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\begin_body

\begin_layout Title
Teorija linearne algebre za ustni izpit —
 IŠRM 2023/24
\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
Povzeto po zapiskih s predavanj prof.
 Cimpriča.
\end_layout

\begin_layout Part
Teorija
\end_layout

\begin_layout Section
Prvi semester
\end_layout

\begin_layout Subsection
Vektorji v 
\begin_inset Formula $\mathbb{R}^{n}$
\end_inset


\end_layout

\begin_layout Standard
Identificaramo 
\begin_inset Formula $n-$
\end_inset

terice realnih števil,
 točke v 
\begin_inset Formula $\mathbb{R}^{n}$
\end_inset

,
 množice paroma enakih geometrijskih vektorjev.
\end_layout

\begin_layout Standard
Osnovne operacije z vektorji:
 Vsota (po komponentah) in množenje s skalarjem (po komponentah),
 kjer je skalar realno število.
\end_layout

\begin_layout Standard
Lastnosti teh računskih operacij:
 asociativnost in komutativnost vsote,
 aditivna enota,
 
\begin_inset Formula $-\vec{a}=\left(-1\right)\cdot\vec{a}$
\end_inset

,
 leva in desna distributivnost,
 homogenost,
 multiplikativna enota.
\end_layout

\begin_layout Subsubsection
Linearna kombinacija vektorjev
\end_layout

\begin_layout Definition*
Linearna kombinacija vektorjev 
\begin_inset Formula $\vec{v_{1}},\dots,\vec{v_{n}}$
\end_inset

 je izraz oblike 
\begin_inset Formula $\alpha_{1}\vec{v_{1}}+\cdots+\alpha_{n}\vec{v_{n}}$
\end_inset

,
 kjer so 
\begin_inset Formula $\alpha_{1},\dots,\alpha_{n}$
\end_inset

 skalarji.
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Definition*
Množico vseh linearnih kombinacij vektorjev 
\begin_inset Formula $\vec{v_{1}},\dots,\vec{v_{n}}$
\end_inset

 označimo z 
\begin_inset Formula $\Lin\left\{ \vec{v_{1}},\dots,\vec{v_{n}}\right\} $
\end_inset

 in ji pravimo linearna ogrinjača (angl.
 span).
 
\begin_inset Formula $\Lin\left\{ \vec{v_{1}},\dots,\vec{v_{n}}\right\} =\left\{ \alpha_{1}\vec{v_{1}}+\cdots+\alpha_{n}\vec{v_{n}};\forall\alpha_{1},\dots,\alpha_{n}\in\mathbb{R}\right\} $
\end_inset


\end_layout

\begin_layout Subsubsection
Linearna neodvisnost vektorjev
\end_layout

\begin_layout Paragraph*
Ideja
\end_layout

\begin_layout Standard
En vektor je linearno neodvisen,
 če ni enak 
\begin_inset Formula $\vec{0}$
\end_inset

.
 Dva,
 če ne ležita na isti premici.
 Trije,
 če ne ležijo na isti ravnini.
\end_layout

\begin_layout Definition
\begin_inset CommandInset label
LatexCommand label
name "def:odvisni"

\end_inset

Vektorji 
\begin_inset Formula $\vec{v_{1}},\dots,\vec{v_{n}}$
\end_inset

 so linearno odvisni,
 če se da enega izmed njih izraziti z linearno kombinacijo preostalih 
\begin_inset Formula $n-1$
\end_inset

 vektorjev.
 Vektorji so linearno neodvisni,
 če niso linearno odvisni (in obratno).
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Definition
\begin_inset CommandInset label
LatexCommand label
name "def:vsi0"

\end_inset

Vektorji 
\begin_inset Formula $v_{1},\dots,v_{n}$
\end_inset

 so linearno neodvisni,
 če za vsake skalarje,
 ki zadoščajo 
\begin_inset Formula $\alpha_{1}v_{1}+\cdots+\alpha_{n}v_{n}=0$
\end_inset

,
 velja 
\begin_inset Formula $\alpha_{1}=\cdots=\alpha_{n}=0$
\end_inset

.
 ZDB poleg 
\begin_inset Formula $\alpha_{1}=\cdots=\alpha_{n}=0$
\end_inset

 ne obstajajo nobeni drugi 
\begin_inset Formula $\alpha_{1},\dots,\alpha_{n}$
\end_inset

,
 kjer bi veljalo 
\begin_inset Formula $\alpha_{1}v_{1}+\cdots+\alpha_{n}v_{n}=0$
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Definition
\begin_inset CommandInset label
LatexCommand label
name "def:kvečjemu1"

\end_inset


\begin_inset Formula $v_{1},\dots,v_{n}$
\end_inset

 so linearno neodvisni,
 če se da vsak vektor na kvečjemu en način izraziti kot linearno kombinacijo 
\begin_inset Formula $v_{1},\dots,v_{n}$
\end_inset

.
\end_layout

\begin_layout Theorem*
Te tri definicije so ekvivalentne.
\end_layout

\begin_layout Proof
Dokazujemo ekvivalenco:
\end_layout

\begin_deeper
\begin_layout Labeling
\labelwidthstring 00.00.0000
\begin_inset Formula $\left(\ref{def:odvisni}\Rightarrow\ref{def:vsi0}\right)$
\end_inset

 Recimo,
 da so 
\begin_inset Formula $v_{1},\dots,v_{n}$
\end_inset

 linearno odvisni v smislu 
\begin_inset CommandInset ref
LatexCommand ref
reference "def:odvisni"
plural "false"
caps "false"
noprefix "false"
nolink "false"

\end_inset

.
 Dokažimo,
 da so tedaj linearno odvisni tudi v smislu 
\begin_inset Formula $\ref{def:vsi0}$
\end_inset

.
 Obstaja tak 
\begin_inset Formula $i$
\end_inset

,
 da lahko 
\begin_inset Formula $v_{i}$
\end_inset

 izrazimo z linearno kombinacijo preostalih,
 torej 
\begin_inset Formula $v_{i}=\alpha_{1}v_{1}+\cdots+\alpha_{i-1}v_{i-1}+\alpha_{i+1}v_{i+1}+\cdots+\alpha_{n}v_{n}$
\end_inset

 za neke 
\begin_inset Formula $\alpha$
\end_inset

.
 Sledi 
\begin_inset Formula $0=\alpha_{1}v_{1}+\cdots+\alpha_{i-1}v_{i-1}+\left(-1\right)v_{i}+\alpha_{i+1}v_{i+1}+\cdots+\alpha_{n}v_{n}$
\end_inset

,
 kar pomeni,
 da obstaja linearna kombinacija,
 ki je enaka 0,
 toda niso vsi koeficienti 0 (že koeficient pred 
\begin_inset Formula $v_{i}$
\end_inset

 je 
\begin_inset Formula $-1$
\end_inset

),
 tedaj so vektorji po definiciji 
\begin_inset CommandInset ref
LatexCommand ref
reference "def:vsi0"
plural "false"
caps "false"
noprefix "false"
nolink "false"

