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+#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
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+\index_command default
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+\spacing single
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+\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
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+\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 četrte 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}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Reši enačbo
+\begin_inset Formula 
+\[
+\left|\begin{array}{cccc}
+1 & 2 & 3 & 4\\
+x+1 & 2 & x+3 & 4\\
+1 & x+2 & x+4 & x+5\\
+1 & -3 & -4 & -5
+\end{array}\right|=\left|\begin{array}{cc}
+3x & -1\\
+6 & x+1
+\end{array}\right|
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
+\begin_layout Paragraph
+Rešitev
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+\left|\begin{array}{cc}
+3x & -1\\
+6 & x+1
+\end{array}\right|=3x\left(x+1\right)+6=3x^{2}+3x+6
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+\left|\begin{array}{cccc}
+1 & 2 & 3 & 4\\
+x+1 & 2 & x+3 & 4\\
+1 & x+2 & x+4 & x+5\\
+1 & -3 & -4 & -5
+\end{array}\right|=\left|\begin{array}{cccc}
+1 & 2 & 3 & 4\\
+x+1 & 2 & x+3 & 4\\
+1 & x+2 & x+4 & x+5\\
+0 & -5 & -7 & -9
+\end{array}\right|=
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+=\left|\begin{array}{cccc}
+0 & -x & -x-1 & -x-1\\
+x+1 & 2 & x+3 & 4\\
+1 & x+2 & x+4 & x+5\\
+0 & -5 & -7 & -9
+\end{array}\right|=
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+=-\left(x+1\right)\left|\begin{array}{ccc}
+-x & -x-1 & -x-1\\
+x+2 & x+4 & x+5\\
+-5 & -7 & -9
+\end{array}\right|+\left|\begin{array}{ccc}
+-x & -x-1 & -x-1\\
+2 & x+3 & 4\\
+-5 & -7 & -9
+\end{array}\right|=
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+=x-1+4x^{2}-5x+1=4x^{2}-6x
+\]
+
+\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
+
+4x^2-6x&=3x^2+3x+6
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+x^2-9x-6&=0
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{align*}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+x_{1,2}=\frac{9\pm\sqrt{81+24}}{2}=\frac{9\pm\sqrt{105}}{2},\quad x_{1}=\frac{9+\sqrt{105}}{2},x_{2}=\frac{9-\sqrt{105}}{2}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Dokaži, da je preslikava 
+\begin_inset Formula $x\mapsto x^{-1}$
+\end_inset
+
+ avtomorfizem grupe natanko tedaj, ko je grupa komutativna.
+\end_layout
+
+\begin_deeper
+\begin_layout Paragraph
+Rešitev
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+udensdash{$f
+\backslash
+left(x
+\backslash
+right)=x^{-1} 
+\backslash
+text{ je avtomorfizem}
+\backslash
+Longleftrightarrow
+\backslash
+forall a,b
+\backslash
+in M:a
+\backslash
+cdot b=b
+\backslash
+cdot a$}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Dokaz
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset CommandInset label
+LatexCommand label
+name "enu:Enota-se-preslika"
+
+\end_inset
+
+Enota se preslika v enoto.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{align*}
+\end_layout
+
+\begin_layout Plain Layout
+
+e
+\backslash
+cdot e^{-1}&=e&&
+\backslash
+text{(definicija inverza $a
+\backslash
+cdot a^{-1}=e$)}
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+e
+\backslash
+cdot e^{-1}&=e^{-1}&&
+\backslash
+text{(definicija enote $e
+\backslash
+cdot a=a$)}
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{align*}
+\end_layout
+
+\begin_layout Plain Layout
+
+$$
+\backslash
+Longrightarrow e=e
+\backslash
+cdot e^{-1}=e^{-1}$$
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+\begin_inset CommandInset label
+LatexCommand label
+name "enu:Da-je-preslikava"
+
+\end_inset
+
+Da je preslikava bijektivna, moramo dokazati, da je injektivna, torej, da
+ so v komutativni grupi inverzi enolični — da dva različna elementa nimata
+ istega inverza, in da je surjektivna, torej, da je kodomena enaka zalogi
+ vrednosti.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Naj bo 
+\begin_inset Formula $\left(M,\cdot\right)$
+\end_inset
+
+ grupa.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+udensdash{$
+\backslash
+forall a,b
+\backslash
+in M:
+\backslash
+left(a^{-1}=b^{-1}
+\backslash
+Longrightarrow a=b
+\backslash
+right)$}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Naj bo 
+\begin_inset Formula $a^{-1}=b^{-1}$
+\end_inset
+
+.
