path: root/šola/la/kolokvij2.lyx

#LyX 2.3 created this file. For more info see http://www.lyx.org/
\lyxformat 544
\begin_document
\begin_header
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\textclass article
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\usepackage{pgfplots}
\usepackage{listings}
\usepackage{multicol}
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\end_preamble
\use_default_options true
\begin_modules
enumitem
\end_modules
\maintain_unincluded_children false
\language slovene
\language_package default
\inputencoding auto
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\index Index
\shortcut idx
\color #008000
\end_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