktna unija grafov 
+\begin_inset Formula $\sim$
+\end_inset
+
+ 
+\begin_inset Formula $VG\cap VH=\emptyset$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Komplement grafa: 
+\begin_inset Formula $\overline{G}$
+\end_inset
+
+ (obratna povezanost)
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $0\leq\left|EG\right|\leq{\left|VG\right| \choose 2}\quad\quad\quad$
+\end_inset
+
+Za padajoče 
+\begin_inset Formula $d_{i}$
+\end_inset
+
+ velja:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\left(d_{1},\dots d_{n}\right)$
+\end_inset
+
+ graf 
+\begin_inset Formula $\Leftrightarrow\left(d_{2}-1,\dots,d_{d_{1}+1}-1,\dots,d_{n}\right)$
+\end_inset
+
+ graf
+\end_layout
+
+\begin_layout Standard
+Če je 
+\begin_inset Formula $AG$
+\end_inset
+
+ matrika sosednosti, 
+\begin_inset Formula $\left(\left(AG\right)^{n}\right)_{i,j}$
+\end_inset
+
+ pove št.
+ 
+\begin_inset Formula $i,j-$
+\end_inset
+
+poti.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+\text{Število trikotnikov: }\frac{\sled\left(\left(AG\right)^{3}\right)}{2\cdot3}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Sprehod
+\end_layout
+
+\begin_layout Standard
+je zaporedje vozlišč, ki so verižno povezana.
+\end_layout
+
+\begin_layout Standard
+Dolžina sprehoda je število prehojenih povezav.
+\end_layout
+
+\begin_layout Standard
+Sklenjen sprehod dolžine 
+\begin_inset Formula $k$
+\end_inset
+
+: 
+\begin_inset Formula $v_{0}=v_{k}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Enostaven sprehod ima disjunktna vozlišča razen 
+\begin_inset Formula $\left(v_{0},v_{k}\right)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Pot v grafu: podgraf 
+\begin_inset Formula $P_{k}$
+\end_inset
+
+ 
+\begin_inset Formula $\sim$
+\end_inset
+
+ enostaven nesklenjen sprehod.
+\end_layout
+
+\begin_layout Paragraph
+Cikel
+\end_layout
+
+\begin_layout Standard
+podgraf, ki je enostaven sklenjen sprehod dolžine 
+\begin_inset Formula $>3$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Če v 
+\begin_inset Formula $G$
+\end_inset
+
+ 
+\begin_inset Formula $\exists$
+\end_inset
+
+ dve različni 
+\begin_inset Formula $u,v-$
+\end_inset
+
+poti 
+\begin_inset Formula $\Leftrightarrow$
+\end_inset
+
+ 
+\begin_inset Formula $G$
+\end_inset
+
+ premore cikel
+\end_layout
+
+\begin_layout Standard
+Sklenjen sprehod lihe dolžine 
+\begin_inset Formula $\in G$
+\end_inset
+
+ 
+\begin_inset Formula $\Leftrightarrow$
+\end_inset
+
+ lih cikel 
+\begin_inset Formula $\in G$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Povezanost
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $u,v$
+\end_inset
+
+ sta v isti komponenti 
+\begin_inset Formula $\text{\ensuremath{\sim}}$
+\end_inset
+
+ 
+\begin_inset Formula $\text{\ensuremath{\exists}}$
+\end_inset
+
+ 
+\begin_inset Formula $u,v-$
+\end_inset
+
+pot
+\end_layout
+
+\begin_layout Standard
+Število komponent grafa: 
+\begin_inset Formula $\Omega G$
+\end_inset
+
+.
+ 
+\begin_inset Formula $G$
+\end_inset
+
+ povezan 
+\begin_inset Formula $\sim\Omega G=1$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Komponenta je maksimalen povezan podgraf.
+\end_layout
+
+\begin_layout Standard
+Premer: 
+\begin_inset Formula $\diam G=\max\left\{ d_{G}\left(v,u\right);\forall v,u\in VG\right\} $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Ekscentričnost: 
+\begin_inset Formula $\ecc_{G}u=max\left\{ d_{G}\left(u,x\right);\forall x\in VG\right\} $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Polmer: 
+\begin_inset Formula $\rad G=\min\left\{ \ecc u;\forall u\in VG\right\} $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\diam C_{n}=\rad C_{n}=\lfloor\frac{n}{2}\rfloor$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+(Liha) ožina (
+\begin_inset Formula $\g G$
+\end_inset
+
+) je dolžina najkrajšega (lihega) cikla.
+\end_layout
+
+\begin_layout Standard
+Vsaj en od 
+\begin_inset Formula $G$
+\end_inset
+
+ in 
+\begin_inset Formula $\overline{G}$
+\end_inset
+
+ je povezan.
+\end_layout
+
+\begin_layout Standard
+Povezava 
+\begin_inset Formula $e$
+\end_inset
+
+ je most 
+\begin_inset Formula $\Leftrightarrow$
+\end_inset
+
+ 
+\begin_inset Formula $\Omega\left(G-e\right)>\Omega G$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $u$
+\end_inset
+
+ je prerezno vozlišče 
+\begin_inset Formula $\Leftrightarrow$
+\end_inset
+
+ 
+\begin_inset Formula $\Omega\left(G-u\right)>\Omega G$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Za nepovezan 
+\begin_inset Formula $G$
+\end_inset
+
+ velja 
+\begin_inset Formula $\diam\overline{G}\leq2$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Dvodelni
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\sim V=A\cup B,A\cap B=\emptyset,\forall uv\in E:u\in A\oplus v\in A$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $K_{m,n}$
+\end_inset
+
+ je poln dvodelni graf, 
+\begin_inset Formula $\left|A\right|=m$
+\end_inset
+
+, 
+\begin_inset Formula $\left|B\right|=n$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G$
+\end_inset
+
+ dvodelen 
+\begin_inset Formula $\Leftrightarrow\forall$
+\end_inset
+
+ komponenta 
+\begin_inset Formula $G$
+\end_inset
+
+ dvodelna
+\end_layout
+
+\begin_layout Standard
+Pot, sod cikel, hiperkocka so dvodelni, petersenov ni.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G$
+\end_inset
+
+ dvodelen 
+\begin_inset Formula $\Leftrightarrow G$
+\end_inset
+
+ ne vsebuje lihega cikla.
+\end_layout
+
+\begin_layout Standard
+Biparticija glede na parnost 
+\begin_inset Formula $d_{G}\left(u,x_{0}\right)$
+\end_inset
+
+, 
+\begin_inset Formula $x_{0}$
+\end_inset
+
+ fiksen.
+\end_layout
+
+\begin_layout Standard
+Dvodelen 
+\begin_inset Formula $k-$
+\end_inset
+
+regularen, 
+\begin_inset Formula $\left|E\right|\ge1\Rightarrow$
+\end_inset
+
+
+\begin_inset Formula $\left|A\right|=\left|B\right|$
+\end_inset
+
+.
+ Dokaz: 
+\begin_inset Formula $\sum_{u\in A}\deg u=\left|E\right|=\cancel{k}\left|A\right|=\cancel{k}\left|B\right|=\sum_{u\in B}\deg u$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Krožni
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\Cir\left(n,\left\{ s_{1},\dots,s_{k}\right\} \right):\left|V\right|=n,$
+\end_inset
+
+ množica preskokov
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\Omega\Cir\left(n,\left\{ s,n-s\right\} \right)=\gcd\left\{ n,s\right\} $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $W_{n}$
+\end_inset
+
+ (kolo) pa je cikel z univerzalnim vozliščem.
+\end_layout
+
+\begin_layout Paragraph
+Homomorfizem
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\varphi:VG\to VH$
+\end_inset
+
+ slika povezave v povezave
+\end_layout
+
+\begin_layout Standard
+Primer: 
+\begin_inset Formula $K_{2}$
+\end_inset
+
+ je homomorfna slika 
+\begin_inset Formula $\forall$
+\end_inset
+
+ bipartitnega grafa.
+\end_layout
+
+\begin_layout Standard
+V povezavah in vozliščih surjektiven 
+\begin_inset Formula $hm\varphi$
+\end_inset
+
+ je epimorfizem.
+\end_layout
+
+\begin_layout Standard
+V vozliščih injektiven 
+\begin_inset Formula $hm\varphi$
+\end_inset
+
+ je monomorfizem/vložitev.
+\end_layout
+
+\begin_layout Standard
+Vložitev, ki ohranja razdalje, je izometrična.
+\end_layout
+
+\begin_layout Standard
+Kompozitum homomorfizmov je spet homomorfizem.
+\end_layout
+
+\begin_layout Standard
+Izomorfizem je bijektivni 
+\begin_inset Formula $hm\varphi$
+\end_inset
+
+ s homomorfnim inverzom.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $im\varphi$
+\end_inset
+
+ 
+\begin_inset Formula $f:VG\to VH$
+\end_inset
+
+ 
+\begin_inset Formula $\forall u,v\in VG:uv\in EG\Leftrightarrow fufv\in EH$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Nad množico vseh grafov 
+\begin_inset Formula $\mathcal{G}$
+\end_inset
+
+ je izomorfizem (
+\begin_inset Formula $\cong$
+\end_inset
+
+) ekv.
+ rel.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $im\varphi$
+\end_inset
+
+ 
+\begin_inset Formula $G\to G$
+\end_inset
+
+ je avtomorfizem.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\Aut G$
+\end_inset
+
+ je grupa avtomorfizmov s komponiranjem.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\Aut K_{n}=\left\{ \pi\in S_{n}=\text{permutacije }n\text{ elementov}\right\} $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\Aut P_{n}=\left\{ \text{trivialni }id,f\left(i\right)=n-i-1\right\} $
+\end_inset
+
+, 
+\begin_inset Formula $\Aut G\cong\Aut\overline{G}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Izomorfizem ohranja stopnje, št.
+ 
+\begin_inset Formula $C_{4}$
+\end_inset
+
+, povezanost, 
+\begin_inset Formula $\left|EG\right|,\dots$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G\cong\overline{G}\Rightarrow\left|VG\right|\%4\in\left\{ 0,1\right\} $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Operacije
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $H$
+\end_inset
+
+ vpeti podgraf 
+\begin_inset Formula $G\Leftrightarrow\exists F\subseteq EG\ni:H=G-F$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $H$
+\end_inset
+
+ inducirani podgraf 
+\begin_inset Formula $G\Leftrightarrow\exists S\subseteq VG\ni:H=G-S$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $H$
+\end_inset
+
+ podgraf 
+\begin_inset Formula $G\Leftrightarrow\exists S\subseteq VG,F\subseteq EG\ni:H=\left(G-F\right)-S$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $uv=e\in EG$
+\end_inset
+
+.
+ 
+\begin_inset Formula $G/e$
+\end_inset
+
+ je skrčitev.
+ (identificiramo 
+\begin_inset Formula $u=v$
+\end_inset
+
+)
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $H$
+\end_inset
+
+ minor 
+\begin_inset Formula $G$
+\end_inset
+
+: 
+\begin_inset Formula $H=f_{1}...f_{k}G$
+\end_inset
+
+ za 
+\begin_inset Formula $f_{i}$
+\end_inset
+
+ skrčitev/odstranjevanje
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $VG^{+}e\coloneqq VG\cup\left\{ x_{e}\right\} $
+\end_inset
+
+, 
+\begin_inset Formula $EG^{+}e\coloneqq EG\setminus e\cup\left\{ x_{e}u,x_{e}v\right\} $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $VG^{+}e$
+\end_inset
+
+ je subdivizija, 
+\begin_inset Formula $e\in EG$
+\end_inset
+
+.