+ 
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+udensdash{$a=b$}
+\end_layout
+
+\end_inset
+
+
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{align*}
+\end_layout
+
+\begin_layout Plain Layout
+
+a
+\backslash
+cdot a^{-1}&=e&&b
+\backslash
+cdot b^{-1}=e
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+a
+\backslash
+cdot b^{-1}&=e&&/
+\backslash
+cdot b
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+a
+\backslash
+cdot e&=e
+\backslash
+cdot b
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{align*}
+\end_layout
+
+\begin_layout Plain Layout
+
+$$a=b$$
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+\begin_inset CommandInset label
+LatexCommand label
+name "enu:Dokaz-ohranjanja-inverzov:"
+
+\end_inset
+
+Dokaz ohranjanja inverzov: 
+\begin_inset Formula $f\left(x\right)^{-1}=f\left(x^{-1}\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+\begin_inset Formula 
+\[
+\left(x^{-1}\right)^{-1}=\left(x^{-1}\right)^{-1}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Ob upoštevanju 
+\begin_inset CommandInset ref
+LatexCommand eqref
+reference "enu:Da-je-preslikava"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+ je to enako kot 
+\begin_inset Formula $x=x$
+\end_inset
+
+, kar drži, torej je preslikava injektivna.
+\end_layout
+
+\begin_layout Standard
+Da je surjektivna, mora veljati 
+\begin_inset Formula $\forall x^{-1}\exists x:x^{-1}=x$
+\end_inset
+
+.
+ Naj bo tak 
+\begin_inset Formula $x$
+\end_inset
+
+ kar 
+\begin_inset Formula $\left(x^{-1}\right)^{-1}$
+\end_inset
+
+.
+ Dokažimo: 
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+udensdash{$
+\backslash
+left(x^{-1}
+\backslash
+right)^{-1}=x$}
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{align*}
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+left(x^{-1}
+\backslash
+right)^{-1}&
+\backslash
+overset{?}{=}x&&/
+\backslash
+cdot x^{-1}
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+e&=e
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{align*}
+\end_layout
+
+\begin_layout Plain Layout
+
+Torej je preslikava bijektivna.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+\begin_inset CommandInset label
+LatexCommand label
+name "enu:Asociativnost-operacije."
+
+\end_inset
+
+Asociativnost operacije.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Zahtevamo, da operacija ostane enaka, zato je asociativna.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+\begin_inset CommandInset label
+LatexCommand label
+name "enu:Po-definiciji-homomorfizma"
+
+\end_inset
+
+Po definiciji homomorfizma je treba dokazati, da 
+\begin_inset Formula 
+\[
+\forall a,b\in M:\left(f\left(a\cdot_{1}b\right)=f\left(a\right)\cdot_{2}f\left(b\right)\right)\Longleftrightarrow\text{grupa je Abelova}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Naj bosta 
+\begin_inset Formula $a,b$
+\end_inset
+
+ poljubna iz grupe 
+\begin_inset Formula $\left(M,\cdot\right)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Lemma
+V grupi 
+\begin_inset Formula $\left(N,\circ\right)$
+\end_inset
+
+ velja za poljubna 
+\begin_inset Formula $x,y\in N$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Lemma
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+udensdash{$
+\backslash
+left(a
+\backslash
+circ b
+\backslash
+right)^{-1}=y^{-1}
+\backslash
+circ x^{-1}$}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Lemma
+Dokaz leme:
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{align*}
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+left(x
+\backslash
+circ y
+\backslash
+right)
+\backslash
+circ
+\backslash
+backslash&&
+\backslash
+left(x
+\backslash
+circ y
+\backslash
+right)^{-1}&
+\backslash
+overset{?}{=}y^{-1}
+\backslash
+circ x^{-1}
