#LyX 2.3 created this file. For more info see http://www.lyx.org/
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\begin_modules
enumitem
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\language slovene
\language_package default
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\index Index
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\end_header
\begin_body
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
newcommand
\backslash
euler{e}
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{multicols}{2}
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\left(AB\right)^{T}=B^{T}+A^{T}$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $E_{ij}\left(\alpha\right)\coloneqq\texttt{i+=\ensuremath{\alpha}j}$
\end_inset
,
\begin_inset Formula $P_{ij}\coloneqq\texttt{i,j=j,i}$
\end_inset
,
\begin_inset Formula $E_{i}\left(\alpha\right)\coloneqq\texttt{i*=\ensuremath{\alpha}}$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $E_{ij}\left(\alpha\right)^{-1}=E_{ij}\left(\alpha\right)$
\end_inset
,
\begin_inset Formula $P_{ij}^{-1}=P_{ji}$
\end_inset
,
\begin_inset Formula $E_{i}\left(\beta\right)^{-1}=E_{i}\left(\beta^{-1}\right)$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\nexists A_{m,n}^{-1}\Leftrightarrow A=0\Leftrightarrow m\not=n\Leftrightarrow\det A=0\Leftrightarrow A$
\end_inset
ima
\begin_inset Formula $\vec{0}$
\end_inset
vrstico/stolpec
\end_layout
\begin_layout Paragraph
Karakterizacija obrnljivih matrik
\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 Itemize
\begin_inset Argument 1
status open
\begin_layout Plain Layout
label=
\begin_inset Formula $\Leftrightarrow$
\end_inset
\end_layout
\end_inset
\begin_inset Formula $\exists A^{-1}$
\end_inset
\end_layout
\begin_layout Itemize
\begin_inset Formula $\exists B\ni:BA=I$
\end_inset
\end_layout
\begin_layout Itemize
\begin_inset Formula $\exists B\ni:AB=I$
\end_inset
\end_layout
\begin_layout Itemize
\begin_inset Formula $\left(AX=0\Longrightarrow X=0\right)$
\end_inset
\end_layout
\begin_layout Itemize
stolpci so ogrodje
\end_layout
\begin_layout Itemize
\begin_inset Formula $\text{RKSO}\left(A\right)=I$
\end_inset
\end_layout
\begin_layout Itemize
\begin_inset Formula $\forall\vec{b}\exists\vec{x}\ni:A\vec{x}=\vec{b}$
\end_inset
\end_layout
\begin_layout Itemize
\begin_inset Formula $A=$
\end_inset
produkt E.
M.
\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 Note Note
status open
\begin_layout Plain Layout
\begin_inset Formula $\exists A^{-1}\Longleftrightarrow\exists B\ni:BA=I\Longleftrightarrow\exists B\ni:AB=I\Longleftrightarrow$
\end_inset
stolpci so LN
\begin_inset Formula $\Longleftrightarrow\left(AX=0\Longrightarrow X=0\right)\Longleftrightarrow$
\end_inset
stolpci so ogrodje
\begin_inset Formula $\Longleftrightarrow\text{RKSO}\left(A\right)=$
\end_inset
\begin_inset Formula $I\Longleftrightarrow\forall\vec{b}\exists\vec{x}\ni:A\vec{x}=\vec{b}\Longleftrightarrow A=$
\end_inset
produkt E.M.
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Matrični zapis sistema:
\begin_inset Formula $A\vec{x}=\vec{b}$
\end_inset
\end_layout
\begin_layout Standard
Najkrajša rešitev sistema
\begin_inset Formula $\vec{x_{0}}\Leftarrow\vert\vert A\vec{x_{0}}-\vec{b}\vert\vert=\min\vert\vert A\vec{x}-\vec{b}\vert\vert$
\end_inset
\end_layout
\begin_layout Standard
...
je običajna rešitev
\begin_inset Formula $A^{T}A\vec{x}=A^{T}\vec{b}$
\end_inset
\end_layout
\begin_layout Standard
Desno množenje z E.
M.
je manipulacija stoplcev.