+ Na 
+\begin_inset Formula $e$
+\end_inset
+
+ dodamo vozlišče.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $H$
+\end_inset
+
+ subdivizija 
+\begin_inset Formula $G$
+\end_inset
+
+ 
+\begin_inset Formula $\Leftrightarrow H=G^{+}\left\{ e_{1},\dots,e_{k}\right\} ^{+}\left\{ f_{1}\dots f_{j}\right\} ^{+}\dots$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Stopnja vozlišč se s subdivizijo ne poveča.
+\end_layout
+
+\begin_layout Standard
+Glajenje 
+\begin_inset Formula $G^{-}v$
+\end_inset
+
+, 
+\begin_inset Formula $v\in VG$
+\end_inset
+
+ je obrat subdivizije.
+ 
+\begin_inset Formula $\deg_{G}v=2$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G$
+\end_inset
+
+ in 
+\begin_inset Formula $H$
+\end_inset
+
+ sta homeomorfna, če sta subdivizija istega grafa.
+\end_layout
+
+\begin_layout Standard
+Kartezični produkt: 
+\begin_inset Formula $V\left(G\square H\right)\coloneqq VG\times VH$
+\end_inset
+
+, 
+\begin_inset Formula $E\left(G\square H\right)\coloneqq\left\{ \left\{ \left(g,h\right),\left(g',h'\right)\right\} ;g=g'\wedge hh'\in EH\vee h=h'\wedge gg'\in EG\right\} $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\left(\mathcal{G},\square\right)$
+\end_inset
+
+ monoid, enota 
+\begin_inset Formula $K_{1}$
+\end_inset
+
+.
+ 
+\begin_inset Formula $Q_{n}\cong Q_{n-1}\square K_{2}=Q_{n-2}\square K_{2}^{\square,2}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Disjunktna unija: 
+\begin_inset Formula $G$
+\end_inset
+
+, 
+\begin_inset Formula $H$
+\end_inset
+
+ disjunktna.
+ 
+\begin_inset Formula $V\left(G\cup H\right)\coloneqq VG\cup VH$
+\end_inset
+
+, 
+\begin_inset Formula $E\left(G\cup H\right)\coloneqq EG\cup EH$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G,H$
+\end_inset
+
+ dvodelna 
+\begin_inset Formula $\Rightarrow G\square H$
+\end_inset
+
+ dvodelen
+\end_layout
+
+\begin_layout Paragraph
+\begin_inset Formula $k-$
+\end_inset
+
+povezan graf
+\end_layout
+
+\begin_layout Standard
+ima 
+\begin_inset Formula $\geq k+1$
+\end_inset
+
+ vozlišč in ne vsebuje prerezne množice moči 
+\begin_inset Formula $<k$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $X\subseteq VG$
+\end_inset
+
+ je prerezna množica 
+\begin_inset Formula $\Leftrightarrow\Omega\left(G-X\right)>\Omega G$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $Y\subseteq EG$
+\end_inset
+
+ prerezna množica povezav 
+\begin_inset Formula $\Leftrightarrow\Omega\left(G-Y\right)>\Omega G$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Povezanost grafa: 
+\begin_inset Formula $\kappa G=\max k$
+\end_inset
+
+, da je 
+\begin_inset Formula $G$
+\end_inset
+
+ 
+\begin_inset Formula $k-$
+\end_inset
+
+povezan.
+ Primeri: 
+\begin_inset Formula $\kappa K_{n}=n-1$
+\end_inset
+
+, 
+\begin_inset Formula $\kappa P_{n}=1$
+\end_inset
+
+, 
+\begin_inset Formula $\kappa C_{n}=2$
+\end_inset
+
+, 
+\begin_inset Formula $\kappa K_{n,m}=\min\left\{ n,m\right\} $
+\end_inset
+
+, 
+\begin_inset Formula $\kappa Q_{n}=n$
+\end_inset
+
+, 
+\begin_inset Formula $\kappa G\text{\ensuremath{\leq}}\delta G$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Izrek (Menger): 
+\begin_inset Formula $\left|VG\right|\geq k+1$
+\end_inset
+
+: 
+\begin_inset Formula $G$
+\end_inset
+
+ 
+\begin_inset Formula $k-$
+\end_inset
+
+povezan 
+\begin_inset Formula $\Leftrightarrow\forall u,v\in VG,uv\not\in EG:\exists k$
+\end_inset
+
+ notranje disjunktnih 
+\begin_inset Formula $u,v-$
+\end_inset
+
+poti
+\end_layout
+
+\begin_layout Standard
+Graf je 
+\begin_inset Formula $k-$
+\end_inset
+
+povezan po povezavah, če ne vsebuje prerezne množice povezav moči 
+\begin_inset Formula $<k$
+\end_inset
+
+.
+ Povezanost grafa po povezavah: 
+\begin_inset Formula $\kappa'G=\max k$
+\end_inset
+
+, da je 
+\begin_inset Formula $G$
+\end_inset
+
+ 
+\begin_inset Formula $k-$
+\end_inset
+
+povezan po povezavah.
+ Primeri: 
+\begin_inset Formula $\kappa'K_{n}=n-1$
+\end_inset
+
+, 
+\begin_inset Formula $\kappa'P_{n}=1$
+\end_inset
+
+, 
+\begin_inset Formula $\kappa'C_{n}=2$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Izrek (Menger'): 
+\begin_inset Formula $G$
+\end_inset
+
+ 
+\begin_inset Formula $k-$
+\end_inset
+
+povezan 
+\begin_inset Formula $\Leftrightarrow\forall u,v\in VG\exists k$
+\end_inset
+
+ po povezavah disjunktnih 
+\begin_inset Formula $u,v-$
+\end_inset
+
+poti
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\forall G\in\mathcal{G}:\kappa G\leq\kappa'G\leq\delta G$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Drevo
+\end_layout
+
+\begin_layout Standard
+je povezan gozd.
+ Gozd je graf brez ciklov.
+\end_layout
+
+\begin_layout Standard
+Drevo z vsaj dvema vozliščema premore dva lista.
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+hspace*{
+\backslash
+fill}
+\end_layout
+
+\end_inset
+
+NTSE:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G$
+\end_inset
+
+ drevo 
+\begin_inset Formula $\Leftrightarrow\forall u,v\in VG\exists!u,v-$
+\end_inset
+
+pot 
+\begin_inset Formula $\Leftrightarrow\Omega G=1\wedge\forall e\in EG:e$
+\end_inset
+
+ most 
+\begin_inset Formula $\Leftrightarrow\Omega G=1\wedge\left|EG\right|=\left|VG\right|-1$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Vpeto drevo grafa je vpet podgraf, ki je drevo.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\tau G\sim$
+\end_inset
+
+ število vpetih dreves.
+ 
+\begin_inset Formula $\Omega G=1\Leftrightarrow\tau G\geq1$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\forall e\in EG:\tau G=\tau\left(G-e\right)+\tau\left(G/e\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+\text{\text{\text{Laplaceova matrika: }}}L\left(G\right)_{ij}=\begin{cases}
+\deg_{G}v_{i} & ;i=j\\
+-\left(\text{št. uv povezav}\right) & ;\text{drugače}
+\end{cases}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Izrek (Kirchoff): za 
+\begin_inset Formula $G$
+\end_inset
+
+ povezan multigraf je 
+\begin_inset Formula $\forall i:\tau G=\det\left(LG\text{ brez \ensuremath{i}te vrstice in \ensuremath{i}tega stolpca}\right)$
+\end_inset
+
+.
+ 
+\begin_inset Formula $\tau K_{n}=n^{n-2}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Prüferjeva koda, če lahko linearno uredimo vozlišča: Ponavljaj, dokler 
+\begin_inset Formula $VG\not=\emptyset$
+\end_inset
+
+: vzemi prvi list, ga odstrani in v vektor dodaj njegovega soseda.
+\end_layout
+
+\begin_layout Standard
+Blok je maksimalen povezan podgraf brez prereznih vozlišč.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\tau G=\tau B_{1}\cdot\cdots\cdot\tau B_{k}$
+\end_inset
+
+ za bloke 
+\begin_inset Formula $\vec{B}$
+\end_inset
+
+ grafa 
+\begin_inset Formula $G$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{multicols}
+\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
+Eulerjev
+\end_layout
+
+\begin_layout Standard
+sprehod v m.grafu vsebuje vse povezave po enkrat.
+\end_layout
+
+\begin_layout Standard
+Eulerjev obhod je sklenjen eulerjev sprehod.
+\end_layout
+
+\begin_layout Standard
+Eulerjev graf premore eulerjev obhod.
+\end_layout
+
+\begin_layout Standard
+Za povezan multigraf 
+\begin_inset Formula $G$
+\end_inset
+
+ eulerjev 
+\begin_inset Formula $\Leftrightarrow\forall v\in VG:\deg_{G}v$
+\end_inset
+
+ sod
+\end_layout
+
+\begin_layout Standard
+Fleuryjev algoritem za eulerjev obhod v eulerjevem grafu: Začnemo v poljubni
+ povezavi, jo izbrišemo, nadaljujemo na mostu le, če nimamo druge možne
+ povezave.
+\end_layout
+
+\begin_layout Standard
+Dekompozicija: delitev na povezavno disjunktne podgrafe.
+\end_layout
+
+\begin_layout Standard
+Dekompozicija je lepa, če so podgrafi izomorfni.
+ (
+\begin_inset Formula $\exists$
+\end_inset
+
+ za 
+\begin_inset Formula $P_{5,2}$
+\end_inset
+
+)
+\end_layout
+
+\begin_layout Standard
+Vsak eulerjev graf premore dekompozicijo v cikle.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $Q_{n}$
+\end_inset
+
+ eulerjev 
+\begin_inset Formula $\Leftrightarrow n$
+\end_inset
+
+ sod
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $K_{m,n,p}$
+\end_inset
+
+ eulerjev 
+\begin_inset Formula $\Leftrightarrow m,n,p$
+\end_inset
+
+ iste parnosti
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G$
+\end_inset
+
+ eulerjev 
+\begin_inset Formula $\wedge H$
+\end_inset
+
+ eulerjev 
+\begin_inset Formula $\Rightarrow G\square H$
+\end_inset
+
+ eulerjev
+\end_layout
+
+\begin_layout Paragraph
+Hamiltonov
+\end_layout
+
+\begin_layout Standard
+cikel vsebuje vsa vozlišča grafa.
+\end_layout
+
+\begin_layout Standard
+Hamilton graf premore Hamiltonov cikel.
+\end_layout
+
+\begin_layout Standard
+Hamiltonova pot vsebuje vsa vozlišča.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G$
+\end_inset
+
+ hamiltonov, 
+\begin_inset Formula $S\subseteq VG\Rightarrow\Omega\left(G-S\right)\leq\left|S\right|$
+\end_inset
+
+ torej:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\exists S\in VG:\Omega\left(G-S\right)>\left|S\right|\Rightarrow G$
+\end_inset
+
+ ni hamiltonov.
+ Primer: 
+\begin_inset Formula $G$
+\end_inset
+
+ vsebuje prerezno vozlišče 
+\begin_inset Formula $\Rightarrow G$
+\end_inset
+
+ ni hamiltonov.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $K_{n,m}$
+\end_inset
+
+ je hamiltonov 
+\begin_inset Formula $\Leftrightarrow m=n$
+\end_inset
+
+ (za 
+\begin_inset Formula $S$
+\end_inset
+
+ vzamemo 
+\begin_inset Formula $\min\left\{ m,n\right\} $
+\end_inset
+
+)
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\left|VG\right|\geq3,\forall u,v\in VG:\deg_{G}u+\deg_{G}v\geq\left|VG\right|\Rightarrow G$
+\end_inset
+
+ hamil.