\end_inset

 linearno odvisni.
\end_layout

\begin_layout Labeling
\labelwidthstring 00.00.0000
\begin_inset Formula $\left(\ref{def:vsi0}\Rightarrow\ref{def:odvisni}\right)$
\end_inset

 Recimo,
 da so 
\begin_inset Formula $v_{1},\dots,v_{n}$
\end_inset

 linearno odvisno v smislu 
\begin_inset Formula $\ref{def:vsi0}$
\end_inset

.
 Tedaj obstajajo 
\begin_inset Formula $\alpha$
\end_inset

,
 ki niso vse 0,
 da velja 
\begin_inset Formula $\alpha_{1}v_{1}+\cdots+\alpha_{n}v_{n}=0$
\end_inset

.
 Tedaj 
\begin_inset Formula $\exists i\ni:\alpha_{i}\not=0$
\end_inset

 in velja
\begin_inset Formula 
\[
\alpha_{i}v_{i}=-\alpha_{1}v_{1}-\cdots-\alpha_{i-1}v_{i-1}-\alpha_{i+1}v_{i+1}-\cdots-\alpha_{n}v_{n}\quad\quad\quad\quad/:\alpha_{i}
\]

\end_inset


\begin_inset Formula 
\[
v_{i}=-\frac{\alpha_{1}}{\alpha_{i}}v_{i}-\cdots-\frac{\alpha_{i-1}}{\alpha_{i}}v_{i-1}-\frac{\alpha_{i+1}}{\alpha_{i}}v_{i+1}-\cdots-\frac{\alpha_{n}}{\alpha_{i}}v_{n}\text{,}
\]

\end_inset

s čimer smo 
\begin_inset Formula $v_{i}$
\end_inset

 izrazili kot linearno kombinacijo preostalih vektorjev.
\end_layout

\begin_layout Labeling
\labelwidthstring 00.00.0000
\begin_inset Formula $\left(\ref{def:vsi0}\Leftrightarrow\ref{def:kvečjemu1}\right)$
\end_inset

 Naj bodo 
\begin_inset Formula $v_{1},\dots,v_{n}$
\end_inset

 LN.
 Recimo,
 da obstaja 
\begin_inset Formula $v$
\end_inset

,
 ki se ga da na dva načina izraziti kot linearno kombinacijo 
\begin_inset Formula $v_{1},\dots,v_{n}$
\end_inset

.
 Naj bo 
\begin_inset Formula $v=\alpha_{1}v_{1}+\cdots+\alpha_{n}v_{n}=\beta_{1}v_{1}+\cdots+\beta_{n}v_{n}$
\end_inset

.
 Sledi 
\begin_inset Formula $0=\left(\alpha_{1}-\beta_{1}\right)v_{1}+\cdots+\left(\alpha_{n}-\beta_{n}\right)v_{n}$
\end_inset

.
 Po definiciji 
\begin_inset CommandInset ref
LatexCommand ref
reference "def:vsi0"
plural "false"
caps "false"
noprefix "false"
nolink "false"

\end_inset

 velja 
\begin_inset Formula $\forall i:\alpha_{i}-\beta_{i}=0\Leftrightarrow\alpha_{i}=\beta_{i}$
\end_inset

,
 torej sta načina,
 s katerima izrazimo 
\begin_inset Formula $v$
\end_inset

,
 enaka,
 torej lahko 
\begin_inset Formula $v$
\end_inset

 izrazimo na kvečjemu en način z 
\begin_inset Formula $v_{1},\dots,v_{n}$
\end_inset

,
 kar ustreza definiciji 
\begin_inset CommandInset ref
LatexCommand ref
reference "def:kvečjemu1"
plural "false"
caps "false"
noprefix "false"
nolink "false"

\end_inset

.
\end_layout

\end_deeper
\begin_layout Subsubsection
Ogrodje in baza
\end_layout

\begin_layout Definition*
Vektorji 
\begin_inset Formula $v_{1},\dots,v_{n}$
\end_inset

 so ogrodje (angl.
 span),
 če 
\begin_inset Formula $\Lin\left\{ v_{1},\dots,v_{n}\right\} =\mathbb{R}^{n}\Leftrightarrow\forall v\in\mathbb{R}^{n}\exists\alpha_{1},\dots,\alpha_{n}\in\mathbb{R}\ni:v=\alpha_{1}v_{1}+\cdots+\alpha_{n}v_{n}$
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Definition*
Vektorji 
\begin_inset Formula $v_{1},\dots,v_{n}$
\end_inset

 so baza,
 če so LN in ogrodje 
\begin_inset Formula $\Leftrightarrow\forall v\in\mathbb{R}^{n}:\exists!\alpha_{1},\dots,\alpha_{n}\in\mathbb{R}\ni:v=\alpha_{1}v_{1}+\cdots+\alpha_{n}v_{n}$
\end_inset

 ZDB vsak vektor 
\begin_inset Formula $\in\mathbb{R}^{n}$
\end_inset

 se da na natanko en način izraziti kot LK 
\begin_inset Formula $v_{1},\dots,v_{n}$
\end_inset

.
\end_layout

\begin_layout Example*
Primer baze je standardna baza 
\begin_inset Formula $\mathbb{R}^{n}$
\end_inset

:
 
\begin_inset Formula $\left\{ \left(1,0,0,\dots,0\right),\left(0,1,0,\dots,0\right),\left(0,0,1,\dots,0\right),\left(0,0,0,\dots,1\right)\right\} $
\end_inset

.
 To pa ni edina baza.
 Primer nestandardne baze v 
\begin_inset Formula $\mathbb{R}^{3}$
\end_inset

 je 
\begin_inset Formula $\left\{ \left(1,1,1\right),\left(0,1,1\right),\left(0,0,1\right)\right\} $
\end_inset

.
\end_layout

\begin_layout Subsubsection
Norma in skalarni produkt
\end_layout

\begin_layout Definition*
Norma vektorja 
\begin_inset Formula $v=\left(\alpha_{1},\dots,\alpha_{n}\right)$
\end_inset

 je definirana z 
\begin_inset Formula $\left|\left|v\right|\right|=\sqrt{\alpha_{1}^{2}+\cdots+\alpha_{n}^{2}}$
\end_inset

.
 Geometrijski pomen norme je dolžina krajevnega vektorja z glavo v 
\begin_inset Formula $v$
\end_inset

.
\end_layout

\begin_layout Standard
Osnovne lastnosti norme:
 
\begin_inset Formula $\left|\left|v\right|\right|\geq0$
\end_inset

,
 
\begin_inset Formula $\left|\left|v\right|\right|=0\Rightarrow v=\vec{0}$
\end_inset

