#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 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 \begin_inset Text \begin_layout Plain Layout r \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout s \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout t \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout k \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $x^{5}+2x^{4}-x^{2}+1$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout 1 \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout 0 \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $x^{4}-1$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout 0 \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout 1 \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $x-2$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $-x^{2}+x+3$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout 1 \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $-x-2$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $-x^{2}-x-4$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $7+11$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $x^{2}+x+4$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $-x^{3}-3x^{2}-6x-7$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $-\frac{1}{7}x+\frac{18}{49}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $-\frac{51}{49}$ \end_inset \end_layout \end_inset \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 \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 \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout 0 \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \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