+\end_layout
+
+\begin_layout Standard
+Dirac: 
+\begin_inset Formula $\left|VG\right|\geq3,\forall u\in VG:\deg_{G}u\geq\frac{\left|VG\right|}{2}\Rightarrow G$
+\end_inset
+
+ hamilton.
+\end_layout
+
+\begin_layout Paragraph
+Ravninski
+\end_layout
+
+\begin_layout Standard
+graf brez sekanja povezav narišemo v ravnino
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $K_{2,3}$
+\end_inset
+
+ je ravninski, 
+\begin_inset Formula $K_{3,3}$
+\end_inset
+
+, 
+\begin_inset Formula $K_{5}$
+\end_inset
+
+, 
+\begin_inset Formula $C_{5}\square C_{5}$
+\end_inset
+
+ in 
+\begin_inset Formula $P_{5,2}$
+\end_inset
+
+ niso ravninski.
+\end_layout
+
+\begin_layout Standard
+Vložitev je ravninski graf z ustrezno risbo v ravnini.
+\end_layout
+
+\begin_layout Standard
+Lica so sklenjena območja vložitve brez vozlišč in povezav.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G$
+\end_inset
+
+ lahko vložimo v ravnino 
+\begin_inset Formula $\Leftrightarrow G$
+\end_inset
+
+ lahko vložimo na sfero.
+\end_layout
+
+\begin_layout Standard
+Dolžina lica 
+\begin_inset Formula $F\sim lF$
+\end_inset
+
+ je št.
+ povezav obhoda lica.
+\end_layout
+
+\begin_layout Standard
+Drevo je ravninski graf z enim licem dolžine 
+\begin_inset Formula $2\left|ET\right|$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\sum_{F\in FG}lF=2\left|EG\right|$
+\end_inset
+
+, 
+\begin_inset Formula $lF\geq gG$
+\end_inset
+
+ za ravninski 
+\begin_inset Formula $G$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $2\left|EG\right|=\sum_{F\in FG}lF\geq\sum_{F\in FG}gG=gG\left|FG\right|$
+\end_inset
+
+ (
+\begin_inset Formula $G$
+\end_inset
+
+ ravn.)
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G$
+\end_inset
+
+ ravninski 
+\begin_inset Formula $\Rightarrow\left|EG\right|\geq\frac{gG}{2}$
+\end_inset
+
+
+\begin_inset Formula $\left|FG\right|$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Euler: 
+\begin_inset Formula $\left|VG\right|-\left|EG\right|+\left|FG\right|=1+\Omega G$
+\end_inset
+
+ za ravninski 
+\begin_inset Formula $G$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G$
+\end_inset
+
+ ravninski, 
+\begin_inset Formula $\left|VG\right|\geq3\Rightarrow\left|EG\right|\leq3\left|VG\right|-6$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G$
+\end_inset
+
+ ravninski brez 
+\begin_inset Formula $C_{3}$
+\end_inset
+
+, 
+\begin_inset Formula $\left|VG\right|\geq3\Rightarrow\left|EG\right|\text{\ensuremath{\leq2\left|VG\right|-4}}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Triangulacija je taka vložitev, da so vsa lica omejena s 
+\begin_inset Formula $C_{3}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+V maksimalen ravninski graf ne moremo dodati povezave, da bi ostal ravninski.
+ 
+\begin_inset Formula $\sim$
+\end_inset
+
+ Ni pravi vpet podgraf ravn.
+ grafa.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $K_{5}-e$
+\end_inset
+
+ je maksimalen ravninski graf 
+\begin_inset Formula $\forall e\in EK_{5}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G$
+\end_inset
+
+ ravninski.
+ NTSE: 
+\begin_inset Formula $G$
+\end_inset
+
+ triangulacija 
+\begin_inset Formula $\Leftrightarrow G$
+\end_inset
+
+ maksimalen ravninski 
+\begin_inset Formula $\Leftrightarrow\left|EG\right|=3\left|VG\right|-6$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G$
+\end_inset
+
+ ravninski 
+\begin_inset Formula $\Leftrightarrow$
+\end_inset
+
+ vsaka subdivizija 
+\begin_inset Formula $G$
+\end_inset
+
+ ravninska.
+\end_layout
+
+\begin_layout Standard
+Kuratovski: 
+\begin_inset Formula $G$
+\end_inset
+
+ ravn.
+ 
+\begin_inset Formula $\Leftrightarrow$
+\end_inset
+
+ ne vsebuje subdivizije 
+\begin_inset Formula $K_{5}$
+\end_inset
+
+ ali 
+\begin_inset Formula $K_{3,3}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Wagner: 
+\begin_inset Formula $G$
+\end_inset
+
+ ravn.
+ 
+\begin_inset Formula $\Leftrightarrow$
+\end_inset
+
+ niti 
+\begin_inset Formula $K_{5}$
+\end_inset
+
+ niti 
+\begin_inset Formula $K_{3,3}$
+\end_inset
+
+ nista njegova minorja
+\end_layout
+
+\begin_layout Standard
+Zunanje-ravninski ima vsa vozlišča na robu zunanjega lica.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $2-$
+\end_inset
+
+povezan zunanje-ravninski je 
+\series bold
+hamiltonov
+\series default
+.
+\end_layout
+
+\begin_layout Standard
+Zunanje-ravninski 
+\begin_inset Formula $G$
+\end_inset
+
+, 
+\begin_inset Formula $\left|VG\right|\geq2$
+\end_inset
+
+ ima vozlišče stopnje 
+\begin_inset Formula $\leq2$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Paragraph
+Barvanje
+\end_layout
+
+\begin_layout Standard
+je taka 
+\begin_inset Formula $C:VG\to\left\{ 1..k\right\} \Leftrightarrow\forall uv\in EG:Cu\not=Cv$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Kromatično število 
+\begin_inset Formula $\chi G$
+\end_inset
+
+ je najmanjši 
+\begin_inset Formula $k$
+\end_inset
+
+, 
+\begin_inset Formula $\ni:\exists$
+\end_inset
+
+ 
+\begin_inset Formula $k-$
+\end_inset
+
+barvanje 
+\begin_inset Formula $G$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Primer: 
+\begin_inset Formula $\chi K_{n}=n$
+\end_inset
+
+, 
+\begin_inset Formula $\chi C_{n}=\begin{cases}
+2 & ;n\text{ sod}\\
+3 & ;n\text{ lih}
+\end{cases}$
+\end_inset
+
+
+\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
diff --git a/šola/ds2/teorija.lyx b/šola/ds2/teorija.lyx
new file mode 100644
index 0000000..f442310
--- /dev/null
+++ b/šola/ds2/teorija.lyx
@@ -0,0 +1,1911 @@
+#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 1cm
+\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
+setlength{
+\backslash
+columnseprule}{0.2pt}
+\backslash
+begin{multicols}{2}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Izjavni račun
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\forall\exists$
+\end_inset
+
+,
+\begin_inset Formula $\neg$
+\end_inset
+
+, 
+\begin_inset Formula $\wedge\uparrow\downarrow$
+\end_inset
+
+, 
+\begin_inset Formula $\vee\oplus$
+\end_inset
+
+, 
+\begin_inset Formula $\Rightarrow$
+\end_inset
+
+ (left to right), 
+\begin_inset Formula $\Leftrightarrow$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+absorbcija: 
+\begin_inset Formula $a\wedge\left(b\vee a\right)\sim a,\quad a\vee\left(b\wedge a\right)\sim a$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+kontrapozicija: 
+\begin_inset Formula $a\Rightarrow b\quad\sim\quad\neg a\vee b$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+osnovna konjunkcija 
+\begin_inset Formula $\coloneqq$
+\end_inset
+
+ minterm
+\end_layout
+
+\begin_layout Standard
+globina 
+\begin_inset Formula $\coloneqq$
+\end_inset
+
+ 
+\begin_inset Formula $\begin{cases}
+1 & \text{izraz nima veznikov}\\
+1+\max\left\{ A_{1}\dots A_{n}\right\}  & A_{i}\text{ param. zun. vezn.}
+\end{cases}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $A_{1},\dots,A_{n},B$
+\end_inset
+
+ je pravilen sklep, če 
+\begin_inset Formula $\vDash\bigwedge_{k=1}^{n}A_{k}\Rightarrow B$
+\end_inset
+
+.
+ 
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+zaključek 
+\begin_inset Formula $B$
+\end_inset
+
+ drži pri vseh tistih naborih vrednostih spremenljivk, pri katerih hkrati
+ držijo vse predpostavke 
+\begin_inset Formula $A_{i}$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+
+\series bold
+Pravila sklepanja
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{align*}
+\end_layout
+
+\begin_layout Plain Layout
+
+&& A, A
+\backslash
+Rightarrow B & 
+\backslash
+vDash B && 
+\backslash
+text{
+\backslash
+emph{modus ponens}} && 
+\backslash
+text{M.
+ P.}
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+&& A
+\backslash
+Rightarrow B, 
+\backslash
+neg B & 
+\backslash
+vDash 
+\backslash
+neg A && 
+\backslash
+text{
+\backslash
+emph{modus tollens}} && 
+\backslash
+text{M.
+ T.}
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+&& A
+\backslash
+wedge B, 
+\backslash
+neg B & 
+\backslash
+vDash A && 
+\backslash
+text{
+\backslash
+emph{disjunktivni silogizem}} && 
+\backslash
+text{D.
+ S.}
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+&& A
+\backslash
+Rightarrow B, B
+\backslash
+Rightarrow C & 
+\backslash
+vDash A
+\backslash
+Rightarrow C && 
+\backslash
+text{
+\backslash
+emph{hipotetični silogizem}} && 
+\backslash
+text{H.
+ S}
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+&& A, B & 
+\backslash
+vDash A
+\backslash
+wedge B && 
+\backslash
+text{
+\backslash
+emph{združitev}} && 
+\backslash
+text{Zd.}
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+&& A
+\backslash
+wedge B & 
+\backslash
+vDash A && 
+\backslash
+text{
+\backslash
+emph{poenostavitev}} && 
+\backslash
+text{Po.}
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{align*}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Protiprimer 
+\begin_inset Formula $1,\dots,1\vDash0$
+\end_inset
+
+ dokaže nepravilnost sklepa.
+\end_layout
+
+\begin_layout Paragraph
+
+\series bold
+Pomožni sklepi
+\series default
+:
+\end_layout
+
+\begin_layout Itemize
+Pogojni sklep (P.S.):
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+newline
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula $A_{1},\dots,A_{n}\vDash B\Rightarrow C\quad\sim\quad A_{1},\dots,A_{n},B\vDash C$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+S protislovjem (R.A.
+ – 
+\emph on
+reduction ad absurdum
+\emph default
+): 
+\begin_inset Formula $A_{1},\dots,A_{n}\vDash B\quad\sim\quad A_{1},\dots,A_{n},\neg B\vDash0$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+Analiza primerov (A.
+ P.): 
+\begin_inset Formula $A_{1},\dots,A_{n},B_{1}\vee B_{2}\vDash C\sim\left(A_{1},\dots,A_{n},B_{1}\vDash C\right)\wedge\left(A_{1},\dots,A_{n},B_{2}\vDash C\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $A_{1},\dots,A_{n},B_{1}\wedge B_{2}\vDash C\quad\sim\quad A_{1},\dots,A_{n},B_{1},B_{2}\vDash C$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Predikatni račun
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $P:D^{n}\longrightarrow\left\{ 0,1\right\} $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+De Morganov zakon negacije:
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $\forall x:\neg P\left(x\right)\quad\sim\quad\neg\exists x:P\left(x\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $\exists x:\neg P\left(x\right)\quad\sim\quad\neg\forall x:P\left(x\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Izjava je zaprta izjavna formula, torej taka, ki ne vsebuje prostih (
+\begin_inset Formula $=$
+\end_inset
+
+nevezanih) nastopov spremenljivk.