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+&&
+\backslash
+left(x
+\backslash
+circ y
+\backslash
+right)
+\backslash
+circ
+\backslash
+left(x
+\backslash
+circ y
+\backslash
+right)^{-1}&
+\backslash
+overset{?}{=}
+\backslash
+left(x
+\backslash
+circ y
+\backslash
+right)
+\backslash
+circ
+\backslash
+left(y^{-1}
+\backslash
+circ x^{-1}
+\backslash
+right)=x
+\backslash
+circ
+\backslash
+left(y
+\backslash
+circ y^{-1}
+\backslash
+right)
+\backslash
+circ x^{-1}
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+&&e&
+\backslash
+overset{?}{=}x
+\backslash
+circ e
+\backslash
+circ x^{-1}
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+&&e&
+\backslash
+overset{?}{=}x
+\backslash
+circ x^{-1}
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+&&e&=e
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{align*}
+\end_layout
+
+\end_inset
+
+Konec leme — lema je dokazana.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{align*}
+\end_layout
+
+\begin_layout Plain Layout
+
+f
+\backslash
+left(a
+\backslash
+cdot b
+\backslash
+right)&
+\backslash
+overset{?}{=}f
+\backslash
+left(a
+\backslash
+right)
+\backslash
+cdot f
+\backslash
+left(b
+\backslash
+right)
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+b^{-1}
+\backslash
+cdot a^{-1}
+\backslash
+overset{
+\backslash
+text{lema}}{=}
+\backslash
+left(a
+\backslash
+cdot b
+\backslash
+right)^{-1}&
+\backslash
+overset{?}{=}a^{-1}
+\backslash
+cdot b^{-1}
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+b^{-1}
+\backslash
+cdot a^{-1}&=a^{-1}
+\backslash
+cdot b^{-1}&&
+\backslash
+text{velja natanko tedaj, ko je grupa Abelova.}
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{align*}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Standard
+\begin_inset CommandInset ref
+LatexCommand eqref
+reference "enu:Enota-se-preslika"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+, 
+\begin_inset CommandInset ref
+LatexCommand eqref
+reference "enu:Da-je-preslikava"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+, 
+\begin_inset CommandInset ref
+LatexCommand eqref
+reference "enu:Dokaz-ohranjanja-inverzov:"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+, 
+\begin_inset CommandInset ref
+LatexCommand eqref
+reference "enu:Asociativnost-operacije."
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+ veljajo ne glede na to, ali je grupa komutativna ali ne, 
+\begin_inset CommandInset ref
+LatexCommand eqref
+reference "enu:Po-definiciji-homomorfizma"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+ pa velja natanko tedaj, ko je grupa komutativna.
+\begin_inset Formula $\qed$
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Prepričaj se, da je množica 
+\begin_inset Formula $\mathbb{Z}\times\mathbb{Z}$
+\end_inset
+
+ komutativen kolobar za operaciji
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{align*}
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+left(a, b
+\backslash
+right)
+\backslash
+oplus
+\backslash
+left(c, d
+\backslash
+right)&=
+\backslash
+left(a+c, b+d
+\backslash
+right)
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+left(a, b
+\backslash
+right)
+\backslash
+otimes
+\backslash
+left(c, d
+\backslash
+right)&=
+\backslash
+left(ac, bd
+\backslash
+right)
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{align*}
+\end_layout
+
+\end_inset
+
+Poišči tudi vse delitelje niča, tj.
+ neničelne elemente 
+\begin_inset Formula $\left(a,b\right)$
+\end_inset
+
+, da velja 
+\begin_inset Formula $\left(a,b\right)\otimes\left(c,d\right)=0\left(=e_{\oplus}\right)$
+\end_inset
+
+ za nek neničeln 
+\begin_inset Formula $\left(c,d\right)$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Paragraph
+Rešitev
+\end_layout
+
+\begin_layout Itemize
+Dokažimo distributivnost!