\end_layout
\begin_layout Standard
\begin_inset Formula $\det\left[\begin{array}{cc}
a & b\\
c & d
\end{array}\right]=ad-bc$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $A_{i,j}\coloneqq A$
\end_inset
brez
\begin_inset Formula $i$
\end_inset
te vrstice in
\begin_inset Formula $j$
\end_inset
tega stolpca
\end_layout
\begin_layout Standard
\begin_inset Formula $\det[a]=a$
\end_inset
,
\begin_inset Formula $\det A=\sum_{k=1}^{n}\left(-1\right)^{k+1}a_{1,k}\det A_{1,j}$
\end_inset
\end_layout
\begin_layout Standard
Razvoj po
\begin_inset Formula $i$
\end_inset
ti vrstici:
\begin_inset Formula $\det A=\sum_{j=1}^{n}\left(-1\right)^{i+j}a_{ij}\det A_{ij}$
\end_inset
\end_layout
\begin_layout Standard
Razvoj po
\begin_inset Formula $j$
\end_inset
tem stolpcu:
\begin_inset Formula $\det A=\sum_{i=1}^{n}\left(-1\right)^{i+j}a_{ij}\det A_{ij}$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\det$
\end_inset
trikotne matrike:
\begin_inset Formula $\prod_{i=1}^{n}a_{ii}$
\end_inset
\end_layout
\begin_layout Standard
Trikotna matrika ima pod ali nad diagonalo same ničle.
\end_layout
\begin_layout Standard
\begin_inset Formula $\det\left(P_{ij}A\right)=-detA,\quad\det\left(E_{i}\alpha A\right)=\alpha\det A$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\det\left(E_{ij}\alpha A\right)=\det A,\quad\det\left(AB\right)=\det A\det B$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\det\left[\begin{array}{cc}
A & B\\
0 & C
\end{array}\right]=\det A\det C,\quad\det A^{T}=\det A$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
\det A^{n}=\left(\det A\right)^{n}\text{ velja tudi za inverz}
\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\det P_{ij}=-1,\quad\det E_{i}\left(\alpha\right)=\alpha,\quad\det E_{ij}\left(\alpha\right)=1$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\det\mathbb{R}^{3}$
\end_inset
: negativne diagonale prištejemo, pozitivne odštejemo
\end_layout
\begin_layout Paragraph
Cramerjevo pravilo
\end_layout
\begin_layout Standard
za rešitev sistema s kvadratno matriko koeficientov:
\begin_inset Formula $x_{i}=\frac{\det A_{i}\left(\vec{b}\right)}{\det A}$
\end_inset
, kjer je
\begin_inset Formula $A_{i}\left(\vec{b}\right)$
\end_inset
matrika
\begin_inset Formula $A$
\end_inset
, ki ima namesto
\begin_inset Formula $i$
\end_inset
-tega stolpca
\begin_inset Formula $\vec{b}$
\end_inset
.
\end_layout
\begin_layout Paragraph
Inverz matrike
\end_layout
\begin_layout Standard
\begin_inset Formula $A_{ij}^{-1}=\frac{\det A_{ji}\left(-1\right)^{j+i}}{\det A}=\frac{1}{\det A}\tilde{A}^{T}$
\end_inset
, kjer je
\begin_inset Formula $\tilde{A}$
\end_inset
kofaktorska matrika:
\begin_inset Formula $\tilde{A_{ij}}=\det A_{ji}\left(-1\right)^{i+j}$
\end_inset
.
\end_layout
\begin_layout Paragraph
Algebrske strukture
\end_layout
\begin_layout Standard
grupoid:
\begin_inset Formula $\left(M\not=\emptyset,\circ:\text{M\ensuremath{\times M\to M}}\right)$
\end_inset
,
\series bold
polgrupa
\series default
je asociativen grupoid,
\series bold
monoid
\series default
je polgrupa z enoto,
\series bold
grupa
\series default
je monoid z inverzom za vsak element,
\series bold
abelova grupa
\series default
je komutativna.
\end_layout
\begin_layout Standard
Desna enota:
\begin_inset Formula $a\circ e=a$
\end_inset
.
Če je leva in desna, je enota.
Grupoid ima kvečjemu eno enoto.
Če je več levih, desne ni.
\end_layout
\begin_layout Standard
Desni inverz:
\begin_inset Formula $a\circ a^{-1}=e$
\end_inset
.
Če je levi in desni, je inverz.
Inverz je enoličen.
V monoidu je levi tudi desni.
\end_layout
\begin_layout Standard
Ko je
\begin_inset Formula $\left(M,\circ\right)$
\end_inset
grupoid in
\begin_inset Formula $N\subset M,N\not=\emptyset$
\end_inset
, je
\begin_inset Formula $N$
\end_inset
\series bold
podgrupoid
\series default
, če
\begin_inset Formula $\forall a,b\in N:a\circ b\in N$
\end_inset
.
\begin_inset Formula $N$
\end_inset
podeduje
\begin_inset Formula $\circ$
\end_inset
v
\begin_inset Formula $\circ_{N}:N\times N\to N$
\end_inset
.
\begin_inset Formula $\circ_{N}$
\end_inset
ohrani komutativnost in asociativnost.
Enota se ne ohrani vedno, inverzi se ne ohranijo vedno.