+\end_layout
+
+\begin_layout Paragraph
+Množice
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $^{\mathcal{C}},\cap\backslash,\cup\oplus$
+\end_inset
+
+ (left to right)
+\end_layout
+
+\begin_layout Standard
+Distributivnost: 
+\begin_inset Formula $\cup\cap$
+\end_inset
+
+, 
+\begin_inset Formula $\cap\cup$
+\end_inset
+
+, 
+\begin_inset Formula $\left(\mathcal{A}\oplus\mathcal{B}\right)\cap\mathcal{C}=\left(\mathcal{A\cap\mathcal{C}}\right)\oplus\left(\mathcal{B}\cap\mathcal{C}\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Asociativnost: 
+\begin_inset Formula $\oplus\cup\cap$
+\end_inset
+
+.
+ Distributivnost: 
+\begin_inset Formula $\oplus\cup\cap$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Absorbcija: 
+\begin_inset Formula $\mathcal{A}\cup\left(\mathcal{A}\cap\mathcal{B}\right)=\mathcal{A}=A\cap\left(\mathcal{A}\cup\mathcal{B}\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\mathcal{A}\subseteq\mathcal{B}\Leftrightarrow\mathcal{A}\cup\mathcal{B}=\mathcal{B}\Leftrightarrow\mathcal{A}\cup\mathcal{B}=\mathcal{A}\Leftrightarrow\mathcal{A}\backslash\mathcal{B}=\emptyset\Leftrightarrow\mathcal{B}^{\mathcal{C}}\subseteq\mathcal{A^{\mathcal{C}}}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\mathcal{A}=\mathcal{B}\Longleftrightarrow\mathcal{A\oplus\mathcal{B}}=\emptyset$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\mathcal{A}=\emptyset\wedge\mathcal{B}=\emptyset\Longleftrightarrow\mathcal{A}\cup\mathcal{B}=\emptyset$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\left(\mathcal{X}\cap\mathcal{P}\right)\cup\left(\mathcal{X^{C}}\cap\mathcal{Q}\right)=\emptyset\Longleftrightarrow\text{\ensuremath{\mathcal{Q\subseteq X}\subseteq\mathcal{P^{C}}}}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\mathcal{A}\backslash\mathcal{B}\sim\mathcal{A}\cap\mathcal{B}^{C}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\mathcal{X}\cup\mathcal{X^{C}}=\emptyset$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\mathcal{W}=\mathcal{W}\cap\mathcal{U}=\mathcal{W\cap}\left(\mathcal{X}\cup\mathcal{X^{C}}\right)=\left(\mathcal{W}\cap\mathcal{X}\right)\cup\left(\mathcal{W}\cap\mathcal{X^{C}}\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\mathcal{A}\oplus\mathcal{B}=\left(\mathcal{A}\backslash\mathcal{B}\right)\cup\left(\mathcal{B\backslash\mathcal{A}}\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+
+\series bold
+Lastnosti binarnih relacij
+\end_layout
+
+\begin_layout Paragraph
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{align*}
+\end_layout
+
+\begin_layout Plain Layout
+
+&& 
+\backslash
+forall a
+\backslash
+in A : & 
+\backslash
+left(a R a
+\backslash
+right) && 
+\backslash
+text{refleksivnost}
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+&& 
+\backslash
+forall a,b
+\backslash
+in A : & 
+\backslash
+left(a R b
+\backslash
+Rightarrow b R a
+\backslash
+right)&&
+\backslash
+text{simetričnost}
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+&& 
+\backslash
+forall a,b
+\backslash
+in A : & 
+\backslash
+left(a R b
+\backslash
+wedge b R a
+\backslash
+Rightarrow a=b
+\backslash
+right) && 
+\backslash
+text{antisimetričnost}
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+&& 
+\backslash
+forall a,b,c
+\backslash
+in A : & 
+\backslash
+left(a R b
+\backslash
+wedge b R c
+\backslash
+Rightarrow a R c
+\backslash
+right) && 
+\backslash
+text{tranzitivnost}
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+&& 
+\backslash
+forall a
+\backslash
+in A : & 
+\backslash
+neg
+\backslash
+left(a R a
+\backslash
+right) && 
+\backslash
+text{irefleksivnost}
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+&& 
+\backslash
+forall a,b
+\backslash
+in A: &
+\backslash
+left(a R b
+\backslash
+Rightarrow 
+\backslash
+neg
+\backslash
+left(b R a
+\backslash
+right)
+\backslash
+right) && 
+\backslash
+text{asimetričnost}
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+&& 
+\backslash
+forall a,b,c
+\backslash
+in A:&
+\backslash
+left(a R b
+\backslash
+wedge b R c
+\backslash
+Rightarrow 
+\backslash
+neg
+\backslash
+left(a R c
+\backslash
+right)
+\backslash
+right) && 
+\backslash
+text{itranzitivnost}
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+&& 
+\backslash
+forall a,b
+\backslash
+in A:&
+\backslash
+left(a
+\backslash
+not=b
+\backslash
+Rightarrow
+\backslash
+left(a R b
+\backslash
+vee b R a
+\backslash
+right)
+\backslash
+right) && 
+\backslash
+text{sovisnost}
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+&& 
+\backslash
+forall a,b
+\backslash
+in A:&
+\backslash
+left(a R b
+\backslash
+vee b R a
+\backslash
+right)&&
+\backslash
+text{stroga sovisnost}
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+&&
+\backslash
+forall a,b,c
+\backslash
+in A:&
+\backslash
+left(aRb
+\backslash
+wedge aRc
+\backslash
+Rightarrow b=c
+\backslash
+right)&&
+\backslash
+text{enoličnost}
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{align*}
+\end_layout
+
+\end_inset
+
+Sklepanje s kvantifikatorji
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{align*}
+\end_layout
+
+\begin_layout Plain Layout
+
+&& 
+\backslash
+exists x:P
+\backslash
+left(x
+\backslash
+right) 
+\backslash
+longrightarrow& x_0
+\backslash
+coloneqq x, P
+\backslash
+left(x
+\backslash
+right) && 
+\backslash
+text{eksistenčna specifikacija}
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+&& P
+\backslash
+left(x_0
+\backslash
+right)
+\backslash
+longrightarrow&
+\backslash
+exists x:P
+\backslash
+left(x
+\backslash
+right)&&
+\backslash
+text{eksistenčna generalizacija}
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+&& 
+\backslash
+forall x:P
+\backslash
+left(x
+\backslash
+right)
+\backslash
+longrightarrow& x_0
+\backslash
+coloneqq x, P
+\backslash
+left(x
+\backslash
+right)&&
+\backslash
+text{univerzalna specifikacija}
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+&& 
+\backslash
+text{poljub.
+ } x_0, P
+\backslash
+left(x_0
+\backslash
+right)
+\backslash
+longrightarrow& 
+\backslash
+forall x:P
+\backslash
+left(x
+\backslash
+right)&&
+\backslash
+text{univerzalna generalizacija}
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{align*}
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula $R\subseteq A\times B:aR\oplus Sb\sim\left(a,b\right)\in R\backslash S\vee\left(a,b\right)\in S\backslash R\sim aRb\oplus aSb$
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Formula $R^{-1}\coloneqq\left\{ \left(b,a\right);\left(a,b\right)\in R\right\} :\quad aRb\sim bR^{-1}a$
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Formula $R*S\coloneqq\left\{ \left(a,c\right);\exists b:\left(aRb\wedge bSc\right)\right\} :R^{2}\coloneqq R*R,R^{n+1}\coloneqq R^{n}*R$
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Formula $\left(R^{-1}\right)^{-1}=R,\left(R\cup S\right)^{-1}=R^{-1}\cup S^{-1},\left(R\cap S\right)^{-1}=R^{-1}\cap S^{-1}$
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Formula $\left(R*S\right)=R^{-1}*S^{-1}$
+\end_inset
+
+.
+ 
+\begin_inset Formula $*\cup$
+\end_inset
+
+ in 
+\begin_inset Formula $\cup*$
+\end_inset
+
+ sta distributivni.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $R^{+}=R\cup R^{2}\cup R^{3}\cup\dots,\quad R^{*}=I\cup R^{+}$
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+Ovojnica 
+\begin_inset Formula $R^{L}\supseteq R$
+\end_inset
+
+ je najmanjša razširitev 
+\begin_inset Formula $R$
+\end_inset
+
+, ki ima lastnost 
+\begin_inset Formula $L$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $R^{\text{ref}}\coloneqq I\cup R,R^{\text{sim}}\coloneqq R\cup R^{-1},R^{\text{tranz}}=R^{+},R^{\text{tranz+ref}}=R^{*}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Ekvivalenčna rel.
+ je simetrična, tranzitivna in refleksivna.
+\end_layout
+
+\begin_layout Standard
+Ekvivalenčni razred: 
+\begin_inset Formula $R\left[x\right]\coloneqq\left\{ y;xRy\right\} $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Faktorska množica: 
+\begin_inset Formula $A/R\coloneqq\left\{ R\left[x\right];x\in A\right\} $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\vec{\mathcal{B}}\text{ razbitje}A\Longleftrightarrow\bigcup_{i}\mathcal{B}_{i}=A\wedge\forall i\mathcal{B}_{i}\not=\emptyset\wedge\mathcal{B}_{i}\cap\mathcal{B}_{j}=\emptyset,i\not=j$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Urejenosti
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\left(M,\preccurlyeq\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Delna: refl., antisim.
+ in tranz.
+ Linearna: delna, sovisna
+\end_layout
+
+\begin_layout Standard
+def.: 
+\begin_inset Formula $x\prec y\Longleftrightarrow x\preccurlyeq y\wedge x\not=y$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $x\text{ je nepo. predh. }y\Longleftrightarrow x\prec y\wedge\neg\exists z\in M:\left(x\prec z\wedge z\prec y\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $a\in M\text{ je minimalen}\Longleftrightarrow\forall x\in M\left(x\preccurlyeq a\Rightarrow x=a\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $a\in M\text{ je maksimalen}\Longleftrightarrow\forall x\in M\left(a\preccurlyeq x\Rightarrow x=a\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $a\in M\text{ je prvi}\Longleftrightarrow\forall x\in M:\left(a\preccurlyeq x\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $a\in M\text{ je zadnji}\Longleftrightarrow\forall x\in M:\left(x\preccurlyeq a\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $M_{1}\times M_{2}$
+\end_inset
+
+: 
+\begin_inset Formula $\left(a_{1},b_{1}\right)\preccurlyeq\left(a_{2},b_{2}\right)\coloneqq a_{1}\preccurlyeq a_{2}\wedge b_{1}\preccurlyeq b_{2}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Srečno!