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{align*}
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+left(a,b
+\backslash
+right)
+\backslash
+otimes
+\backslash
+left(
+\backslash
+left(c,d
+\backslash
+right)
+\backslash
+oplus
+\backslash
+left(e,f
+\backslash
+right)
+\backslash
+right)&
+\backslash
+overset{?}{=}
+\backslash
+left(a,b
+\backslash
+right)
+\backslash
+otimes
+\backslash
+left(c,d
+\backslash
+right)
+\backslash
+oplus
+\backslash
+left(a,b
+\backslash
+right)
+\backslash
+otimes
+\backslash
+left(e,f
+\backslash
+right)
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+left(a,b
+\backslash
+right)
+\backslash
+otimes
+\backslash
+left(c+e,d+f
+\backslash
+right)&
+\backslash
+overset{?}{=}
+\backslash
+left(ac,bd
+\backslash
+right)
+\backslash
+oplus
+\backslash
+left(ae,bf
+\backslash
+right)
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+left(a
+\backslash
+cdot
+\backslash
+left(c+e
+\backslash
+right),b
+\backslash
+cdot
+\backslash
+left(d+f
+\backslash
+right)
+\backslash
+right)&=
+\backslash
+left(a
+\backslash
+cdot c+a
+\backslash
+cdot e,b
+\backslash
+cdot d+b
+\backslash
+cdot f
+\backslash
+right)
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{align*}
+\end_layout
+
+\begin_layout Plain Layout
+
+Velja, ker je $
+\backslash
+left(
+\backslash
+mathbb{Z},+,
+\backslash
+cdot
+\backslash
+right)$ distributiven bigrupoid.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+Dokažimo 
+\begin_inset Formula $\left(\mathbb{Z}\times\mathbb{Z},\oplus\right)$
+\end_inset
+
+ je Abelova grupa!
+\end_layout
+
+\begin_deeper
+\begin_layout Itemize
+Komutativnost:
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{align*}
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+forall
+\backslash
+left(a,b
+\backslash
+right),
+\backslash
+left(c,d
+\backslash
+right)
+\backslash
+in
+\backslash
+mathbb{Z}
+\backslash
+times
+\backslash
+mathbb{Z}:&&
+\backslash
+left(a,b
+\backslash
+right)
+\backslash
+oplus
+\backslash
+left(c,d
+\backslash
+right)&=
+\backslash
+left(c,d
+\backslash
+right)
+\backslash
+oplus
+\backslash
+left(a,b
+\backslash
+right)
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+&&
+\backslash
+left(a+c,b+d
+\backslash
+right)&=
+\backslash
+left(c+a,d+b
+\backslash
+right)
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{align*}
+\end_layout
+
+\begin_layout Plain Layout
+
+Velja, ker je $
+\backslash
+left(
+\backslash
+mathbb{Z},+
+\backslash
+right)$ komutativen grupoid.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+Notranja operacija:
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{align*}
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+forall
+\backslash
+left(a,b
+\backslash
+right),
+\backslash
+left(c,d
+\backslash
+right)
+\backslash
+in
+\backslash
+mathbb{Z}
+\backslash
+times
+\backslash
+mathbb{Z}:&&
+\backslash
+left(a,b
+\backslash
+right)
+\backslash
+oplus
+\backslash
+left(c,d
+\backslash
+right)&
+\backslash
+in
+\backslash
+mathbb{Z}
+\backslash
+times
+\backslash
+mathbb{Z}
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+&&
+\backslash
+left(a+c,b+d
+\backslash
+right)&
+\backslash
+in
+\backslash
+mathbb{Z}
+\backslash
+times
+\backslash
+mathbb{Z}
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{align*}
+\end_layout
+
+\begin_layout Plain Layout
+
+Velja, ker je $
+\backslash
+left(
+\backslash
+mathbb{Z},+
+\backslash
+right)$ grupoid.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+Asociativnost:
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{align*}
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+forall
+\backslash
+left(a,b
+\backslash
+right),
+\backslash
+left(c,d
+\backslash
+right),
+\backslash
+left(e,f
+\backslash
+right)
+\backslash
+in
+\backslash
+mathbb{Z}
+\backslash
+times
+\backslash
+mathbb{Z}:&&
+\backslash
+left(a,b
+\backslash
+right)
+\backslash
+oplus
+\backslash
+left(
+\backslash
+left(c,d
+\backslash
+right)
+\backslash
+oplus
+\backslash
+left(e,f
+\backslash