,
 
\begin_inset Formula $\left|\left|\alpha v\right|\right|=\left|\alpha\right|\cdot\left|\left|v\right|\right|$
\end_inset

,
 
\begin_inset Formula $\left|\left|u+v\right|\right|\leq\left|\left|u\right|\right|+\left|\left|v\right|\right|$
\end_inset

 (trikotniška neenakost)
\end_layout

\begin_layout Definition*
Skalarni produkt 
\begin_inset Formula $u=\left(\alpha_{1},\dots,\alpha_{n}\right),v=\left(\beta_{1},\dots,\beta_{n}\right)$
\end_inset

 označimo z 
\begin_inset Formula $\left\langle u,v\right\rangle \coloneqq\alpha_{1}\beta_{1}+\cdots+\alpha_{n}\beta_{n}$
\end_inset

.
 Obstaja tudi druga oznaka in pripadajoča drugačna definicija 
\begin_inset Formula $u\cdot v\coloneqq\left|\left|u\right|\right|\cdot\left|\left|v\right|\right|\cdot\cos\varphi$
\end_inset

,
 kjer je 
\begin_inset Formula $\varphi$
\end_inset

 kot med 
\begin_inset Formula $u,v$
\end_inset

.
\end_layout

\begin_layout Claim*
Velja 
\begin_inset Formula $\left\langle u,v\right\rangle =u\cdot v$
\end_inset

.
\end_layout

\begin_layout Proof
Uporabimo kosinusni izrek,
 ki pravi,
 da v trikotniku s stranicami dolžin 
\begin_inset Formula $a,b,c$
\end_inset

 velja 
\begin_inset Formula $c^{2}=a^{2}+b^{2}-2ab\cos\varphi$
\end_inset

,
 kjer je 
\begin_inset Formula $\varphi$
\end_inset

 kot med 
\begin_inset Formula $b$
\end_inset

 in 
\begin_inset Formula $c$
\end_inset

.
 Za vektorja 
\begin_inset Formula $v$
\end_inset

 in 
\begin_inset Formula $u$
\end_inset

 z vmesnim kotom 
\begin_inset Formula $\varphi$
\end_inset

 torej velja 
\begin_inset Formula 
\[
\left|\left|u-v\right|\right|^{2}=\left|\left|u\right|\right|^{2}+\left|\left|v\right|\right|^{2}-2\left|\left|u\right|\right|\cdot\left|\left|v\right|\right|\cdot\cos\varphi.
\]

\end_inset

Obenem velja 
\begin_inset Formula $\left|\left|u\right|\right|^{2}=\alpha_{1}^{2}+\cdots+\alpha_{n}^{2}=\left\langle u,u\right\rangle $
\end_inset

,
 torej lahko zgornjo enačbo prepišemo v
\begin_inset Formula 
\[
\left\langle u-v,u-v\right\rangle =\left\langle u,u\right\rangle +\left\langle v,v\right\rangle -2\left|\left|u\right|\right|\cdot\left|\left|v\right|\right|\cdot\cos\varphi.
\]

\end_inset

Naj bo 
\begin_inset Formula $w=u,v$
\end_inset

.
 Iz prihodnosti si izposodimo obe linearnosti in simetričnost.
 
\begin_inset Formula 
\[
\left\langle u-v,u-v\right\rangle =\left\langle u-v,w\right\rangle =\left\langle u,w\right\rangle -\left\langle v,w\right\rangle =\left\langle u,u-v\right\rangle -\left\langle v,u-v\right\rangle =\left\langle u,u\right\rangle -\left\langle u,v\right\rangle -\left\langle v,u\right\rangle +\left\langle v,v\right\rangle 
\]

\end_inset

 Prišli smo do enačbe
\begin_inset Formula 
\[
\cancel{\left\langle u,u\right\rangle }-2\left\langle u,v\right\rangle +\cancel{\left\langle v,v\right\rangle }=\cancel{\left\langle u,u\right\rangle }+\cancel{\left\langle v,v\right\rangle }-2\left|\left|u\right|\right|\cdot\left|\left|v\right|\right|\cdot\cos\varphi\quad\quad\quad\quad/:-2
\]

\end_inset


\begin_inset Formula 
\[
\left\langle u,v\right\rangle =\left|\left|u\right|\right|\cdot\left|\left|v\right|\right|\cdot\cos\varphi.
\]

\end_inset


\end_layout

\begin_layout Claim*
Paralelogramska identiteta.
 
\begin_inset Formula $\left|\left|u+v\right|\right|^{2}+\left|\left|u-v\right|\right|^{2}=2\left|\left|u\right|\right|^{2}+2\left|\left|v\right|\right|^{2}$
\end_inset

 ZDB vsota kvadratov dolžin obeh diagonal je enota vsoti kvadratov dolžin vseh štirih stranic.
\end_layout

\begin_layout Proof
\begin_inset Formula 
\[
\left|\left|u+v\right|\right|^{2}=\left\langle u+v,u+v\right\rangle =\left\langle u,u+v\right\rangle +\left\langle v,u+v\right\rangle =\left\langle u,u\right\rangle +\left\langle u,v\right\rangle +\left\langle v,u\right\rangle +\left\langle v,v\right\rangle 
\]

\end_inset

 
\begin_inset Formula 
\[
\left|\left|u-v\right|\right|^{2}=\left\langle u-v,u-v\right\rangle =\left\langle u,u-v\right\rangle -\left\langle v,u-v\right\rangle =\left\langle u,u\right\rangle -\left\langle u,v\right\rangle -\left\langle v,u\right\rangle +\left\langle v,v\right\rangle 
\]

\end_inset


\begin_inset Formula 
\[
\left|\left|u+v\right|\right|^{2}+\left|\left|u-v\right|\right|^{2}=2\left\langle u,u\right\rangle +2\left\langle v,v\right\rangle =2\left|\left|u\right|\right|^{2}+2\left|\left|v\right|\right|^{2}
\]

\end_inset


\end_layout

\begin_layout Claim*
Cauchy-Schwarzova neenakost.
 
\begin_inset Formula $\left|\left\langle u,v\right\rangle \right|\leq\left|\left|u\right|\right|\cdot\left|\left|v\right|\right|$
\end_inset


\end_layout

\begin_layout Proof
\begin_inset Formula $\left|\left\langle u,v\right\rangle \right|=\left|\left|\left|u\right|\right|\cdot\left|\left|v\right|\right|\cdot\cos\varphi\right|=\left|\left|u\right|\right|\cdot\left|\left|v\right|\right|\cdot\left|\cos\varphi\right|\leq\left|\left|u\right|\right|\cdot\left|\left|v\right|\right|$
\end_inset

,
 kajti 
\begin_inset Formula $\left|\cos\varphi\right|\in\left[0,1\right]$
\end_inset

.
\end_layout

\begin_layout Claim*
Trikotniška neenakost.
 
