path: root/šola/la/kolokvij1.lyx

#LyX 2.3 created this file. For more info see http://www.lyx.org/
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\end_header

\begin_body

\begin_layout Title
List s formulami za 1.
 kolokvij 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 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 $\vec{u}\cdot\vec{u}=\vert\vert\vec{u}\vert\vert^{2}$
\end_inset

, 
\begin_inset Formula $\left(\alpha\vec{u}+\beta\vec{v}\right)\cdot\vec{w}=\alpha\left(\vec{u}\cdot\vec{w}\right)+\beta\left(\vec{v},\vec{w}\right)$
\end_inset


\end_layout

\begin_layout Standard
Paralelogramska ident.: 
\begin_inset Formula $\vert\vert\vec{u}+\vec{v}\vert\vert^{2}+\vert\vert\vec{u}-\vec{v}\vert\vert^{2}=2\vert\vert\vec{u}\vert\vert^{2}+2\vert\vert\vec{v}\vert\vert^{2}$
\end_inset


\end_layout

\begin_layout Standard
Ploščina paralelograma: 
\begin_inset Formula $\vert\vert\vec{u}\times\vec{v}\vert\vert=\vert\vert\vec{u}\vert\vert\vert\vert\vec{v}\vert\vert\sin\phi$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $<\vec{u},\vec{v}>=\vert\vert\vec{u}\vert\vert\vert\vert\vec{v}\vert\vert\cos\phi$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $<\vec{u}\times\vec{v},\vec{u}>=0$
\end_inset

, 
\begin_inset Formula $\vert\vert\vec{u}\times\vec{v}\vert\vert^{2}+<\vec{u},\vec{v}>^{2}=\vert\vert\vec{u}\vert\vert^{2}\vert\vert\vec{v}\vert\vert^{2}$
\end_inset


\end_layout

\begin_layout Standard
Vol.
 ppd.: 
\begin_inset Formula $[u,v,w]=<u\times v,w>=\vert\vert u\times v\vert\vert\cdot\vert\vert w\vert\vert\cos\phi$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $\vec{u}\times\vec{u}=0$
\end_inset

, 
\begin_inset Formula $\vec{u}\times\vec{v}=-\left(\vec{v}\times\vec{u}\right)$
\end_inset


\end_layout

\begin_layout Standard
Linearnost 
\begin_inset Formula $\times$
\end_inset

: 
\begin_inset Formula $\left(\alpha\vec{u}+\beta\vec{v}\right)\times\vec{w}=\alpha\left(\vec{u}\times\vec{w}\right)+\beta\left(\vec{v}\times\vec{w}\right)$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $[u,v,w]=-[u,w,v]$
\end_inset

, 
\begin_inset Formula $[u,v,w]=[w,u,v]=[v,w,u]$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $\vec{r}=\vec{r_{0}}+t\vec{p},t\in\mathbb{R}\Longleftrightarrow x=x_{0}+tp_{1},y=y_{0}+tp_{2},z=z_{0}+tp_{3}$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $\Longleftrightarrow t=\frac{x-x_{0}}{p_{1}}=\frac{y-y_{0}}{p_{2}}=\frac{z-z_{0}}{p_{3}}$
\end_inset

 :Normalna enačba 
\begin_inset Formula $\mathbb{R}^{3}$
\end_inset

 premice
\end_layout

\begin_layout Standard
Projekcija 
\begin_inset Formula $\vec{r_{1}}$
\end_inset

 na 
\begin_inset Formula $\vec{r_{0}}+t\vec{p}$
\end_inset

 
\begin_inset Formula $\coloneqq$
\end_inset

 
\begin_inset Formula $\vec{r_{1}'}=\vec{r_{0}}+t'\vec{p}$
\end_inset

 in 
\begin_inset Formula $<\vec{r_{1}'}-\vec{r_{1}},\vec{p}>=0$
\end_inset


\end_layout

\begin_layout Standard
Dvojni 
\begin_inset Formula $\times$
\end_inset

: 
\begin_inset Formula $\vec{a}\times\left(\vec{b}\times\vec{c}\right)=\vec{b}\left(\vec{a}\cdot\vec{c}\right)-\vec{c}\left(\vec{a}\cdot\vec{b}\right)$
\end_inset


\end_layout

\begin_layout Standard
Dvojni 
\begin_inset Formula $\times$
\end_inset

: 
\begin_inset Formula $\left(\vec{a}\times\vec{b}\right)\times\vec{c}=\vec{b}\left(\vec{a}\cdot\vec{c}\right)-\vec{a}\left(\vec{b}\cdot\vec{c}\right)$
\end_inset