+right)
+\backslash
+right)&=
+\backslash
+left(
+\backslash
+left(a,b
+\backslash
+right)
+\backslash
+oplus
+\backslash
+left(c,d
+\backslash
+right)
+\backslash
+right)
+\backslash
+oplus
+\backslash
+left(e,f
+\backslash
+right)
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+&&
+\backslash
+left(a+
+\backslash
+left(c+e
+\backslash
+right),b+
+\backslash
+left(d,f
+\backslash
+right)
+\backslash
+right)&=
+\backslash
+left(
+\backslash
+left(a+c
+\backslash
+right)+e,
+\backslash
+left(b+d
+\backslash
+right)+f
+\backslash
+right)
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{align*}
+\end_layout
+
+\begin_layout Plain Layout
+
+Velja, ker je $
+\backslash
+left(
+\backslash
+mathbb{Z},+
+\backslash
+right)$ grupoid.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+Enota:
+\begin_inset Formula 
+\[
+\exists e\in\mathbb{Z}\times\mathbb{Z}\ni:\forall\left(a,b\right)\in\mathbb{Z}\times\mathbb{Z}:\left(a,b\right)\oplus e=\left(a,b\right)
+\]
+
+\end_inset
+
+naj bo 
+\begin_inset Formula $e\coloneqq\left(0,0\right)$
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\left(a,b\right)\oplus\left(0,0\right)=\left(a+b,b+0\right)=\left(a,b\right)
+\]
+
+\end_inset
+
+Velja, ker je 
+\begin_inset Formula $0$
+\end_inset
+
+ enota v 
+\begin_inset Formula $\left(\mathbb{Z},+\right)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+Inverzi:
+\begin_inset Formula 
+\[
+\forall\left(a,b\right)\in\mathbb{Z}\times\mathbb{Z}\exists t\in\mathbb{Z}\times\mathbb{Z}\ni:\left(a,b\right)\oplus t=e_{\oplus}=\left(0,0\right)
+\]
+
+\end_inset
+
+naj bo 
+\begin_inset Formula $t\coloneqq\left(-a,-b\right)$
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\left(a,b\right)\oplus\left(-a,-b\right)=\left(a-a,b-b\right)=\left(0,0\right)=e_{\oplus}
+\]
+
+\end_inset
+
+Velja, ker je 
+\begin_inset Formula $\left(\mathbb{Z},+\right)$
+\end_inset
+
+ grupa.
+\end_layout
+
+\end_deeper
+\begin_layout Itemize
+Dokažimo komutativnost 
+\begin_inset Formula $\left(\mathbb{Z}\times\mathbb{Z},\otimes\right)$
+\end_inset
+
+!
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{align*}
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+forall
+\backslash
+left(a,b
+\backslash
+right),
+\backslash
+left(c,d
+\backslash
+right)
+\backslash
+in
+\backslash
+mathbb{Z}
+\backslash
+times
+\backslash
+mathbb{Z}:&&
+\backslash
+left(a,b
+\backslash
+right)
+\backslash
+otimes
+\backslash
+left(c,d
+\backslash
+right)&
+\backslash
+overset{?}{=}
+\backslash
+left(c,d
+\backslash
+right)
+\backslash
+otimes
+\backslash
+left(a,b
+\backslash
+right)
+\backslash
+
+\backslash
+
+\end_layout
+
+\begin_layout Plain Layout
+
+&&
+\backslash
+left(ac,bd
+\backslash
+right)=
+\backslash
+left(ca,db
+\backslash
+right)
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{align*}
+\end_layout
+
+\begin_layout Plain Layout
+
+Velja, ker je $
+\backslash
+left(
+\backslash
+mathbb{Z},
+\backslash
+cdot
+\backslash
+right)$ komutativen grupoid.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+\qed
+\]
+
+\end_inset
+
+Vsi delitelji niča 
+\begin_inset Formula $=\left\{ \left(a,b\right)\in\mathbb{Z}\times\mathbb{Z};\left(a,b\right)\otimes\left(c,d\right)=e_{\oplus}=\left(0,0\right)\right\} $
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Itemize
+Če je 
+\begin_inset Formula $c=0$
+\end_inset
+
+ in 
+\begin_inset Formula $d\not=0$
+\end_inset
+
+:
+\begin_inset Formula 
+\[
+\left(a,b\right)=\left\{ \left(a,0\right);a\in\mathbb{Z}\right\} \sim\mathbb{Z}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+Če je 
+\begin_inset Formula $c\not=0$
+\end_inset
+
+ in 
+\begin_inset Formula $d=0$
+\end_inset
+
+:
+\begin_inset Formula 
+\[
+\left(a,b\right)=\left\{ \left(0,a\right);a\in\mathbb{Z}\right\} \sim\mathbb{Z}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+S pomočjo (razširjenega) Evklidovega algoritma izračunaj 
+\begin_inset Formula $\gcd\left(x^{5}+2x^{4}-x^{2}+1,x^{4}-1\right)$
+\end_inset
+
+ in ga izrazi kot linearno kombinacijo teh dveh polinomov.