\end_layout
\begin_layout Standard
Ko je
\begin_inset Formula $\left(M,\circ\right)$
\end_inset
polgrupa,
\begin_inset Formula $N$
\end_inset
podgrupoid, je
\series bold
\begin_inset Formula $N$
\end_inset
podpolgrupa
\series default
.
\end_layout
\begin_layout Standard
Ko je
\begin_inset Formula $\left(M,\circ\right)$
\end_inset
monoid in
\begin_inset Formula $N$
\end_inset
podgrupoid, je
\begin_inset Formula $N$
\end_inset
\series bold
podmonoid
\series default
, če vsebuje enoto
\begin_inset Formula $\left(M,\circ\right)$
\end_inset
(da, prav tisto).
\end_layout
\begin_layout Standard
Ko je
\begin_inset Formula $\left(M,\circ\right)$
\end_inset
grupa in
\begin_inset Formula $N$
\end_inset
podmonoid, je
\begin_inset Formula $N$
\end_inset
\series bold
podgrupa
\series default
, če vsebuje inverze vseh svojih elementov.
\end_layout
\begin_layout Standard
\begin_inset Formula $N\not=\emptyset$
\end_inset
je
\series bold
podgrupa
\series default
\begin_inset Formula $\left(M,\circ\right)$
\end_inset
, ko
\begin_inset Formula $a,b\in N\Rightarrow a\circ b^{-1}\in N$
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset Formula $GL_{n}$
\end_inset
je grupa vseh obrnljivih matrik z množenjem matrik,
\begin_inset Formula $O_{n}$
\end_inset
je grupa matrik, kjer
\begin_inset Formula $A^{T}=A^{-1}$
\end_inset
(ortogonalne),
\begin_inset Formula $SL_{n}$
\end_inset
je grupa matrik z
\begin_inset Formula $\det A=1$
\end_inset
,
\begin_inset Formula $SO_{n}$
\end_inset
je grupa ortogonalnih matrik z
\begin_inset Formula $\det A=1$
\end_inset
.
\end_layout
\begin_layout Paragraph
Homomorfizem
\end_layout
\begin_layout Standard
grupoidov in polgrup
\begin_inset Formula $\left(M_{1},\circ_{1}\right),\left(M_{2},\circ_{2}\right)$
\end_inset
je
\begin_inset Formula $f:M_{1}\to M_{2}\ni:\forall a,b\in M_{1}:\left(f\left(a\circ_{1}b\right)=f\left(a\right)\circ_{2}f\left(b\right)\right)$
\end_inset
.
\end_layout
\begin_layout Standard
Homomorfizem monoidov mora imeti še lastnost
\begin_inset Formula $f\left(e_{1}\right)=e_{2}$
\end_inset
, homomorfizem grup pa lastnost
\begin_inset Formula $f\left(a^{-1}\right)=f\left(a\right)^{-1}$
\end_inset
.
\end_layout
\begin_layout Standard
Kompozitum homomorfizmov je homomorfizem.
\end_layout
\begin_layout Standard
\series bold
Izomorfizem
\series default
je bijektiven homomorfizem.
Med izomorfnima grupama obstaja izomorfizem.
\end_layout
\begin_layout Standard
\begin_inset Formula $\left(M,+,\cdot\right)$
\end_inset
je
\series bold
bigrupoid
\series default
, ko sta
\begin_inset Formula $\left(M,+\right)$
\end_inset
in
\begin_inset Formula $\left(M,\cdot\right)$
\end_inset
grupoida.
\end_layout
\begin_layout Standard
\series bold
Distributiven bigrupoid
\series default
ima
\series bold
po eno
\series default
L in D distributivnost in je
\series bold
polkolobar
\series default
, če je
\begin_inset Formula $\left(M,+\right)$
\end_inset
komutativna polgrupa.
\end_layout
\begin_layout Standard
\series bold
Kolobar
\series default
je distri.
bigrupoid, kjer je
\series bold
\begin_inset Formula $\left(M,+\right)$
\end_inset
\series default
abelova grupa.
\end_layout
\begin_layout Standard
Pri
\series bold
asociativnem kolobarju
\series default
je
\begin_inset Formula $\left(M,\cdot\right)$
\end_inset
polgrupa.
Lemut pravi, da je to pogoj že za kolobarje, Cimprič pa ne.
\end_layout
\begin_layout Standard
Pri
\series bold
asociativnem kolobarju z enoto
\series default
je
\begin_inset Formula $\left(M,\cdot\right)$
\end_inset
monoid.
\end_layout
\begin_layout Standard
\series bold
Obseg
\series default
je kolobar z enoto za množenje
\series bold
\begin_inset Formula $1$
\end_inset
\series default
in inverzom za množenje za vsak neničeln element (
\begin_inset Formula $0$
\end_inset
je enota za
\begin_inset Formula $+$
\end_inset
).