+\end_layout
+
+\begin_layout Paragraph
+Funkcijska polnost
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $T_{0},$
+\end_inset
+
+
+\begin_inset Formula $T_{1}$
+\end_inset
+
+, 
+\begin_inset Formula $S$
+\end_inset
+
+ – 
+\begin_inset Formula $f\left(\vec{x}\right)=\neg f\left(\vec{x}\oplus\vec{1}\right)$
+\end_inset
+
+, 
+\begin_inset Formula $L$
+\end_inset
+
+, 
+\begin_inset Formula $M$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $L$
+\end_inset
+
+ – 
+\begin_inset Formula $f\left(\vec{x}\right)=\left[\begin{array}{ccc}
+a_{0} & \dots & a_{n}\end{array}\right]^{T}\oplus\wedge\left[\begin{array}{cccc}
+1 & x_{1} & \dots & x_{n}\end{array}\right]$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $M$
+\end_inset
+
+ – 
+\begin_inset Formula $\forall i,j:\vec{w_{i}}<\vec{w_{j}}\Rightarrow f\left(\vec{w_{i}}\right)\leq f\left(\vec{w_{j}}\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Newpage pagebreak
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Supermum in infimum
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\sup\left(a,b\right)$
+\end_inset
+
+ in 
+\begin_inset Formula $\inf\left(a,b\right)$
+\end_inset
+
+ v 
+\begin_inset Formula $\left(M,\preceq\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\sup\left(a,b\right)\coloneqq j\ni:a\preceq j\wedge b\preceq j\wedge\forall x:a\preceq x\wedge b\preceq x\Rightarrow j\preceq x$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\inf\left(a,b\right)\coloneqq j\ni:j\preceq a\wedge j\preceq b\wedge\forall x:x\preceq a\wedge x\preceq b\Rightarrow x\preceq j$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Relacijska 
+\series bold
+def.
+ mreže
+\series default
+: Delna urejenost je mreža 
+\begin_inset Formula $\Leftrightarrow\forall a,b\in M:\exists\sup\left(a,b\right)\wedge\exists\inf\left(a,b\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Algebrajska 
+\series bold
+def.
+ mreže
+\series default
+: 
+\begin_inset Formula $\left(M,\wedge,\vee\right)$
+\end_inset
+
+ je mreža, če veljata idempotentnosti 
+\begin_inset Formula $a\vee a=a\wedge a=a$
+\end_inset
+
+, komutativnosti, asociativnosti in absorpciji.
+\end_layout
+
+\begin_layout Standard
+Mreža je 
+\series bold
+omejena
+\series default
+ 
+\begin_inset Formula $\Leftrightarrow\exists0,1\in M\ni:\forall x\in M:0\preceq x\preceq1$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Mreža je 
+\series bold
+komplementarna
+\series default
+ 
+\begin_inset Formula $\Leftrightarrow\forall a\in M\exists a^{-1}\in M\ni:a\wedge a^{-1}\sim0\text{ in }a\vee a^{-1}\sim1$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+V 
+\series bold
+distributivni mreži
+\series default
+ veljata obe distributivnosti.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\sup\left(a,b\right)\sim a\wedge b,\quad\inf\left(a,b\right)\sim a\vee b$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+V delni urejenosti velja: 
+\begin_inset Formula $a\preceq b\Leftrightarrow a=\inf\left(a,b\right)\Leftrightarrow b=\sup\left(a,b\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $M_{5},N_{5}$
+\end_inset
+
+ nista distributivni.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+\begin_inset Formula $\left(N,\wedge,\vee\right)$
+\end_inset
+
+ 
+\series default
+je
+\series bold
+ podmreža
+\series default
+ 
+\begin_inset Formula $\left(M,\wedge,\vee\right)\Leftrightarrow\emptyset\not=N\subseteq M,\forall a,b\in N:a\vee b\in N\text{ in }a\wedge b\in N$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Boolova algebra
+\series default
+ je komplementarna distributivna mreža.
+ Tedaj ima vsak element natanko en komplement in velja dualnost ter De Morganova
+ zakona.
+\end_layout
+
+\begin_layout Paragraph
+Funkcije
+\end_layout
+
+\begin_layout Standard
+Funkcija 
+\begin_inset Formula $f$
+\end_inset
+
+ je preslikava, če je 
+\begin_inset Formula $D_{f}$
+\end_inset
+
+ domena.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $f,g\text{ injekciji }\Rightarrow g\circ f\text{ injekcija}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $f,g\text{ surjekciji }\Rightarrow g\circ f\text{ surjekcija}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $g\circ f\text{ injekcija }\Rightarrow f\text{ injekcija}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $g\circ f\text{ surjekcija }\Rightarrow g\text{ surjekcija}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Slika množice 
+\begin_inset Formula $A_{1}\subseteq A$
+\end_inset
+
+: 
+\begin_inset Formula $f\left(A_{1}\right)\coloneqq\left\{ y\in B;\exists x\in A_{1}\ni:f\left(x\right)=y\right\} $
+\end_inset
+
+.
+ Praslika 
+\begin_inset Formula $B_{1}\subseteq B$
+\end_inset
+
+: 
+\begin_inset Formula $f^{-1}\left(B_{1}\right)=\left\{ x\in A:\exists y\in B_{1}\ni:f\left(x\right)=y\right\} $
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Paragraph
+Permutacije
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\pi=\pi^{-1}\Leftrightarrow\pi$
+\end_inset
+
+ je konvolucija.
+\end_layout
+
+\begin_layout Standard
+V disjunktnih ciklih velja: 
+\begin_inset Formula $C_{1}C_{2}=C_{2}C_{1}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+V ciklih velja: 
+\begin_inset Formula $C_{2}^{-1}C_{1}^{-1}=\left(C_{1}C_{2}\right)^{-1}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Razcep na disjunktne cikle je enoličen.
+\end_layout
+
+\begin_layout Standard
+Neenolično razbitje cikla dolžine 
+\begin_inset Formula $n$
+\end_inset
+
+ na produkt 
+\begin_inset Formula $n-1$
+\end_inset
+
+ transpozicij: 
+\begin_inset Formula $\left(a_{1}a_{2}a_{3}a_{4}a_{5}\right)=\left(a_{1}a_{2}\right)\left(a_{1}a_{3}\right)\left(a_{1}a_{4}\right)\left(a_{1}a_{5}\right)$
+\end_inset
+
+.
+ Parnost števila transpozicij je enolična.
+ 
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+newcommand
+\backslash
+red{
+\backslash
+text{red}}
+\backslash
+newcommand
+\backslash
+sgn{
+\backslash
+text{sgn}}
+\backslash
+newcommand
+\backslash
+lcm{
+\backslash
+text{lcm}}
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula $\sgn\pi=\sgn\pi^{-1}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\red\pi$
+\end_inset
+
+ je najmanjše 
+\begin_inset Formula $k\ni:\pi^{k}=id$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Za cikel 
+\begin_inset Formula $C$
+\end_inset
+
+ dolžine 
+\begin_inset Formula $n$
+\end_inset
+
+ velja: 
+\begin_inset Formula $C^{n}=id$
+\end_inset
+
+ — 
+\begin_inset Formula $\red C=n$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Red produkta disjunktnih ciklov dolžin 
+\begin_inset Formula $\vec{n}$
+\end_inset
+
+ je 
+\begin_inset Formula $\lcm\left(\vec{n}\right)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Paragraph
+Moči končnih množic
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\left|A\times B\right|=\left|A\right|\left|B\right|$
+\end_inset
+
+, 
+\begin_inset Formula $\left|\mathcal{P}\left(A\right)\right|=2^{\left|A\right|}$
+\end_inset
+
+, 
+\begin_inset Formula $\left|B^{A}\right|=\left|B\right|^{\left|A\right|}$
+\end_inset
+
+, 
+\begin_inset Formula $\left|B\backslash A\right|=\left|B\right|-\left|A\cap B\right|$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Princip vključitve in izključitve: 
+\begin_inset Formula $\left|A_{1}\cup A_{2}\cup\cdots\cup A_{n}\right|=\sum_{i=1}^{n}\left(-1\right)^{i+1}S_{i}$
+\end_inset
+
+, kjer 
+\begin_inset Formula $S_{k}\coloneqq\sum_{I\subseteq\left\{ 1,\dots,n\right\} ,\left|I\right|=k}\bigcap_{i\in I}A_{i}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Neskončne množice
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $A$
+\end_inset
+
+ in 
+\begin_inset Formula $B$
+\end_inset
+
+ sta enakomočni: 
+\begin_inset Formula $A\sim B\Leftrightarrow\exists\text{bijekcija }f:A\to B$
+\end_inset
+
+.
+ 
+\begin_inset Formula $\sim$
+\end_inset
+
+ je ekvivalenčna relacija.
+\end_layout
+
+\begin_layout Standard
+Ekvivalenčni razredi: 
+\begin_inset Formula $0,1,2,\dots,\aleph_{0},c$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $A$
+\end_inset
+
+ je neskončna 
+\begin_inset Formula $\Leftrightarrow\exists B\subset A\ni:A\sim B$
+\end_inset
+
+, drugače je končna.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $A$
+\end_inset
+
+ ima manjšo ali enako moč kot 
+\begin_inset Formula $B$
+\end_inset
+
+ zapišemo: 
+\begin_inset Formula $A\leq B\Leftrightarrow\exists\text{injekcija }f:A\to B$
+\end_inset
+
+.
+ Označimo 
+\begin_inset Formula $A<B\Leftrightarrow A\leq B\wedge A\not\sim B$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $A\leq B\wedge B\leq A\Leftrightarrow A\sim B,\quad\forall A,B:A<B\vee B<A\vee A\sim B$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\forall A\not=\emptyset,B:A\leq B\Leftrightarrow\exists\text{surjekcija }g:B\to A$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\forall$
+\end_inset
+
+neskončna množica vsebuje števno neskončno podmnožico.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $A<\mathcal{P}\left(A\right)$
+\end_inset
+
+, posledično 
+\begin_inset Formula $A<\mathcal{P}\left(A\right)<\mathcal{P}^{2}\left(A\right)<\mathcal{P}^{2}\left(A\right)<\cdots$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\mathbb{N}<\mathcal{P}\left(\mathbb{N}\right)=c<\mathcal{P}^{2}\left(\mathbb{N}\right)<\cdots$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Za neskončno 
+\begin_inset Formula $A$
+\end_inset
+
+ in končno 
+\begin_inset Formula $B$
+\end_inset
+
+ velja 
+\begin_inset Formula $A\backslash B\sim A$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Za neskončno 
+\begin_inset Formula $A$
+\end_inset
+
+ in števno neskončno 
+\begin_inset Formula $B$
+\end_inset
+
+ velja 
+\begin_inset Formula $A\sim A\cup B$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Paragraph
+Teorija števil
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $-\lfloor-x\rfloor=\lceil x\rceil,\quad-\lceil-x\rceil=\lfloor x\rfloor$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\lceil x\rceil=\min\left\{ k\in\mathbb{Z};k\geq x\right\} ,\quad\lfloor x\rfloor=\max\left\{ k\in\mathbb{Z};k\leq x\right\} $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $m\vert n\Leftrightarrow\exists k\in\mathbb{Z}\ni:n=km$
+\end_inset
+
+.
+ 
+\begin_inset Formula $\vert$
+\end_inset
+
+ je antisimetrična.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $m\vert a\wedge m\vert b\Rightarrow m\vert\left(a+b\right)$
+\end_inset
+
+, 
+\begin_inset Formula $m\vert a\Rightarrow m\vert ak$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $a\bot b\Leftrightarrow\gcd\left(a,b\right)=1\Leftrightarrow m\bot\left(a\mod b\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $ab=\gcd\left(a,b\right)\lcm\left(a,b\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $p\in\mathbb{N}$
+\end_inset
+
+ je praštevilo 
+\begin_inset Formula $\Leftrightarrow\left|\text{D}\left(p\right)\right|=2$
+\end_inset
+
+ (število deliteljev): 
+\begin_inset Formula $p\in\mathbb{P}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $a,b\in\mathbb{N},p\in\mathbb{P}$
+\end_inset
+
+: 
+\begin_inset Formula $a\bot b\vee a\vert b,\quad p\vert ab\Rightarrow p\vert a\vee p\vert b$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $m\vert a-b$
+\end_inset
+
+ označimo 
+\begin_inset Formula $a\equiv b\pmod m$
+\end_inset
+
+, 
+\begin_inset Formula $a\mod m=b\mod m$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $a\equiv b\pmod m\Rightarrow\forall k\in\mathbb{Z}:a\overset{+}{\cdot}k\equiv b\overset{+}{\cdot}k\pmod m$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $a\equiv b\pmod m\wedge c\equiv d\pmod m\Rightarrow a\overset{+}{\overset{-}{\cdot}}c\equiv b\overset{+}{\overset{-}{\cdot}}d\pmod m$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Mali fermatov izrek: 
+\begin_inset Formula $a\in\mathbb{N},p\in\mathbb{P}$
+\end_inset
+
+ velja 
+\begin_inset Formula $a\equiv a^{p}\pmod p$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $p,q\in\mathbb{P}:a\equiv b\pmod p\wedge a\equiv b\pmod p\Rightarrow a\equiv b\pmod{pq}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Eulerjeva funkcija 
+\begin_inset Formula $\varphi\left(n\right)\coloneqq\left|\left\{ k\in n;1\leq k<n\wedge k\bot n\right\} \right|$
+\end_inset
+
+ — število tujih števil, manjših od n.