\begin_inset Formula $\left|\left|u+v\right|\right|\leq\left|\left|u\right|\right|+\left|\left|v\right|\right|$
\end_inset


\end_layout

\begin_layout Proof
Sledi iz Cauchy-Schwarzove.
 Velja 
\begin_inset Formula 
\[
-\left|\left|u\right|\right|\cdot\left|\left|v\right|\right|\leq\left|\left\langle u,v\right\rangle \right|\leq\left|\left|u\right|\right|\cdot\left|\left|v\right|\right|\quad\quad\quad\quad/\cdot2
\]

\end_inset


\begin_inset Formula 
\[
-2\left|\left|u\right|\right|\cdot\left|\left|v\right|\right|\leq2\left|\left\langle u,v\right\rangle \right|\leq2\left|\left|u\right|\right|\cdot\left|\left|v\right|\right|\quad\quad\quad\quad/+\left|\left|u\right|\right|^{2}+\left|\left|v\right|\right|^{2}
\]

\end_inset


\begin_inset Formula 
\[
-2\left|\left|u\right|\right|\cdot\left|\left|v\right|\right|+\left|\left|u\right|\right|^{2}+\left|\left|v\right|\right|^{2}\leq\cancel{2\left|\left\langle u,v\right\rangle \right|+\left|\left|u\right|\right|^{2}+\left|\left|v\right|\right|^{2}\leq}2\left|\left|u\right|\right|\cdot\left|\left|v\right|\right|+\left|\left|u\right|\right|^{2}+\left|\left|v\right|\right|^{2}
\]

\end_inset

uporabimo kosinusni izrek na levi strani enačbe,
 desno pa zložimo v kvadrat:
\begin_inset Formula 
\[
\left|\left|u+v\right|\right|^{2}\leq\left(\left|\left|u\right|\right|+\left|\left|v\right|\right|\right)^{2}\quad\quad\quad\quad/\sqrt{}
\]

\end_inset


\begin_inset Formula 
\[
\left|\left|u+v\right|\right|\leq\left|\left|u\right|\right|+\left|\left|v\right|\right|
\]

\end_inset


\end_layout

\begin_layout Claim*
Za neničelna vektorja velja 
\begin_inset Formula $u\perp v\Leftrightarrow\left\langle u,v\right\rangle =0$
\end_inset

.
\end_layout

\begin_layout Proof
\begin_inset Formula $\left\langle u,v\right\rangle =u\cdot v=\left|\left|u\right|\right|\cdot\left|\left|v\right|\right|\cdot\cos\varphi$
\end_inset

,
 kar je 0 
\begin_inset Formula $\Leftrightarrow\varphi=\pi=90°$
\end_inset

.
\end_layout

\begin_layout Subsubsection
Vektorski in mešani produkt
\end_layout

\begin_layout Standard
Definirana sta le za vektorje v 
\begin_inset Formula $\mathbb{R}^{3}$
\end_inset

.
\end_layout

\begin_layout Definition*
Naj bo 
\begin_inset Formula $u=\left(\alpha_{1},\alpha_{2},\alpha_{3}\right),v=\left(\beta_{1},\beta_{2},\beta_{3}\right)$
\end_inset

.
 
\begin_inset Formula $u\times v=\left(\alpha_{2}\beta_{3}-\alpha_{3}\beta_{2},\alpha_{3}\beta_{1}-\alpha_{1}\beta_{3},\alpha_{1}\beta_{2}-\alpha_{2}\beta_{1}\right)$
\end_inset

.
\end_layout

\begin_layout Paragraph
Geometrijski pomen
\end_layout

\begin_layout Standard
Vektor 
\begin_inset Formula $u\times v$
\end_inset

 je pravokoten na 
\begin_inset Formula $u$
\end_inset

 in 
\begin_inset Formula $v$
\end_inset

,
 njegova dolžina je 
\begin_inset Formula $\left|\left|u\right|\right|\cdot\left|\left|v\right|\right|\cdot\sin\varphi$
\end_inset

,
 kar je ploščina paralelograma,
 ki ga oklepata 
\begin_inset Formula $u$
\end_inset

 in 
\begin_inset Formula $v$
\end_inset

.
\end_layout

\begin_layout Standard
Pravilo desnega vijaka nam je v pomoč pri doložanju usmeritve vektorskega produkta.
 Če iztegnjen kazalec desne roke predstavlja 
\begin_inset Formula $u$
\end_inset

 in iztegnjen sredinec 
\begin_inset Formula $v$
\end_inset

,
 iztegnjen palec kaže v smeri 
\begin_inset Formula $u\times v$
\end_inset

.
\end_layout

\begin_layout Claim*
Lagrangeva identiteta.
 
\begin_inset Formula $\left|\left|u\times v\right|\right|+\left\langle u,v\right\rangle ^{2}=\left|\left|u\right|\right|^{2}\cdot\left|\left|v\right|\right|^{2}$
\end_inset


\begin_inset Note Note
status open

\begin_layout Plain Layout
DOKAZ???????
\end_layout

\end_inset


\end_layout

\begin_layout Definition*
Mešani produkt vektorjev 
\begin_inset Formula $u,v,w$
\end_inset

 je skalar 
\begin_inset Formula $\left\langle u\times v,w\right\rangle $
\end_inset

.
 Oznaka:
 
\begin_inset Formula $\left[u,v,w\right]=\left\langle u\times v,w\right\rangle $
\end_inset

.
\end_layout

\begin_layout Paragraph*
Geometrijski pomen
\end_layout

\begin_layout Standard
Volumen paralelpipeda,
 ki ga določajo 
\begin_inset Formula $u,v,w$
\end_inset

.
 Razlaga:
 
\begin_inset Formula $\left[u,v,w\right]=\left\langle u\times v,w\right\rangle =\left|\left|u\times v\right|\right|\cdot\left|\left|w\right|\right|\cdot\cos\varphi$
\end_inset

;
 
\begin_inset Formula $\left|\left|u\times v\right|\right|$
\end_inset

 je namreč ploščina osnovne ploskve,
 
\begin_inset Formula $\left|\left|w\right|\right|\cdot\cos\varphi$
\end_inset

 pa je višina paralelpipeda.
\end_layout

\begin_layout Claim*
Osnovne lastnosti vektorskega produkta so 
\begin_inset Formula $u\times u=0$
\end_inset

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

,
 
\begin_inset Formula $\left(\alpha u+\beta v\right)\times w=\alpha\left(u\times w\right)+\beta\left(v\times w\right)$
\end_inset

 (linearnost)
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Claim*
Osnovne lastnosti mešanega produkta so linearnost v vsakem faktorju,
 menjava dveh faktorjev spremeni predznak (
\begin_inset Formula $\left[u,v,w\right]=-\left[v,u,w\right]$
\end_inset

