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\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
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\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.
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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*}
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\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.
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status open
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\backslash
begin{align*}
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e
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text{(definicija inverza $a
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cdot a^{-1}=e$)}
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
e
\backslash
cdot e^{-1}&=e^{-1}&&
\backslash
text{(definicija enote $e
\backslash
cdot a=a$)}
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\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.
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\begin_deeper
\begin_layout Standard
Naj bo
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grupa.
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\begin_layout Standard
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status open
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\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*}
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\begin_layout Plain Layout
$$a=b$$
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\end_inset
\end_layout
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\begin_layout Enumerate
\begin_inset CommandInset label
LatexCommand label
name "enu:Dokaz-ohranjanja-inverzov:"
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Dokaz ohranjanja inverzov:
\begin_inset Formula $f\left(x\right)^{-1}=f\left(x^{-1}\right)$
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\begin_deeper
\begin_layout Standard
\begin_inset Formula
\[
\left(x^{-1}\right)^{-1}=\left(x^{-1}\right)^{-1}
\]
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\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$}
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\backslash
begin{align*}
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\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
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\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
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\backslash
begin{align*}
\end_layout
\begin_layout Plain Layout
\backslash
left(x
\backslash
circ y
\backslash
right)
\backslash
circ
\backslash
backslash&&
\backslash
left(x
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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
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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}}{=}
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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