+ 
+\begin_inset Formula $p\in\mathbb{P}\Rightarrow\varphi\left(p\right)=p-1$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $p\in\mathbb{P},n\in\mathbb{N}\Rightarrow\varphi\left(p\right)=p^{n}-p^{n-1},\quad\varphi\left(a\right)\varphi\left(b\right)=\varphi\left(ab\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+REA: 
+\begin_inset Formula $ax+by=d$
+\end_inset
+
+
+\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
diff --git a/šola/krožek/spremenljivke.odp b/šola/krožek/spremenljivke.odp
new file mode 100644
index 0000000..8bb71b6
--- /dev/null
+++ b/šola/krožek/spremenljivke.odp
diff --git a/šola/krožek/uvodno.odp b/šola/krožek/uvodno.odp
new file mode 100644
index 0000000..fe941ee
--- /dev/null
+++ b/šola/krožek/uvodno.odp
diff --git a/šola/la/dn6/dokument.lyx b/šola/la/dn6/dokument.lyx
new file mode 100644
index 0000000..a75e37f
--- /dev/null
+++ b/šola/la/dn6/dokument.lyx
@@ -0,0 +1,1514 @@
+#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}}}
+\usepackage{amsmath}
+\usepackage{tikz}
+\newcommand{\udensdash}[1]{%
+    \tikz[baseline=(todotted.base)]{
+        \node[inner sep=1pt,outer sep=0pt] (todotted) {#1};
+        \draw[densely dashed] (todotted.south west) -- (todotted.south east);
+    }%
+}%
+\end_preamble
+\use_default_options true
+\begin_modules
+enumitem
+theorems-ams
+\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 false
+\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 0cm
+\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 Title
+Rešitev šeste domače naloge Linearne Algebre
+\end_layout
+
+\begin_layout Author
+
+\noun on
+Anton Luka Šijanec
+\end_layout
+
+\begin_layout Date
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+today
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Abstract
+Za boljšo preglednost sem svoje rešitve domače naloge prepisal na računalnik.
+ Dokumentu sledi še rokopis.
+ Naloge je izdelala asistentka Ajda Lemut.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+newcommand
+\backslash
+euler{e}
+\backslash
+newcommand
+\backslash
+rang{
+\backslash
+text{rang}}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Naj bosta 
+\begin_inset Formula $V,W$
+\end_inset
+
+ vektorska prostora.
+ Pokaži, da je množica vseh linearnih preslikav 
+\begin_inset Formula $\mathcal{L}\left(V,W\right)=\left\{ A:V\to W:A\text{ linearna}\right\} $
+\end_inset
+
+ vektorski prostor.
+\end_layout
+
+\begin_deeper
+\begin_layout Paragraph
+Rešitev
+\end_layout
+
+\begin_layout Standard
+Definirali smo, da za linearno preslikavo velja aditivnost 
+\begin_inset Formula $L\left(v_{1}+v_{2}\right)=Lv_{1}+Lv_{2}$
+\end_inset
+
+ in homogenost 
+\begin_inset Formula $L\alpha v=\alpha Lv$
+\end_inset
+
+, skupaj 
+\begin_inset Formula $L\left(\alpha_{1}v_{1}+\alpha_{2}v_{2}\right)=\alpha_{1}Lv_{2}+\alpha_{2}Lv_{2}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Vektorski prostor pa smo definirali kot urejeno trojico 
+\begin_inset Formula $\left(V,+,\cdot\right)\ni:$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\left(V,+\right)$
+\end_inset
+
+ je Abelova grupa: komutativnost, asociativnost, inverzi, enota, notranjost
+\end_layout
+
+\begin_layout Enumerate
+aksiomi množenja s skalarjem iz polja 
+\begin_inset Formula $F$
+\end_inset
+
+: 
+\begin_inset Formula $\forall\alpha,\beta\in F\forall a,b\in V:$
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+\begin_inset Formula $\alpha\cdot\left(a+b\right)=\alpha\cdot a+\alpha\cdot b$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\left(\alpha+\beta\right)\cdot a=\alpha\cdot a+\beta\cdot a$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\left(\alpha\cdot\beta\right)\cdot a=\alpha\cdot\left(\beta\cdot a\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $1\cdot a=a$
+\end_inset
+
+, kjer je 
+\begin_inset Formula $1$
+\end_inset
+
+ enota 
+\begin_inset Formula $F$
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Standard
+Da linearne preslikave 
+\begin_inset Formula $L:V\to W$
+\end_inset
+
+ sploh obstajajo, privzemam, da sta 
+\begin_inset Formula $V$
+\end_inset
+
+ in 
+\begin_inset Formula $W$
+\end_inset
+
+ vektorska prostora nad istim poljem.
+\end_layout
+
+\begin_layout Standard
+Treba je definirati 
+\begin_inset Formula $+$
+\end_inset
+
+, 
+\begin_inset Formula $F$
+\end_inset
+
+ in 
+\begin_inset Formula $\cdot$
+\end_inset
+
+ ter dokazati, da je pri izbranih 
+\begin_inset Formula $+$
+\end_inset
+
+, 
+\begin_inset Formula $F$
+\end_inset
+
+ in 
+\begin_inset Formula $\cdot$
+\end_inset
+
+ 
+\begin_inset Formula $\left(\mathcal{L},+,\cdot\right)$
+\end_inset
+
+ vektorski prostor po tej definiciji.
+ Vzemimo za 
+\begin_inset Formula $+$
+\end_inset
+
+ operacijo 
+\begin_inset Formula $+$
+\end_inset
+
+ iz vektorskega prostora 
+\begin_inset Formula $W$
+\end_inset
+
+ in definirajmo operacijo na 
+\begin_inset Formula $\mathcal{L}$
+\end_inset
+
+: 
+\begin_inset Formula $\forall L_{1},L_{2}\in\mathcal{L}:\quad\left(L_{1}+L_{2}\right)v\coloneqq L_{1}v+L_{2}v$
+\end_inset
+
+.
+ Dokažimo, da je 
+\begin_inset Formula $\left(\mathcal{L},+\right)$
+\end_inset
+
+ abelova grupa:
+\end_layout
+
+\begin_layout Enumerate
+Notranjost operacije: Trdimo, da je 
+\begin_inset Formula $L_{1}+L_{2}$
+\end_inset
+
+ linearna transformacija.
+ Dokaz: 
+\begin_inset Formula $\forall v\in V:$
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+Aditivnost: 
+\begin_inset Formula $\left(L_{1}+L_{2}\right)\left(v_{1}+v_{2}\right)\overset{\text{def}+}{=}L_{1}\left(v_{1}+v_{2}\right)+L_{2}\left(v_{1}+v_{2}\right)\overset{\text{aditivnost}}{=}L_{1}v_{1}+L_{1}v_{2}+L_{2}v_{1}+L_{2}v_{2}\overset{W\text{V.P.}}{=}L_{1}v_{1}+L_{2}v_{1}+L_{1}v_{2}+L_{2}v_{2}\overset{def+}{=}\left(L_{1}+L_{2}\right)v_{1}+\left(L_{1}+L_{2}\right)v_{2}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Homogenost: 
+\begin_inset Formula $\alpha\left(L_{1}+L_{2}\right)v\overset{\text{def}+}{=}\alpha\left(L_{1}v+L_{2}v\right)\overset{W\text{V.P.}}{=}\alpha L_{1}v+\alpha L_{2}v\overset{\text{homogenost}}{=}L_{1}\alpha v+L_{2}\alpha v\overset{\text{def}+}{=}\left(L_{1}+L_{2}\right)\left(\alpha v\right)$
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Enote: Enota naj bo tista linearna preslikava 
+\begin_inset Formula $L_{0}$
+\end_inset
+
+, ki slika ves 
+\begin_inset Formula $V$
+\end_inset
+
+ v 
+\begin_inset Formula $0\in W$
+\end_inset
+
+.
+ Dokaz: 
+\begin_inset Formula $\forall L\in\mathcal{L}:\quad$
+\end_inset
+
+
+\begin_inset Formula $Lv+L_{0}v\text{\ensuremath{\overset{\text{def}L_{0}}{=}}}Lv+0\ensuremath{\overset{\left(W,+\right)\text{abelova grupa}}{=}}Lv$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Inverzi
+\begin_inset CommandInset label
+LatexCommand label
+name "enu:Inverzi:-Ker-je"
+
+\end_inset
+
+: Ker je 
+\begin_inset Formula $W$
+\end_inset
+
+ V.
+ P., 
+\begin_inset Formula $\forall w\in W\exists!-w\in W\ni:w+\left(-w\right)=0$
+\end_inset
+
+, zato 
+\begin_inset Formula $\forall L\in\mathcal{L}\exists-L\in\mathcal{L}\ni:-L+L=L_{0}$
+\end_inset
+
+ s predpisom 
+\begin_inset Formula $-L$
+\end_inset
+
+ slika element 
+\begin_inset Formula $v\in V$
+\end_inset
+
+ v tisti 
+\begin_inset Formula $w\in W$
+\end_inset
+
+, ki je inverz 
+\begin_inset Formula $Lv\in W$
+\end_inset
+
+.
+ 
+\begin_inset Formula $-L$
+\end_inset
+
+ je res 
+\begin_inset Formula $\in\mathcal{L}$
+\end_inset
+
+.
+ Velja 
+\begin_inset Formula $-L\coloneqq$
+\end_inset
+
+
+\begin_inset Formula $\left(-1\right)\cdot L$
+\end_inset
+
+, kjer je 
+\begin_inset Formula $-1$
+\end_inset
+
+ inverz enote polja, ki ga izberemo kasneje.
+ 
+\begin_inset Formula $\forall v\in V,L\in\mathcal{L}:\left(\left(-1\right)\cdot L\right)v\overset{\text{def}\cdot\text{sledi}}{=}\left(-1\right)\left(Lv\right)\overset{\text{def\ensuremath{\cdot},homogenost}}{=}L\left(-1v\right)\overset{\text{karakteristika F}\not=0}{L\left(-v\right)}$
+\end_inset
+
+.
+ Ta dokaz se sklicuje na določitev polja in skalarnega množenja, ki ga podam
+ kasneje.
+\end_layout
+
+\begin_layout Enumerate
+Asociativnost: 
+\begin_inset Formula $\forall L_{1},L_{2},L_{3}\in\mathcal{L}:L_{1}+\left(L_{2}+L_{3}\right)=\left(L_{1}+L_{2}\right)+L_{3}$
+\end_inset
+
+ velja očitno iz definicije 
+\begin_inset Formula $+$
+\end_inset
+
+, saj je 
+\begin_inset Formula $W$
+\end_inset
+
+ vektorski prostor.
+ Komutativnost spet iz istih razlogov.
+\end_layout
+
+\begin_layout Standard
+Določiti moramo še polje in množenje s skalarjem.