),
 cikličen pomik ne spremeni vrednosti (
\begin_inset Formula $\left[u,v,w\right]=\left[v,w,u\right]=\left[w,u,v\right]$
\end_inset

).
\end_layout

\begin_layout Subsubsection
Premica v 
\begin_inset Formula $\mathbb{R}^{n}$
\end_inset


\end_layout

\begin_layout Standard
Premico lahko podamo z
\end_layout

\begin_layout Itemize
dvema različnima točkama
\end_layout

\begin_layout Itemize
s točko 
\begin_inset Formula $\vec{r_{0}}$
\end_inset

 in neničelnim smernim vektorjem 
\begin_inset Formula $\vec{p}$
\end_inset

.
 Premica je tako množica točk 
\begin_inset Formula $\left\{ \vec{r}=\vec{r_{0}}+t\vec{p};\forall t\in\mathbb{R}\right\} $
\end_inset

.
 Taki enačbi premice rečemo parametrična.
\end_layout

\begin_layout Itemize
s točko in normalo (v 
\begin_inset Formula $\mathbb{R}^{2}$
\end_inset

;
 v 
\begin_inset Formula $\mathbb{R}^{n}$
\end_inset

 potrebujemo točko in 
\begin_inset Formula $n-1$
\end_inset

 normal)
\end_layout

\begin_layout Standard
Nadaljujmo s parametričnim zapisom 
\begin_inset Formula $\vec{r}=\vec{r_{0}}+t\vec{p}$
\end_inset

.
 Če točke zapišemo po komponentah,
 dobimo parametrično enačbo premice po komponentah:
 
\begin_inset Formula $\left(x,y,z\right)=\left(x_{0},y_{0},z_{0}\right)+t\left(p_{1},p_{2},p_{3}\right)$
\end_inset

.
\begin_inset Formula 
\[
x=x_{0}+tp_{1}
\]

\end_inset


\begin_inset Formula 
\[
y=y_{0}+tp_{2}
\]

\end_inset


\begin_inset Formula 
\[
z=z_{0}+tp_{3}
\]

\end_inset


\end_layout

\begin_layout Standard
Sedaj lahko iz vsake enačbe izrazimo 
\begin_inset Formula $t$
\end_inset

 in dobimo normalno enačbo premice v 
\begin_inset Formula $\mathbb{R}^{n}$
\end_inset

:
\begin_inset Formula 
\[
t=\frac{x-x_{0}}{p_{1}}=\frac{y-y_{0}}{p_{2}}=\frac{z-z_{0}}{p_{3}}\text{, oziroma v splošnem za premico v \ensuremath{\mathbb{R}^{n}}: }t=\frac{x_{1_{0}}-x_{1}}{p_{1}}=\cdots=\frac{x_{n_{0}}-x_{n}}{p_{n}}
\]

\end_inset


\end_layout

\begin_layout Standard
Osnovne naloge s premicami so projekcija točke na premico,
 zrcaljenje točke čez premico in razdalja med točko in premico.
\end_layout

\begin_layout Paragraph*
Iskanje projekcije dane točke na dano premico
\end_layout

\begin_layout Standard
(skica prepuščena bralcu) 
\begin_inset Formula $\vec{r_{1}}$
\end_inset

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

 in dobimo 
\begin_inset Formula $\vec{r_{1}'}$
\end_inset

.
 Za 
\begin_inset Formula $\vec{r_{1}'}$
\end_inset

 vemo,
 da leži na premici,
 torej 
\begin_inset Formula $\exists t\in\mathbb{R}\ni:\vec{r_{1}'}=\vec{r_{0}}+t\vec{p}$
\end_inset

.
 Poleg tega vemo,
 da je 
\begin_inset Formula $\vec{r_{1}'}-\vec{r_{1}}$
\end_inset

 pravokoten na premico oz.
 njen smerni vektor 
\begin_inset Formula $\vec{p}$
\end_inset

,
 torej 
\begin_inset Formula $\left\langle \vec{r_{1}'}-\vec{r_{1}},\vec{p}\right\rangle =0$
\end_inset

.
 Ti dve enačbi združimo,
 da dobimo 
\begin_inset Formula $t$
\end_inset

,
 ki ga nato vstavimo v prvo enačbo:
\begin_inset Formula 
\[
\left\langle \vec{r_{0}}+t\vec{p}-\vec{r_{1},}\vec{p}\right\rangle =0\Longrightarrow\left\langle \vec{r_{0}},\vec{p}\right\rangle +t\left\langle \vec{p},\vec{p}\right\rangle -\left\langle \vec{r_{1}},\vec{p}\right\rangle =0\Longrightarrow t=\frac{\left\langle \vec{r_{1}},\vec{p}\right\rangle -\left\langle \vec{r_{0}},\vec{p}\right\rangle }{\left\langle \vec{p},\vec{p}\right\rangle }
\]

\end_inset


\begin_inset Formula 
\[
\vec{r_{1}'}=\vec{r_{0}}+t\vec{p}=\vec{r_{0}}+\frac{\left\langle \vec{r_{1}},\vec{p}\right\rangle -\left\langle \vec{r_{0}},\vec{p}\right\rangle }{\left\langle \vec{p},\vec{p}\right\rangle }\vec{p}
\]

\end_inset


\end_layout

\begin_layout Standard
Spotoma si lahko izpišemo obrazec za oddaljenost točke od premice:
 
\begin_inset Formula $a=\left|\left|\vec{r_{1}'}-\vec{r_{1}}\right|\right|$
\end_inset

 in obrazec za zrcalno sliko (
\begin_inset Formula $\vec{r_{1}''}$
\end_inset

):
 
\begin_inset Formula $\vec{r_{1}'}=\frac{\vec{r_{1}''}+\vec{r_{1}}}{2}\Longrightarrow\vec{r_{1}''}=2\vec{r_{1}'}-\vec{r_{1}}$
\end_inset

.
\end_layout

\begin_layout Subsubsection
Ravnine v 
\begin_inset Formula $\mathbb{R}^{n}$
\end_inset


\end_layout

\begin_layout Standard
Ravnino lahko podamo
\end_layout

\begin_layout Itemize
s tremi nekolinearnimi točkami
\end_layout

\begin_layout Itemize
s točko na ravnini in dvema neničelnima smernima vektorjema,
 ki sta linarno neodvisna.
 Ravnina je tako množica točk 
\begin_inset Formula $\left\{ \vec{r}=\vec{r_{0}}+s\vec{p}+t\vec{q};\forall s,t\in\mathbb{R}\right\} $
\end_inset

.
 Taki enačbi ravnine rečemo parametrična.
\end_layout

\begin_layout Itemize
s točko in na ravnini in normalo (v 
\begin_inset Formula $\mathbb{R}^{3}$
\end_inset

