2 files changed, 3097 insertions, 0 deletions
diff --git a/šola/ana2/teor.lyx b/šola/ana2/teor.lyx
new file mode 100644
index 0000000..e1ee169
--- /dev/null
+++ b/šola/ana2/teor.lyx
@@ -0,0 +1,864 @@
+#LyX 2.4 created this file. For more info see https://www.lyx.org/
+\lyxformat 620
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass article
+\begin_preamble
+\usepackage{hyperref}
+\usepackage{siunitx}
+\usepackage{pgfplots}
+\usepackage{listings}
+\usepackage{multicol}
+\sisetup{output-decimal-marker = {,}, quotient-mode=fraction, output-exponent-marker=\ensuremath{\mathrm{3}}}
+\usepackage{amsmath}
+\usepackage{tikz}
+\newcommand{\udensdash}[1]{%
+    \tikz[baseline=(todotted.base)]{
+        \node[inner sep=1pt,outer sep=0pt] (todotted) {#1};
+        \draw[densely dashed] (todotted.south west) -- (todotted.south east);
+    }%
+}%
+\DeclareMathOperator{\Lin}{Lin}
+\DeclareMathOperator{\rang}{rang}
+\DeclareMathOperator{\sled}{sled}
+\DeclareMathOperator{\Aut}{Aut}
+\DeclareMathOperator{\red}{red}
+\DeclareMathOperator{\karakteristika}{char}
+\usepackage{algorithm,algpseudocode}
+\providecommand{\corollaryname}{Posledica}
+\end_preamble
+\use_default_options true
+\begin_modules
+enumitem
+theorems-ams
+\end_modules
+\maintain_unincluded_children no
+\language slovene
+\language_package default
+\inputencoding auto-legacy
+\fontencoding auto
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_roman_osf false
+\font_sans_osf false
+\font_typewriter_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry true
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification false
+\use_refstyle 1
+\use_formatted_ref 0
+\use_minted 0
+\use_lineno 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\leftmargin 2cm
+\topmargin 2cm
+\rightmargin 2cm
+\bottommargin 2cm
+\headheight 2cm
+\headsep 2cm
+\footskip 1cm
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style german
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tablestyle default
+\tracking_changes false
+\output_changes false
+\change_bars false
+\postpone_fragile_content false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\docbook_table_output 0
+\docbook_mathml_prefix 1
+\end_header
+
+\begin_body
+
+\begin_layout Title
+ANA2 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 Section
+Množice v 
+\begin_inset Formula $\mathbb{R}^{n}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Definition*
+Razdalja točk v 
+\begin_inset Formula $\mathbb{R}^{n}$
+\end_inset
+
+ je norma njune razlike.
+ 
+\begin_inset Formula $\varepsilon-$
+\end_inset
+
+okolica točke 
+\begin_inset Formula $a\in\mathbb{R}^{n}$
+\end_inset
+
+ so take točke,
+ ki so od 
+\begin_inset Formula $a$
+\end_inset
+
+ oddaljene manj od 
+\begin_inset Formula $\varepsilon\in\mathbb{R}$
+\end_inset
+
+.
+ Robna točka množice 
+\begin_inset Formula $A$
+\end_inset
+
+ je taka točka,
+ katere poljubno majhna okolica vsebuje tako točke iz 
+\begin_inset Formula $A$
+\end_inset
+
+ kot tudi točke,
+ ki niso iz 
+\begin_inset Formula $A$
+\end_inset
+
+.
+ Odprta množica ne vsebuje robnih točk.
+ Zaprta množica je komplement neke odprte množice.
+\end_layout
+
+\begin_layout Claim*
+\begin_inset Formula $A\subset\mathbb{R}$
+\end_inset
+
+ zaprta 
+\begin_inset Formula $\Leftrightarrow$
+\end_inset
+
+ za vsako zaporedje s členi v 
+\begin_inset Formula $A$
+\end_inset
+
+ velja,
+ da so vsa njegova stekališča,
+ čim obstajajo,
+ v 
+\begin_inset Formula $A$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Proof
+Dokazujemo ekvivalenco
+\end_layout
+
+\begin_deeper
+\begin_layout Description
+\begin_inset Formula $\left(\Rightarrow\right)$
+\end_inset
+
+ Naj bo 
+\begin_inset Formula $s$
+\end_inset
+
+ stekališče 
+\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
+\end_inset
+
+,
+ 
+\begin_inset Formula $a_{n}\in A$
+\end_inset
+
+ in 
+\begin_inset Formula $s\not\in A$
+\end_inset
+
+ (RAAPDD).