+\end_layout
+
+\begin_deeper
+\begin_layout Paragraph
+Rešitev
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float table
+placement h
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\begin_inset Tabular
+<lyxtabular version="3" rows="7" columns="4">
+<features tabularvalignment="middle">
+<column alignment="center" valignment="top">
+<column alignment="center" valignment="top">
+<column alignment="center" valignment="top">
+<column alignment="center" valignment="top">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+r
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+s
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+t
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+k
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $x^{5}+2x^{4}-x^{2}+1$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+1
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+0
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $x^{4}-1$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+0
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+1
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $x-2$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $-x^{2}+x+3$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+1
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $-x-2$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $-x^{2}-x-4$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $7+11$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $x^{2}+x+4$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $-x^{3}-3x^{2}-6x-7$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $-\frac{1}{7}x+\frac{18}{49}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $-\frac{51}{49}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\frac{1}{7}x^{3}-\frac{11}{49}x^{2}+\frac{10}{49}x-\frac{23}{49}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $-\frac{1}{7}x^{4}-\frac{3}{49}x^{3}+\frac{12}{49}x^{2}+\frac{10}{49}x+\frac{4}{7}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+0
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+Koraki razširjenega Evklidovega algoritma.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Tako dobljen polinom 
+\begin_inset Formula $-\frac{51}{49}$
+\end_inset
+
+ normiramo (delimo 
+\begin_inset Formula $r,s,t$
+\end_inset
+
+ z 
+\begin_inset Formula $-\frac{51}{49}$
+\end_inset
+
+).
+\begin_inset Formula 
+\[
+\gcd\left(x^{5}+2x^{4}-x^{2}+1,x^{4}-1\right)=1
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+-\frac{49}{51}\left(\frac{1}{7}x^{3}-\frac{11}{49}x^{2}+\frac{10}{49}x-\frac{23}{49}\right)\left(x^{5}+2x^{4}-x^{2}+1\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+-\frac{49}{51}\left(-\frac{1}{7}x^{4}-\frac{3}{49}x^{3}+\frac{12}{49}x^{2}+\frac{10}{49}x+\frac{4}{7}\right)\left(x^{4}-1\right)=1=
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+=\left(-\frac{7}{51}x^{3}+\frac{11}{51}x^{2}-\frac{10}{51}x+\frac{23}{51}\right)\left(x^{5}+2x^{4}-x^{2}+1\right)+
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
++\left(\frac{7}{51}x^{4}+\frac{3}{51}x^{3}-\frac{12}{51}x^{2}-\frac{10}{51}x-\frac{28}{51}\right)\left(x^{4}-1\right)
+\]
+
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset External
+	template PDFPages
+	filename /home/z/www/dir/zapiski/LA1DN4 FMF 2023-12-26.pdf
+	extra LaTeX "pages=-"
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document
diff --git a/šola/la/dn4/gcd.r b/šola/la/dn4/gcd.r
new file mode 100644
index 0000000..5d10a40
--- /dev/null
+++ b/šola/la/dn4/gcd.r
@@ -0,0 +1,29 @@
+polygcd <- function (a, b) {
+	rp <- a
+	sp <- 1
+	tp <- 0
+	r <- b
+	s <- 0
+	t <- 1
+	while (any(r != 0)) {
+		print(list("debug"=list("r"=r, "s"=s, "t"=t)))
+		rs <- r
+		ss <- s
+		ts <- t
+		korak <- function (t, p) {
+			odstrani_ničle <- function (v) {
+				if (all(v == 0))
+					return(v)
+				return(v[min(which(v!=0)):max(which(v!=0))])
+			}
+			return(p-polymul(odstrani_ničle(t), polydiv(odstrani_ničle(r), odstrani_ničle(rp))$d))
+		}
+		r <- korak(r, rp)
+		s <- korak(s, sp)
+		t <- korak(t, tp)
+		rp <- rs
+		sp <- ss
+		tp <- ts
+	}
+	return(list("r"=rp, "s"=sp, "t"=tp))
+}