;
 v 
\begin_inset Formula $\mathbb{R}^{n}$
\end_inset

 poleg točke potrebujemo 
\begin_inset Formula $n-2$
\end_inset

 normal)
\end_layout

\begin_layout Standard
Nadaljujmo s parametričnim zapisom 
\begin_inset Formula $\vec{r}=\vec{r_{0}}+s\vec{p}+t\vec{q}$
\end_inset

.
 Če točke zapišemo po komponentah,
 dobimo parametrično enačbo ravnine po komponentah:
 
\begin_inset Formula $\left(x,y,z\right)=\left(x_{0},y_{0},z_{0}\right)+s\left(p_{1},p_{2},p_{3}\right)+t\left(q_{1},q_{2},q_{3}\right)$
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
x=x_{0}+sp_{1}+tq_{1}
\]

\end_inset


\begin_inset Formula 
\[
y=y_{0}+sp_{2}+tq_{2}
\]

\end_inset


\begin_inset Formula 
\[
z=y_{0}+sp_{3}+tq_{3}
\]

\end_inset


\end_layout

\begin_layout Paragraph
Normalna enačba ravnine v 
\begin_inset Formula $\mathbb{R}^{3}$
\end_inset


\end_layout

\begin_layout Standard
(skica prepuščena bralcu) Vemo,
 da je 
\begin_inset Formula $\vec{n}$
\end_inset

 (normala) pravokotna na vse vektorje v ravnini,
 tudi na 
\begin_inset Formula $\vec{r}-\vec{r_{0}}$
\end_inset

 za poljuben 
\begin_inset Formula $\vec{r}$
\end_inset

 na ravnini.
 Velja torej normalna enačba ravnine:
 
\begin_inset Formula $\left\langle \vec{r}-\vec{r_{0}},\vec{n}\right\rangle =0$
\end_inset

.
 Razpišimo jo po komponentah,
 da na koncu dobimo normalno enačbo ravnine po komponentah:
\begin_inset Formula 
\[
\left\langle \left(x,y,z\right)-\left(x_{0},y_{0},z_{0}\right),\left(n_{1},n_{2},n_{3}\right)\right\rangle =0=\left\langle \left(x-x_{0},y-y_{0},z-z_{0}\right),\left(n_{1},n_{2},n_{3}\right)\right\rangle 
\]

\end_inset


\begin_inset Formula 
\[
n_{1}\left(x-x_{0}\right)+n_{2}\left(y-y_{0}\right)+n_{3}\left(z-z_{0}\right)=0=n_{1}x-n_{1}x_{0}+n_{2}y-n_{2}y_{0}+n_{3}z-n_{3}z_{0}=0
\]

\end_inset


\begin_inset Formula 
\[
n_{1}x+n_{2}y+n_{3}z=n_{1}x_{0}+n_{2}y_{0}+n_{3}z_{0}=d
\]

\end_inset


\end_layout

\begin_layout Paragraph
Iskanje pravokotne projekcije dane točke na dano ravnino
\end_layout

\begin_layout Standard
(skica prepuščena bralcu) Projicirati želimo 
\begin_inset Formula $\vec{r_{1}}$
\end_inset

 v 
\begin_inset Formula $\vec{r_{1}'}$
\end_inset

 na ravnini 
\begin_inset Formula $\vec{r}=\vec{r_{0}}+s\vec{p}+t\vec{q}$
\end_inset

.
 Vemo,
 da 
\begin_inset Formula $\vec{r_{1}'}$
\end_inset

 leži na ravnini,
 zato 
\begin_inset Formula $\exists s,t\in\mathbb{R}\ni:\vec{r_{1}'}=\vec{r_{0}}+s\vec{p}+t\vec{q}$
\end_inset

.
 Poleg tega vemo,
 da je 
\begin_inset Formula $\vec{r_{1}'}-\vec{r_{1}}$
\end_inset

 pravokoten na ravnino oz.
 na 
\begin_inset Formula $\vec{p}$
\end_inset

 in na 
\begin_inset Formula $\vec{q}$
\end_inset

 hkrati,
 torej 
\begin_inset Formula $\left\langle \vec{r_{1}'}-\vec{r_{1}},\vec{p}\right\rangle =0=\left\langle \vec{r_{1}'}-\vec{r_{1}},\vec{q}\right\rangle $
\end_inset

.
 Vstavimo 
\begin_inset Formula $\vec{r_{1}'}$
\end_inset

 iz prve enačbe v drugo in dobimo
\begin_inset Formula 
\[
\left\langle \vec{r_{0}}+s\vec{p}+t\vec{q}-\vec{r_{1}},\vec{p}\right\rangle =0=\left\langle \vec{r_{0}}+s\vec{p}+t\vec{q}-\vec{r_{1}},\vec{q}\right\rangle 
\]

\end_inset


\begin_inset Formula 
\[
\left\langle \vec{r_{0}},\vec{p}\right\rangle +s\left\langle \vec{p},\vec{p}\right\rangle +t\left\langle \vec{q},\vec{p}\right\rangle -\left\langle \vec{r_{1}},\vec{p}\right\rangle =0=\left\langle \vec{r_{0}},\vec{q}\right\rangle +s\left\langle \vec{p},\vec{q}\right\rangle +t\left\langle \vec{q},\vec{q}\right\rangle -\left\langle \vec{r_{1}},\vec{q}\right\rangle 
\]

\end_inset

dobimo sistem dveh enačb
\begin_inset Formula 
\[
s\left\langle \vec{p},\vec{p}\right\rangle +t\left\langle \vec{q},\vec{p}\right\rangle =\left\langle \vec{r_{1}}-\vec{r_{0}},\vec{p}\right\rangle 
\]

\end_inset


\begin_inset Formula 
\[
s\left\langle \vec{p},\vec{q}\right\rangle +t\left\langle \vec{q},\vec{q}\right\rangle =\left\langle \vec{r_{1}}-\vec{r_{0}},\vec{q}\right\rangle 
\]

\end_inset

sistem rešimo in dobljena 
\begin_inset Formula $s,t$
\end_inset

 vstavimo v prvo enačbo zgoraj,
 da dobimo 
\begin_inset Formula $\vec{r_{1}'}$
\end_inset

.
\end_layout

\begin_layout Subsubsection
Regresijska premica
\end_layout

\begin_layout Standard
Regresijska premica je primer uporabe zgornje naloge.
 V ravnini je danih 
\begin_inset Formula $n$
\end_inset

 točk 
\begin_inset Formula $\left(x_{1},y_{1}\right),\dots,\left(x_{n},y_{n}\right)$
\end_inset