+ Vzemimo za 
+\begin_inset Formula $F$
+\end_inset
+
+ polje vektorskega prostora 
+\begin_inset Formula $W$
+\end_inset
+
+ in množenje s skalarjem definirajmo takole: 
+\begin_inset Formula $\forall v\in V,\alpha\in F:\left(\alpha L\right)v\coloneqq\alpha\left(Lv\right)$
+\end_inset
+
+.
+ Zopet za vsak slučaj dokažimo še linearnost dobljene preslikave 
+\begin_inset Formula $\forall\alpha,\beta\in F\forall L\in\mathcal{L}\forall v_{1},v_{2}\in V:$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Aditivnost: 
+\begin_inset Formula $\left(\alpha L\right)\left(v_{1}+v_{2}\right)\overset{\text{def}\cdot}{=}\alpha\left(L\left(v_{1}+v_{2}\right)\right)\overset{\text{aditivnost}}{=}\alpha\left(Lv_{1}+Lv_{2}\right)\overset{W\text{V.P.}}{=}\alpha\left(Lv_{1}\right)+\alpha\left(Lv_{2}\right)\overset{\text{def}\cdot}{=}\left(\alpha L\right)v_{1}+\left(\alpha L\right)v_{2}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Homogenost: 
+\begin_inset Formula $\left(\alpha L\right)\left(\beta v\right)\overset{\text{def}\cdot}{=}\alpha\left(L\left(\beta v\right)\right)\overset{\text{homogenost}}{=}\alpha$
+\end_inset
+
+
+\begin_inset Formula $\beta Lv\overset{\text{F\text{polje}}}{=}\beta\alpha\left(Lv\right)\overset{\text{def}\cdot}{=}\beta\left(\alpha L\right)v$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Iz tega dokaza sledi tudi obstoj inverzov (
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "enu:Inverzi:-Ker-je"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+).
+\end_layout
+
+\begin_layout Standard
+Sedaj lahko dokažemo še štiri aksiome vektorskih prostorov za množenje s
+ skalarjem.
+ 
+\begin_inset Formula $\forall\alpha,\beta\in F\forall L_{1},L_{2}\in V:$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+udensdash{$
+\backslash
+alpha
+\backslash
+left(L_1+L_2
+\backslash
+right)
+\backslash
+overset{?}{=}
+\backslash
+alpha L_1+
+\backslash
+alpha L_2$}
+\end_layout
+
+\end_inset
+
+: 
+\begin_inset Formula $\left(\alpha\left(L_{1}+L_{2}\right)\right)v\overset{\text{def}+\cdot}{=}\alpha\left(L_{1}v+L_{2}v\right)\overset{W\text{V.P.}}{=}\alpha L_{1}+\alpha L_{2}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+udensdash{$
+\backslash
+left(
+\backslash
+alpha+
+\backslash
+beta
+\backslash
+right)L_1=
+\backslash
+alpha L_1+
+\backslash
+beta L_1$}
+\end_layout
+
+\end_inset
+
+: Po definiciji našega 
+\begin_inset Formula $\cdot$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+udensdash{$
+\backslash
+alpha
+\backslash
+left(
+\backslash
+beta L_1
+\backslash
+right)=
+\backslash
+left(
+\backslash
+alpha
+\backslash
+beta
+\backslash
+right)L_1$}
+\end_layout
+
+\end_inset
+
+: Po definiciji našega 
+\begin_inset Formula $\cdot$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+udensdash{$1
+\backslash
+cdot L_1=L_1$}
+\end_layout
+
+\end_inset
+
+: 
+\begin_inset Formula $\left(1\cdot L_{1}\right)v\overset{\text{def}\cdot}{=}1\cdot\left(L_{1}v\right)\overset{W\text{V.P.}}{=}L_{1}v$
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Naj bo 
+\begin_inset Formula $Z:\mathbb{R}^{3}\to\mathbb{R}^{3}$
+\end_inset
+
+ zrcaljenje preko ravnine 
+\begin_inset Formula $x+y+z=0$
+\end_inset
+
+.
+ Določi matriko 
+\begin_inset Formula $Z$
+\end_inset
+
+ v standardni bazi.
+\end_layout
+
+\begin_deeper
+\begin_layout Paragraph
+Rešitev
+\end_layout
+
+\begin_layout Standard
+Tri točke na taki ravnini so 
+\begin_inset Formula $\left(0,0,0\right)$
+\end_inset
+
+, 
+\begin_inset Formula $\left(1,0,-1\right)$
+\end_inset
+
+ in 
+\begin_inset Formula $\left(0,1,-1\right)$
+\end_inset
+
+.
+ Normala ravnine je 
+\begin_inset Formula $\left(1,0,-1\right)\times\left(0,1,-1\right)=\left(1,1,1\right)$
+\end_inset
+
+.
+ Parametrično to ravnino zapišemo kot 
+\begin_inset Formula $\left\{ s\vec{r}+p\vec{q};s,p\in\mathbb{R}\right\} $
+\end_inset
+
+, kjer 
+\begin_inset Formula $\vec{r}=\left(1,0,-1\right)$
+\end_inset
+
+ in 
+\begin_inset Formula $\vec{q}=\left(0,1,-1\right)$
+\end_inset
+
+.
+ Za določitev matrike linearne preslikave 
+\begin_inset Formula $Z$
+\end_inset
+
+ bomo zrcalili vektorje standardne baze 
+\begin_inset Formula $\left(1,0,0\right)$
+\end_inset
+
+, 
+\begin_inset Formula $\left(0,1,0\right)$
+\end_inset
+
+ in 
+\begin_inset Formula $\left(0,0,1\right)$
+\end_inset
+
+ čez to ravnino.
+ Zrcaljenje 
+\begin_inset Formula $\vec{t}$
+\end_inset
+
+ v 
+\begin_inset Formula $Z\vec{t}$
+\end_inset
+
+ čez ravnino je opisano z enačbo 
+\begin_inset Formula $Z\vec{t}=\vec{t}+2\left(\hat{t}-\vec{t}\right)=2\hat{t}-\vec{t}$
+\end_inset
+
+, kjer s 
+\begin_inset Formula $\hat{t}$
+\end_inset
+
+ označim pravokotno projekcijo točke 
+\begin_inset Formula $\vec{t}$
+\end_inset
+
+ na ravnino.
+ Torej najprej tako projicirajmo standardno bazo na ravnino.
+\begin_inset Formula 
+\[
+\langle\hat{t}-\vec{t},\vec{q}\rangle=0=\langle\hat{t}-\vec{t},\vec{r}\rangle\quad\text{(pravokotna projekcija)}
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\langle s\vec{r}+p\vec{q}-\vec{t},\vec{q}\rangle=0=\langle s\vec{r}+p\vec{q}-\vec{t},\vec{r}\rangle\quad\text{(parametrični zapis \ensuremath{\hat{t}})}
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+s\langle\vec{r},\vec{q}\rangle+p\langle\text{\ensuremath{\vec{q},\vec{q}\rangle-\langle\vec{t},\vec{q}\rangle=0=s\langle\vec{r},\vec{r}\rangle+p\langle\vec{q},\vec{r}\rangle-\langle\vec{t},\vec{r}\rangle}}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{align*}
+\end_layout
+
+\begin_layout Plain Layout
+
+s
+\backslash
+langle
+\backslash
+vec{r},
+\backslash
+vec{q}
+\backslash
+rangle+p
+\backslash
+langle
+\backslash
+vec{q},
+\backslash
+vec{q}
+\backslash
+rangle&=
+\backslash
+langle
+\backslash
+vec{t},
+\backslash
+vec{q}
+\backslash
+rangle
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+s
+\backslash
+langle
+\backslash
+vec{r},
+\backslash
+vec{r}
+\backslash
+rangle+p
+\backslash
+langle
+\backslash
+vec{q},
+\backslash
+vec{r}
+\backslash
+rangle&=
+\backslash
+langle
+\backslash
+vec{t},
+\backslash
+vec{r}
+\backslash
+rangle
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{align*}
+\end_layout
+
+\end_inset
+
+Dobimo sistem enačb z neznankama 
+\begin_inset Formula $s$
+\end_inset
+
+ in 
+\begin_inset Formula $p$
+\end_inset
+
+, parametroma projekcije.
+ Vstavimo 
+\begin_inset Formula $\vec{r}=\left(1,0,-1\right)$
+\end_inset
+
+ in 
+\begin_inset Formula $\vec{q}=\left(0,1,-1\right)$
+\end_inset
+
+ ter za 
+\begin_inset Formula $\vec{t}$
+\end_inset
+
+ posamično vse tri točke standardne baze in izračunajmo njihove projekcije.
+\begin_inset Formula 
+\[
+s\cdot1+p\cdot2=\langle\vec{t},\left(0,1,-1\right)\rangle
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+s\cdot2+p\cdot1=\langle\vec{t},\left(1,0,-1\right)\rangle
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Nato izračunamo še njihovo zrcaljenje iz projekcij po enačbi za zrcaljenje
+ zgoraj.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+setlength{
+\backslash
+columnseprule}{0.2pt}
+\backslash
+begin{multicols}{3}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\vec{b_{1}}=\left(1,0,0\right)$
+\end_inset
+
+
+\begin_inset Formula 
+\[
+s+2p=0
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+2s+p=1
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+s=-2p
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+p-4p=1
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+p=-\frac{1}{3},\quad s=\frac{2}{3}
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\hat{t}=\left(\frac{2}{3},-\frac{1}{3},-\frac{1}{3}\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+Z\vec{b_{1}}=\left(\frac{1}{3},-\frac{2}{3},-\frac{2}{3}\right)
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\vec{b_{2}}=\left(0,1,0\right)$
+\end_inset
+
+
+\begin_inset Formula 
+\[
+s+2p=1
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+2s+p=0
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+p=-2s
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+s-4s=1
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+p=\frac{2}{3},\quad s=-\frac{1}{3}
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\hat{b_{2}}=\left(-\frac{1}{3},\frac{2}{3},-\frac{1}{3}\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+Z\vec{b_{2}}=\left(-\frac{2}{3},\frac{1}{3},-\frac{2}{3}\right)
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\vec{b_{3}}=\left(0,0,1\right)$
+\end_inset
+
+
+\begin_inset Formula 
+\[
+s+2p=-1
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+2s+p=-1
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+s=-2p-1
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+2\left(-2p-1\right)+p=-1
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+p=-\frac{1}{3},\quad s=-\frac{1}{3}
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\hat{b_{2}}=\left(-\frac{1}{3},-\frac{1}{3},\frac{2}{3}\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+Z\vec{b_{3}}=\left(-\frac{2}{3},-\frac{2}{3},\frac{1}{3}\right)
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{multicols}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Dobljene z 
+\begin_inset Formula $Z$
+\end_inset
+
+ preslikane (čez ravnino zrcaljene) vektorje po stolpcih zložimo v matriko
+ 
+\begin_inset Formula $Z$
+\end_inset
+
+:
+\begin_inset Formula 
+\[
+Z=\left[\begin{array}{ccc}
+1/3 & -2/3 & -2/3\\
+-2/3 & 1/3 & -2/3\\
+-2/3 & -2/3 & 1/3
+\end{array}\right]
+\]
+
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Določi rang matrike
+\begin_inset Formula 
+\[
+B=\left[\begin{array}{cccc}
+-2-t & 4 & 5+t & 4\\
+1 & -1 & -2 & 1\\
+-t & 3 & 1+t & 4+t
+\end{array}\right]
+\]
+
+\end_inset
+
+v odvisnosti od parametra 
+\begin_inset Formula $t$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Paragraph
+Rešitev
+\end_layout
+
+\begin_layout Standard
+Za 
+\begin_inset Formula $A:V\to U$
+\end_inset
+
+ smo definirali 
+\begin_inset Formula $\rang A\coloneqq\dim\text{Im}A$
+\end_inset
+
+, kjer je 
+\begin_inset Formula $\text{Im}A\coloneqq\left\{ Av;v\in V\right\} $
+\end_inset
+
+.