+ Ker je 
+\begin_inset Formula $A$
+\end_inset
+
+ zaprta,
+ je 
+\begin_inset Formula $\mathbb{R}\setminus A$
+\end_inset
+
+ odprta,
+ zato 
+\begin_inset Formula $\exists\varepsilon>0\ni:\left(s-\varepsilon,s+\varepsilon\right)\subset\mathbb{R}\setminus A$
+\end_inset
+
+,
+ torej v 
+\begin_inset Formula $\left(s-\varepsilon,s+\varepsilon\right)$
+\end_inset
+
+ ni nobenega člena zaporedja,
+ torej 
+\begin_inset Formula $s$
+\end_inset
+
+ ni stekališče 
+\begin_inset Formula $\rightarrow\!\leftarrow$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Description
+\begin_inset Formula $\left(\Leftarrow\right)$
+\end_inset
+
+ Dokazujemo,
+ da je 
+\begin_inset Formula $A$
+\end_inset
+
+ zaprta,
+ torej,
+ da je 
+\begin_inset Formula $B=\mathbb{R}\setminus A$
+\end_inset
+
+ odprta.
+ PDDRAA 
+\begin_inset Formula $B$
+\end_inset
+
+ ni odprta 
+\begin_inset Formula $\Rightarrow\exists x\in B\ni:\forall n\in\mathbb{N}:$
+\end_inset
+
+ 
+\begin_inset Formula $n^{-1}-$
+\end_inset
+
+okolica 
+\begin_inset Formula $x$
+\end_inset
+
+ vsebuje nek element 
+\begin_inset Formula $A$
+\end_inset
+
+.
+ Našli smo torej zaporedje v 
+\begin_inset Formula $A$
+\end_inset
+
+ s stekališčem v 
+\begin_inset Formula $B$
+\end_inset
+
+.
+ 
+\begin_inset Formula $\rightarrow\!\leftarrow$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Definition*
+Stroga podmnožica 
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+ je kompaktna,
+ če ima vsako zaporedje s členi v njej v njej tudi stekališče.
+ Množica je omejena,
+ če je podmnožica neke okolice izhodišča.
+\end_layout
+
+\begin_layout Theorem*
+\begin_inset Formula $A\subset\mathbb{R}$
+\end_inset
+
+ kompaktna 
+\begin_inset Formula $\Leftrightarrow A$
+\end_inset
+
+ zaprta in omejena.
+\end_layout
+
+\begin_layout Proof
+Dokazujemo ekvivalenco
+\end_layout
+
+\begin_deeper
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $\left(\Leftarrow\right)$
+\end_inset
+
+ Naj bo 
+\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
+\end_inset
+
+ zaporedje v 
+\begin_inset Formula $A$
+\end_inset
+
+.
+ Ker je 
+\begin_inset Formula $A$
+\end_inset
+
+ omejena,
+ je zaporedje omejeno,
+ torej premore stekališča.
+ Ker je 
+\begin_inset Formula $A$
+\end_inset
+
+ zaprta,
+ vsebuje vsa ta stekališča.
+ Torej je 
+\begin_inset Formula $A$
+\end_inset
+
+ kompaktna.
+\end_layout
+
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $\left(\Rightarrow\right)$
+\end_inset
+
+ 
+\begin_inset Formula $A$
+\end_inset
+
+ je omejeno,
+ sicer bi našli zaporedje,
+ da velja 
+\begin_inset Formula $a_{i}\geq i$
+\end_inset
+
+,
+ ki nima stekališča.
+ Treba je dokazati še,
+ da je 
+\begin_inset Formula $A$
+\end_inset
+
+ zaprta.
+ Vsa stekališča zaporedij s členi v 
+\begin_inset Formula $A$
+\end_inset
+
+ imajo v 
+\begin_inset Formula $A$
+\end_inset
+
+ stekališče (kompaktnost).
+ Torej za vsako stekališče zaporedja s členi v 
+\begin_inset Formula $A$
+\end_inset
+
+ velja,
+ da ima v 
+\begin_inset Formula $A$
+\end_inset
+
+ stekališče,
+ torej je 
+\begin_inset Formula $A$
+\end_inset
+
+ zaprta.
+\end_layout
+
+\end_deeper
+\begin_layout Remark*
+Vsako zaporedje v kompaktni množici ima stekališče,
+ kar za zaprto množico ni rečeno.
+ Zaprta množica lahko vsebuje zaporedja brez stekališč.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+razloži normo,
+ trikotniško neenakost,
+ itd.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Definition*
+Točka 
+\begin_inset Formula $a\in A\subseteq\mathbb{R}^{n}$
+\end_inset
+
+ je notranja,
+ če obstaja neka njena okolica,
+ ki je podmnožica 
+\begin_inset Formula $A$
+\end_inset
+
+.