.
 Iščemo tako premico 
\begin_inset Formula $y=ax+b$
\end_inset

,
 ki se najbolj prilega tem točkam.
 Prileganje premice točkam merimo z metodo najmanjših kvadratov:
 naj bo 
\begin_inset Formula $d_{i}$
\end_inset

 navpična razdalja med 
\begin_inset Formula $\left(x_{i},y_{i}\right)$
\end_inset

 in premico 
\begin_inset Formula $y=ax+b$
\end_inset

,
 torej razdalja med točkama 
\begin_inset Formula $\left(x_{i},y_{i}\right)$
\end_inset

 in 
\begin_inset Formula $\left(x_{i},ax_{i}+b\right)$
\end_inset

,
 kar je 
\begin_inset Formula $\left|y_{i}-ax_{i}-b\right|$
\end_inset

.
 Minimizirati želimo vsoto kvadratov navpičnih razdalj,
 torej izraz 
\begin_inset Formula $d_{1}^{2}+\cdots+d_{n}^{2}=\left(y_{1}-ax_{1}-b\right)^{2}+\cdots+\left(y_{n}-ax_{n}-b\right)^{2}=\left|\left|\left(y_{1}-ax_{1}-b,\dots,y_{n}-ax_{n}-b\right)\right|\right|^{2}=\left|\left|\left(y_{1},\dots,y_{n}\right)-a\left(x_{1},\dots,x_{n}\right)-b\left(1,\dots,1\right)\right|\right|^{2}$
\end_inset

.
\end_layout

\begin_layout Standard
Če je torej 
\begin_inset Formula $\vec{r}=\vec{0}+a\left(x_{1},\dots,x_{n}\right)+b\left(1,\dots,1\right)$
\end_inset

 hiperravnina v 
\begin_inset Formula $n-$
\end_inset

dimenzionalnem prostoru,
 bo norma,
 ki jo želimo minimizirati,
 najmanjša tedaj,
 ko 
\begin_inset Formula $a,b$
\end_inset

 izberemo tako,
 da najdemo projekcijo 
\begin_inset Formula $\left(y_{1},\dots,y_{n}\right)$
\end_inset

 na to hiperravnino (skica prepuščena bralcu).
 Rešimo sedaj nalogo projekcije točke na ravnino:
\end_layout

\begin_layout Standard
Označimo 
\begin_inset Formula $\vec{y}\coloneqq\left(y_{1},\dots,y_{n}\right)$
\end_inset

,
 
\begin_inset Formula $\vec{x}\coloneqq\left(x_{1},\dots,x_{n}\right)$
\end_inset


\end_layout

\begin_layout Section
Drugi semester
\end_layout

\begin_layout Part
Vaja za ustni izpit
\end_layout

\begin_layout Standard
Ustni izpit je sestavljen iz treh vprašanj.
 Sekcije so zaporedna vprašanja na izpitu,
 podsekcije so učiteljevi naslovi iz Primerov vprašanj,
 podpodsekcije pa so dejanska vprašanja,
 kot so se pojavila na dosedanjih izpitih.
\end_layout

\begin_layout Section
Prvo vprašanje
\end_layout

\begin_layout Standard
Prvo vprašanje je iz 1.
 semestra.
\end_layout

\begin_layout Subsubsection
\begin_inset Formula $\det AB=\det A\det B$
\end_inset


\end_layout

\begin_layout Subsection
Baze vektorskega prostora
\end_layout

\begin_layout Subsubsection
Linearno neodvisne množice
\end_layout

\begin_layout Subsubsection
Ogrodje
\end_layout

\begin_layout Subsubsection
Definicija baze
\end_layout

\begin_layout Subsubsection
Dimenzija prostora
\end_layout

\begin_layout Subsection
Cramerovo pravilo
\end_layout

\begin_layout Subsubsection
Trditev in dokaz
\end_layout

\begin_layout Subsection
Obrnljive matrike
\end_layout

\begin_layout Subsubsection
Definicija obrnljivosti
\end_layout

\begin_layout Subsubsection
Produkt obrnljivih matrik je obrnljiva matrika
\end_layout

\begin_layout Subsubsection
Karakterizacija obrnljivih matrik z dokazom
\end_layout

\begin_layout Subsubsection
\begin_inset Formula $\Ker A=\left\{ 0\right\} \Leftrightarrow A$
\end_inset

 obrnljiva
\end_layout

\begin_layout Subsubsection
\begin_inset Formula $A$
\end_inset

 ima desni inverz 
\begin_inset Formula $\Rightarrow A$
\end_inset

 obrnljiva
\end_layout

\begin_layout Subsubsection
Formula za inverz matrike z dokazom
\end_layout

\begin_layout Subsection
Vektorski podprostori
\end_layout

\begin_layout Subsection
Elementarne matrike
\end_layout

\begin_layout Subsection
Pod-/predoločeni sistem
\end_layout

\begin_layout Subsubsection
Definicija,
 iskanje posplošene rešitve z izpeljavo
\end_layout

\begin_layout Subsubsection
Moč ogrodja 
\begin_inset Formula $\geq$
\end_inset

 moč LN množice
\end_layout

\begin_layout Subsubsection
Vsak poddoločen sistem ima netrivialno rešitev
\end_layout

\begin_layout Standard
Posledica prejšnje trditve.
\end_layout

\begin_layout Subsection
Regresijska premica
\end_layout

\begin_layout Subsubsection
Definicija
\end_layout

\begin_layout Subsection
Vektorski/mešani produkt
\end_layout

\begin_layout Subsection
Grupe/polgrupe
\end_layout

\begin_layout Subsubsection
Definicija in lastnosti grupe
\end_layout

\begin_layout Subsubsection
Definicija homomorfizma
\end_layout

\begin_layout Subsubsection
Primeri homomorfizmov z dokazi
\end_layout

\begin_layout Subsubsection
Definicija permutacijske grupe in dokaz,
 da je grupa
\end_layout

\begin_layout Subsubsection
Primeri grup
\end_layout

\begin_layout Subsubsection
Dokaz,
 da so ortogonalne matrike podgrupa v grupi obrnljivih matrik
\end_layout

\begin_layout Subsubsection
Matrika permutacije
\end_layout

\begin_layout Subsubsection
Dokaz,
 da je preslikava,
 ki permutaciji priredi matriko,
 homomorfizem
\end_layout

\begin_layout Subsection
Projekcija točke na premico/ravnino
\end_layout

\begin_layout Subsection
\begin_inset Formula $\det A=\det A^{T}$
\end_inset


\end_layout

\begin_layout Subsection
Formula za inverz
\end_layout

\begin_layout Subsection
Homogeni sistemi enačb
\end_layout

\begin_layout Section
Drugo vprašanje
\end_layout

\begin_layout Standard
Drugo vprašanje zajema snov linearnih preslikav/lastnih vrednosti.
\end_layout