+ Dokazali smo, da je rang matrike enak številu linearno neodvisnih vrstic
+ matrike in da velja 
+\begin_inset Formula $\rang A=\rang A^{T}$
+\end_inset
+
+.
+\begin_inset Formula 
+\[
+\rang\left[\begin{array}{cccc}
+-2-t & 4 & 5+t & 4\\
+1 & -1 & -2 & 1\\
+-t & 3 & 1+t & 4+t
+\end{array}\right]=\rang\left[\begin{array}{ccc}
+-2-t & 1 & -t\\
+4 & -1 & 3\\
+5+t & -2 & 1+t\\
+4 & 1 & 4+t
+\end{array}\right]=
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+=\rang\left[\begin{array}{ccc}
+4 & 1 & -3\\
+-2-t & 1 & -t\\
+5+t & -2 & 1+t\\
+4 & 1 & 4+t
+\end{array}\right]=\rang\left[\begin{array}{ccc}
+4 & 1 & -3\\
+3 & -1 & 1\\
+5+t & -2 & 1+t\\
+4 & 1 & 4+t
+\end{array}\right]=
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+=\rang\left[\begin{array}{ccc}
+4 & 1 & -3\\
+3 & -1 & 1\\
+1+t & -3 & -3\\
+4 & 1 & 4+t
+\end{array}\right]=\rang\left[\begin{array}{ccc}
+1 & 2 & -4\\
+3 & -1 & 1\\
+1+t & -3 & -3\\
+4 & 1 & 4+t
+\end{array}\right]=
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+=\rang\left[\begin{array}{ccc}
+1 & 2 & -4\\
+0 & -7 & 13\\
+0 & -5-t & 1+4t\\
+0 & -7 & 20+t
+\end{array}\right]=\rang\left[\begin{array}{ccc}
+1 & 2 & -4\\
+0 & -7 & 13\\
+0 & 0 & \frac{-58+15t}{7}\\
+0 & 0 & 7+t
+\end{array}\right]
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Rang je vsaj 2, ker sta 
+\begin_inset Formula $\left(1,2,-4\right)$
+\end_inset
+
+ in 
+\begin_inset Formula $\left(0,-7,13\right)$
+\end_inset
+
+ linearno neodvisna.
+ Rang je kvečjemu 3, ker je manjša izmed stranic matrike dolžine 3.
+ Rang ne more biti 2, ker sistem 
+\begin_inset Formula $\frac{-58+15t}{7}=7+t=0$
+\end_inset
+
+ nima rešitve.
+ 
+\begin_inset Formula $\rang B=3$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Poišči karakteristični in minimalni polinom matrike
+\begin_inset Formula 
+\[
+A=\left[\begin{array}{ccc}
+4 & -5 & 3\\
+2 & -3 & 2\\
+-1 & 1 & 0
+\end{array}\right]
+\]
+
+\end_inset
+
+in s pomočjo Cayley-Hamiltonovega izreka določi njen inverz.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+\begin_inset Formula 
+\[
+\nabla_{P}\left(\lambda\right)=\det\left(A-\lambda I\right)=\left|\begin{array}{ccc}
+4-\lambda & -5 & 3\\
+2 & -3-\lambda & 2\\
+-1 & 1 & -\lambda
+\end{array}\right|=
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+=-3\left(3+\lambda\right)-2\left(4-\lambda\right)-10\lambda+\lambda\left(3+\lambda\right)\left(4-\lambda\right)+10+6=
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+-9-3\lambda-8+2\lambda-10\lambda+\left(3\lambda+\lambda^{2}\right)\left(4-\lambda\right)+16=
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+=-17+16-11\lambda+12\lambda-3\lambda^{2}+4\lambda^{2}-\lambda^{3}=-\lambda^{3}+\lambda^{2}+\lambda-1
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Eno ničlo uganemo (
+\begin_inset Formula $\lambda_{1}=1$
+\end_inset
+
+), nato 
+\begin_inset Formula $-\lambda^{3}+\lambda^{2}+\lambda-1:\lambda-1=-\lambda^{2}+1=\left(1+\lambda\right)\left(1-\lambda\right)$
+\end_inset
+
+.
+ 1 je torej dvojna ničla, 
+\begin_inset Formula $\lambda_{2}=-1$
+\end_inset
+
+ pa enojna.
+ Ker 
+\begin_inset Formula $m_{A}\left(\lambda\right)|\nabla_{A}\left(\lambda\right)$
+\end_inset
+
+, je kandidat za 
+\begin_inset Formula $m_{A}\left(\lambda\right)$
+\end_inset
+
+ poleg 
+\begin_inset Formula $-\nabla_{A}\left(\lambda\right)$
+\end_inset
+
+ še 
+\begin_inset Formula $p\left(\lambda\right)=\left(\lambda-x\right)\left(\lambda+1\right)=1-\lambda^{2}$
+\end_inset
+
+.
+ Po Cayley-Hamiltonovem izreku 
+\begin_inset Formula $m_{A}\left(A\right)=0=\nabla_{A}\left(A\right)$
+\end_inset
+
+.
+ Toda ker 
+\begin_inset Formula $I-A^{2}\not=0$
+\end_inset
+
+, je 
+\begin_inset Formula $m_{A}\left(\lambda\right)=-\nabla_{A}\left(\lambda\right)=\lambda^{3}-\lambda^{2}-\lambda+1$
+\end_inset
+
+.
+ Izračunajmo inverz:
+\begin_inset Formula 
+\[
+m_{A}\left(A\right)=A^{3}-A^{2}-A+I=0\quad\quad/-I
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+A^{3}-A^{2}-A=-I\quad\quad/\cdot A^{-1}
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+A^{3}A^{-1}-A^{2}A^{-1}-AA^{-1}=-IA^{-1}\quad\quad/\cdot\left(-I\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+-A^{2}+A+I=A^{1}=
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+=-\left[\begin{array}{ccc}
+4 & -5 & 3\\
+2 & -3 & 2\\
+-1 & 1 & 0
+\end{array}\right]\left[\begin{array}{ccc}
+4 & -5 & 3\\
+2 & -3 & 2\\
+-1 & 1 & 0
+\end{array}\right]+\left[\begin{array}{ccc}
+4 & -5 & 3\\
+2 & -3 & 2\\
+-1 & 1 & 0
+\end{array}\right]+\left[\begin{array}{ccc}
+1 & 0 & 0\\
+0 & 1 & 0\\
+0 & 0 & 1
+\end{array}\right]=
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+=\left[\begin{array}{ccc}
+2 & -3 & 1\\
+2 & -3 & 2\\
+1 & -1 & 2
+\end{array}\right]\text{, kar je res \ensuremath{A^{-1}.}}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+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/LADN6FMF1.pdf
+	extra LaTeX "pages=-"
+
+\end_inset
+
+
+\begin_inset External
+	template PDFPages
+	filename /mnt/slu/shramba/upload/www/d/LADN6FMF2.pdf
+	extra LaTeX "pages=-"
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document
diff --git a/šola/p2/dn/.gitignore b/šola/p2/dn/.gitignore
new file mode 100644
index 0000000..099718c
--- /dev/null
+++ b/šola/p2/dn/.gitignore
@@ -0,0 +1,6 @@
+*
+!*.c
+!*.txt
+!*_testi
+!.gitignore
+!*_izhodisca/
diff --git a/šola/p2/dn/dn06-naloga1.c b/šola/p2/dn/dn06-naloga1.c
new file mode 100644
index 0000000..9301d71
--- /dev/null
+++ b/šola/p2/dn/dn06-naloga1.c
@@ -0,0 +1,29 @@
+#include <stdio.h>
+#include <stdlib.h>
+#include <stdbool.h>
+#include <string.h>
+#include "naloga1.h"
+char * zdruzi (char ** nizi, char * locilo) {
+	int ll = strlen(locilo);
+	int len = 1-ll;
+	char ** n = nizi;
+	while (*n)
+		len += strlen(*n++)+ll;
+	char * r = malloc(sizeof *r * (len+1));
+	char * rorig = r;
+	while (*nizi) {
+		strcpy(r, *nizi);
+		r += strlen(*nizi);
+		nizi++;
+		if (*nizi) {
+			strcpy(r, locilo);
+			r += ll;
+		}
+	}
+	return rorig;
+}
+#ifndef test
+int main () {
+	return 0;
+}
+#endif
diff --git a/šola/p2/dn/dn06-naloga2.c b/šola/p2/dn/dn06-naloga2.c
new file mode 100644
index 0000000..021930c
--- /dev/null
+++ b/šola/p2/dn/dn06-naloga2.c
@@ -0,0 +1,41 @@
+#include <stdio.h>
+#include <stdlib.h>
+#include <stdbool.h>
+#include <string.h>
+#include "naloga2.h"
+int ** ap2pp (int (*kazalec)[N], int izvornoStVrstic, int ciljnoStVrstic) {
+	int * ka = (int *) kazalec;
+	int ** r = malloc(ciljnoStVrstic*sizeof *r);
+	int outstolpcev = izvornoStVrstic*N/ciljnoStVrstic;
+	for (int i = 0; i < izvornoStVrstic*N; i++) {
+		int j = i/outstolpcev;
+		int k = i%outstolpcev;
+		if (!k) {
+			r[j] = malloc((outstolpcev+1)*sizeof *(r[j]));
+			r[j][outstolpcev] = 0;
+		}
+		r[j][k] = ka[i];
+	}
+	return r;
+}
+int (*pp2ap(int ** kazalec, int izvornoStVrstic, int * ciljnoStVrstic))[N] {
+	int * r = NULL;
+	int rlen = 0;
+	for (int i = 0; i < izvornoStVrstic; i++)
+		for (int j = 0; kazalec[i][j]; j++) {
+			r = realloc(r, ++rlen*sizeof *r);
+			r[rlen-1] = kazalec[i][j];
+		}
+	if (rlen % N != 0) {
+		r = realloc(r, (rlen/N+1)*N * sizeof *r);
+		while (rlen % N)
+			r[rlen++] = 0;
+	}
+	*ciljnoStVrstic = rlen/N;
+	return (int (*)[N]) r;
+}
+#ifndef test
+int main () {
+	return 0;
+}
+#endif
diff --git a/šola/p2/dn/naloga.c b/šola/p2/dn/naloga.c
new file mode 100644
index 0000000..70e5cd6
--- /dev/null
+++ b/šola/p2/dn/naloga.c
@@ -0,0 +1,30 @@
+int steviloZnakov (char * niz, char znak) {
+	int r = 0;
+	while (*niz) {
+		if (*niz++ == znak)
+			r++;
+	return r;
+}
+#include <string.h>
+char * kopirajDoZnaka (char * niz, char znak) {
+	strchr(niz, znak)[0] = '\0';
+	char * r = strdup(niz);
+	niz[strlen(niz)][0] = znak;
+	return r;
+}
+char ** razcleni (char * besedilo, char locilo, int * stOdsekov) {
+	char * p = besedilo;
+	char ** r = NULL;
+	*stOdsekov = 0;
+	while (1) {
+		if (*p == locilo || !*p) {
+			*p = '\0';
+			r = realloc(r, ++*stOdsekov*sizeof *r);
+			r[*stOdsekov-1] = strdup(besedilo);
+			besedilo = p+1;
+			if (!*p)
+				return r;
+		}
+		p++;
+	}
+}