\end_layout
\begin_layout Standard
\series bold
Komutativen kolobar
\series default
ima komutativno množenje.
\end_layout
\begin_layout Standard
\series bold
Polje
\series default
je komutativen obseg.
\end_layout
\begin_layout Standard
\series bold
Podbigrupoid
\series default
je
\begin_inset Formula $N\subset M$
\end_inset
, zaprta za
\begin_inset Formula $+$
\end_inset
in
\begin_inset Formula $\cdot$
\end_inset
.
\end_layout
\begin_layout Standard
\series bold
Podkolobar
\series default
je
\begin_inset Formula $N\subset M$
\end_inset
, da je
\begin_inset Formula $N$
\end_inset
podgrupa
\begin_inset Formula $\left(M,+\right)$
\end_inset
in podgrupoid
\begin_inset Formula $\left(M,\cdot\right)$
\end_inset
–
\begin_inset Formula $N$
\end_inset
zaprta za odštevanje in množenje.
\end_layout
\begin_layout Standard
\series bold
Podobseg
\series default
je podkolobar, kjer je
\begin_inset Formula $N\backslash\left\{ 0\right\} $
\end_inset
podgrupa
\begin_inset Formula $\left(M\backslash\left\{ 0\right\} ,\cdot\right)$
\end_inset
.
\begin_inset Formula $0$
\end_inset
namreč ni obrnljiva –
\begin_inset Formula $N$
\end_inset
zaprta za
\begin_inset Formula $-$
\end_inset
in deljenje.
\end_layout
\begin_layout Standard
\series bold
Homomorfizem kolobarjev
\series default
je
\begin_inset Formula $f:M_{1}\to M_{2}\ni:f\left(a+_{1}b\right)=f\left(a\right)+_{2}f\left(b\right)\wedge f\left(a\cdot_{1}b\right)=f\left(a\right)\cdot_{2}f\left(b\right)$
\end_inset
\end_layout
\begin_layout Standard
\series bold
Homomorfizem kolobarjev z enoto
\series default
dodatno
\begin_inset Formula $f\left(1_{1}\right)=1_{2}$
\end_inset
\end_layout
\begin_layout Paragraph
Vektorski prostor
\end_layout
\begin_layout Standard
je Abelova grupa z množenjem s skalarjem.
\begin_inset Formula $F$
\end_inset
je polje, za prostor
\begin_inset Formula $\left(V,+,\cdot\right)$
\end_inset
nad
\begin_inset Formula $F$
\end_inset
velja:
\end_layout
\begin_layout Itemize
\begin_inset Formula $\left(V,+\right)$
\end_inset
je Abelova grupa
\end_layout
\begin_layout Itemize
\begin_inset Formula $\alpha\cdot\left(a+b\right)=\alpha\cdot a+\alpha\cdot b,\quad\left(\alpha+\beta\right)\cdot a=\alpha\cdot a+\beta\cdot a$
\end_inset
\end_layout
\begin_layout Itemize
\begin_inset Formula $\left(\alpha\cdot\beta\right)\cdot a=\alpha\cdot\left(\beta\cdot a\right),\quad1\cdot a=a$
\end_inset
\end_layout
\begin_layout Standard
\series bold
Direktna vsota vektorskih prostorov
\series default
je vektorski prostor.
\begin_inset Formula $V_{1}\oplus V_{2}$
\end_inset
so pari
\begin_inset Formula $\left(v_{1},v_{2}\right)$
\end_inset
.
\begin_inset Formula $\left(v_{1},v_{2}\right)+\left(v_{1}',v_{2}'\right)=\left(v_{1}+v_{1}',v_{2}+v_{2}'\right)$
\end_inset
,
\begin_inset Formula $\alpha\cdot\left(v_{1},v_{2}\right)=\left(\alpha\cdot v_{1},\alpha\cdot v_{2}\right)$
\end_inset
.
\end_layout
\begin_layout Paragraph
Vektorski podprostor
\end_layout
\begin_layout Standard
je
\begin_inset Formula $W\subseteq V,W\not=\emptyset$
\end_inset
, zaprta za seštevanje in množenje s skalarjem.
Oziroma taka, da vsebuje vse svoje linearne kombinacije —
\begin_inset Formula $\forall a,b\in W\forall\alpha,\beta\in F:\alpha a+\beta b\in W$
\end_inset
.
Vsak podprostor vsebuje 0.
\series bold
Presek podprostorov
\series default
je tudi sam podprostor.
\series bold
Vsota podprostorov
\series default
(
\begin_inset Formula $W_{1}+W_{2}=\left\{ w_{1}+w_{2};w_{1}\in W_{1},w_{2}\in W_{2}\right\} $
\end_inset
) je tudi sama podprostor.
\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