\begin_layout Subsection
Diagonalizacija
\end_layout

\begin_layout Subsubsection
Definicija,
 trditve
\end_layout

\begin_layout Subsection
Prehod na novo bazo
\end_layout

\begin_layout Subsubsection
Prehodna matrika in njene lastnosti
\end_layout

\begin_layout Subsubsection
Predstavitev vektorjev in linearnih preslikav z različnimi bazami
\end_layout

\begin_layout Subsubsection
Razvoj vektorja po eni in drugi bazi (prehod vektorja na drugo bazo)
\end_layout

\begin_layout Subsection
Matrika linearne preslikave
\end_layout

\begin_layout Subsection
Rang matrike
\end_layout

\begin_layout Subsubsection
Definicija
\end_layout

\begin_layout Subsubsection
Dokaz,
 da je rang število LN stolpcev
\end_layout

\begin_layout Subsubsection
Dimenzijska formula za podprostore
\end_layout

\begin_layout Subsection
\begin_inset Formula $\rang A=\rang A^{T}$
\end_inset


\end_layout

\begin_layout Subsection
Ekvivalentnost matrik
\end_layout

\begin_layout Subsubsection
Definicija
\end_layout

\begin_layout Subsubsection
Dokaz,
 da je relacija ekvivalenčna
\end_layout

\begin_layout Subsubsection
Dokaz,
 da je vsaka matrika ekvivalentna matriki 
\begin_inset Formula $I_{r}$
\end_inset

,
 t.
 j.
 bločni matriki,
 katere zgornji levi blok je 
\begin_inset Formula $I$
\end_inset

 dimenzije 
\begin_inset Formula $r$
\end_inset

,
 drugi trije bloki pa so ničelne matrike.
\end_layout

\begin_layout Subsection
Jedro/slika
\end_layout

\begin_layout Subsection
Minimalni poinom
\end_layout

\begin_layout Subsubsection
Definicija karakterističnega in minimalnega polinoma
\end_layout

\begin_layout Subsection
Cayley-Hamiltonov izrek
\end_layout

\begin_layout Subsubsection
Trditev in dokaz
\end_layout

\begin_layout Subsection
Korenski razcep
\end_layout

\begin_layout Subsubsection
Definicija korenskih podprostorov
\end_layout

\begin_layout Subsubsection
Presek različnih korenskih podprostorov je trivialen
\end_layout

\begin_layout Subsubsection
Vsota korenskih podprostorov je direktna (se sklicuje na zgornjo trditev)
\end_layout

\begin_layout Subsection
Osnovna formula rang 
\begin_inset Formula $+$
\end_inset

 ničnost
\end_layout

\begin_layout Subsubsection
Definicija
\end_layout

\begin_layout Subsection
Funkcije matrik
\end_layout

\begin_layout Section
Tretje vprašanje
\end_layout

\begin_layout Standard
Tretje vprašanje zajema naslednje snovi:
\end_layout

\begin_layout Itemize
vektorski prostori s skalarnim produktom,
\end_layout

\begin_layout Itemize
adjungirana preslikava,
\end_layout

\begin_layout Itemize
singularni razcep,
\end_layout

\begin_layout Itemize
kvadratne forme.
\end_layout

\begin_layout Subsubsection
Singularni razcep:
 Konstrukcija 
\begin_inset Formula $Q_{1},Q_{2},D$
\end_inset

 in dokaz 
\begin_inset Formula $A=Q_{1}DQ_{2}^{-1}$
\end_inset

.
\end_layout

\begin_layout Subsection
Ortogonalne/unitarne matrike
\end_layout

\begin_layout Subsubsection
Definicija
\end_layout

\begin_layout Subsubsection
Dokaz 
\begin_inset Formula $AA^{*}=I$
\end_inset


\end_layout

\begin_layout Subsubsection
Lastne vrednosti
\end_layout

\begin_layout Subsubsection
Prehodna matrika iz ONB v drugo ONB ima ortogonalne stolpce (dokaz)
\end_layout

\begin_layout Subsection
Kvadratne krivulje
\end_layout

\begin_layout Subsection
Psevdoinverz
\end_layout

\begin_layout Subsubsection
Definicija
\end_layout

\begin_layout Subsection
Najkrajša posplošena rešitev sistema
\end_layout

\begin_layout Subsubsection
Definicija,
 trditev in dokaz
\end_layout

\begin_layout Subsection
Simetrične matrike
\end_layout

\begin_layout Subsubsection
Vse o simetričnih matrikah
\end_layout

\begin_layout Subsection
Adjungirana linearna preslikava
\end_layout

\begin_layout Subsubsection
Definicija in celotna formulacija
\end_layout

\begin_layout Subsubsection
Rieszov izrek
\end_layout

\begin_layout Subsubsection
Dokaz obstoja in enoličnosti kot posledica Rieszovega izreka
\end_layout

\begin_layout Subsubsection
Formula za matriko linearne preslikave in 
\begin_inset Formula $\left\langle Au,v\right\rangle =v^{*}Au=\left\langle u,A^{*}v\right\rangle $
\end_inset


\end_layout

\begin_layout Subsubsection
Lastne vrednosti adjungirane matrike
\end_layout

\begin_layout Subsection
Klasifikacija skalarnih produktov
\end_layout

\begin_layout Subsection
Normalne matrike
\end_layout

\begin_layout Subsubsection
Definicija,
 lastnosti,
 izreki,
 dokazi
\end_layout

\begin_layout Subsubsection
\begin_inset Formula $A$
\end_inset

 normalna 
\begin_inset Formula $\Rightarrow A$
\end_inset

 in 
\begin_inset Formula $A^{*}$
\end_inset

 imata isto množico lastnih vrednosti
\end_layout

\begin_layout Subsubsection
\begin_inset Formula $\Ker\left(A-xI\right)=\Ker\left(A-\overline{x}I\right)$
\end_inset

 za normalno 
\begin_inset Formula $A$
\end_inset


\end_layout

\begin_layout Subsection
Ortogonalni komplement
\end_layout

\begin_layout Subsubsection
Formula za ortogonalno projekcijo
\end_layout

\begin_layout Subsection
Izrek o reprezentaciji linearnih funkcionalov
\end_layout

\begin_layout Subsection
Pozitivno semidefinitne matrike
\end_layout

\begin_layout Subsubsection
Definicija,
 lastnosti.
\end_layout

\begin_layout Subsubsection
Dokaz,
 da imajo nenegativne lastne vrednosti.
\end_layout

\begin_layout Subsubsection
Kvadratni koren pozitivno semidefinitne matrike.
\end_layout

\begin_layout Subsubsection
\begin_inset Formula $A\geq0\Rightarrow A$
\end_inset

 sebiadjungirana
\end_layout

\begin_layout Subsection
Ortogonalne in ortonormirane baze/Gram-Schmidt
\end_layout

\end_body
\end_document