\end_layout

\begin_layout Standard
Norm.
 e.
 ravnine: 
\begin_inset Formula $n_{1}x+n_{2}y+n_{3}z=d=n_{1}x_{0}+n_{2}y_{0}+n_{3}z_{0}$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $\Longleftrightarrow<\vec{r}-\vec{r_{0}},\vec{n}>=0$
\end_inset

, saj je 
\begin_inset Formula $\forall\vec{r}\in$
\end_inset

ravnine
\begin_inset Formula $:\left(\vec{r}-\vec{r_{0}}\right)\bot\vec{n}$
\end_inset


\end_layout

\begin_layout Standard
Parametrična e.
 ravnine: 
\begin_inset Formula $\vec{r}=\vec{r_{0}}+s\vec{p}+t\vec{q},s\in\mathbb{R},q\in\mathbb{R}$
\end_inset


\end_layout

\begin_layout Standard
Proj.
 
\begin_inset Formula $\vec{r_{1}}=\vec{r_{0}}+s\vec{p}+t\vec{q}$
\end_inset

 velja 
\begin_inset Formula $<\vec{r_{1}'}-\vec{r_{1},\vec{p}}>=0=<\vec{r_{1}'}-\vec{r_{1},\vec{q}}>$
\end_inset


\end_layout

\begin_layout Standard
sistem: 
\begin_inset Formula $s\vec{p}\cdot\vec{p}+t\vec{q}\cdot\vec{p}=\left(\vec{r_{1}}-\vec{r_{0}}\right)\cdot\vec{p}$
\end_inset

; 
\begin_inset Formula $s\vec{p}\cdot\vec{q}+t\vec{q}\cdot\vec{q}=\left(\vec{r_{1}}-\vec{r_{0}}\right)\cdot\vec{q}$
\end_inset


\end_layout

\begin_layout Standard
Hiperravnino v 
\begin_inset Formula $\mathbb{R}^{n}$
\end_inset

 določa 
\begin_inset Formula $n$
\end_inset

 linearno neodvisnih vektorjev.
\end_layout

\begin_layout Standard
\begin_inset Note Note
status open

\begin_layout Plain Layout
Posplošena rešitev: 
\begin_inset Formula $\min\sum_{k=1}^{m}\left(a_{k,1}x_{1}+\ldots+a_{1,n}x_{n}-b_{k}\right)^{2}$
\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Formula $\Longleftrightarrow\min\vert\vert x_{1}\vec{a_{1}}+\ldots+x_{n}\vec{a_{n}}-\vec{b}\vert\vert^{2}$
\end_inset

 (proj 
\begin_inset Formula $\vec{b}$
\end_inset

 na hiperravnino)
\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 $M/A\coloneqq D-CA^{-1}B$
\end_inset

, 
\begin_inset Formula $M/D\coloneqq A-BD^{-1}C$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
M^{-1}=\left[\begin{array}{cc}
A & B\\
C & D
\end{array}\right]^{-1}=
\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
=\left[\begin{array}{cc}
A^{-1}+A^{-1}B\left(M/A\right)^{-1}CA^{-1} & -A^{-1}B\left(M/A\right)^{-1}\\
-\left(M/A\right)^{-1}CA^{-1} & \left(M/A\right)^{-1}
\end{array}\right]=
\]

\end_inset


\begin_inset Formula 
\[
=\left[\begin{array}{cc}
\left(M/D\right)^{-1} & -\left(M/D\right)^{-1}BD^{-1}\\
-D^{-1}C\left(M/D\right)^{-1} & D^{-1}+D^{-1}C\left(M/D\right)^{-1}BD^{-1}
\end{array}\right]
\]

\end_inset


\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 $\vec{a}\times\vec{b}=\left|\begin{array}{ccc}
\vec{i} & \vec{j} & \vec{k}\\
a_{1} & a_{2} & a_{3}\\
b_{1} & b_{2} & b_{3}
\end{array}\right|$
\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 $\det\left(P_{ij}A\right)=-detA$
\end_inset

, 
\begin_inset Formula $\det\left(E_{i}\alpha A\right)=\alpha\det A$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $\det\left(E_{ij}\alpha A\right)=\det A$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $\det\left(AB\right)=\det A\det B$
\end_inset


\end_layout

\begin_layout Standard
Za možne napake ne odgovarjam.
 Srečno!
\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