+ Točka 
+\begin_inset Formula $a\in\mathbb{R}^{n}$
+\end_inset
+
+ je stekališče množice 
+\begin_inset Formula $A$
+\end_inset
+
+,
+ če vsaka njena okolica seka
+\begin_inset Foot
+status open
+
+\begin_layout Plain Layout
+t.
+ j.
+ ima neprazen presek z
+\end_layout
+
+\end_inset
+
+ 
+\begin_inset Formula $A\setminus\left\{ a\right\} $
+\end_inset
+
+.
+ Točka 
+\begin_inset Formula $a\in A$
+\end_inset
+
+,
+ ki ni stekališče 
+\begin_inset Formula $A$
+\end_inset
+
+,
+ je izolirana točka 
+\begin_inset Formula $A$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Definition*
+Zaporedje s členi v 
+\begin_inset Formula $\mathbb{R}^{k}$
+\end_inset
+
+ je funkcija 
+\begin_inset Formula $\mathbb{N}\to\mathbb{R}^{k}$
+\end_inset
+
+,
+ 
+\begin_inset Formula $n\mapsto a_{n}=\left(a_{n}^{\left(1\right)},\dots,a_{n}^{\left(k\right)}\right)$
+\end_inset
+
+.
+ 
+\begin_inset Formula $a\in\mathbb{R}^{k}$
+\end_inset
+
+ je limita zaporedja 
+\begin_inset Formula $\left(a_{n}\right)_{n}$
+\end_inset
+
+ s členi v 
+\begin_inset Formula $\mathbb{R}^{k}$
+\end_inset
+
+,
+ če 
+\begin_inset Formula $\forall\varepsilon>0\exists n_{0}\in\mathbb{N}\forall n>n_{0}:\left|a-a_{n}\right|<\varepsilon$
+\end_inset
+
+ in pišemo 
+\begin_inset Formula $a=\lim_{n\to\infty}a_{n}$
+\end_inset
+
+.
+ Če zaporedje ima limito,
+ je konvergentno,
+ sicer je divergentno.
+ Točka 
+\begin_inset Formula $s\in\mathbb{R}^{k}$
+\end_inset
+
+ je stekališče zaporedja 
+\begin_inset Formula $\left(a_{n}\right)_{n}$
+\end_inset
+
+ s členi v 
+\begin_inset Formula $\mathbb{R}^{k}$
+\end_inset
+
+,
+ če je v vsaki okolici 
+\begin_inset Formula $s$
+\end_inset
+
+ neskončno členov 
+\begin_inset Formula $\left(a_{n}\right)_{n}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Fact*
+Velja:
+\end_layout
+
+\begin_deeper
+\begin_layout Itemize
+Vsako konvergentno zaporedje je omejeno in ima natanko eno limito,
+ ki je njegovo edino stekališče.
+\end_layout
+
+\begin_layout Itemize
+Vsako omejeno zaporedje ima stekališče.
+\end_layout
+
+\begin_layout Itemize
+Stekališče zaporedja je limita nekega podzaporedja in obratno.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $A\subset\mathbb{R}^{k}$
+\end_inset
+
+ je zaprta 
+\begin_inset Formula $\Leftrightarrow$
+\end_inset
+
+ vsako stekališče zaporedja s členi v 
+\begin_inset Formula $A$
+\end_inset
+
+ leži v 
+\begin_inset Formula $A$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Section
+Funkcije več spremenljivk
+\end_layout
+
+\begin_layout Definition*
+Naj bo 
+\begin_inset Formula $D\subseteq\mathbb{R}^{k}$
+\end_inset
+
+ in 
+\begin_inset Formula $f:D\to\mathbb{R}$
+\end_inset
+
+ preslikava.
+ Če je 
+\begin_inset Formula $k\geq2$
+\end_inset
+
+,
+ je 
+\begin_inset Formula $f$
+\end_inset
+
+ funkcija več spremenljivk.
+ 
+\begin_inset Formula $\Gamma_{f}=\left\{ \left(x,fx\right);x\in D\right\} \subset\mathbb{R}^{k}\times\mathbb{R}$
+\end_inset
+
+ je graf funkcije 
+\begin_inset Formula $f$
+\end_inset
+
+.
+ Za 
+\begin_inset Formula $a\in\mathbb{R}^{k}$
+\end_inset
+
+ stekališče 
+\begin_inset Formula $D$
+\end_inset
+
+ je 
+\begin_inset Formula $L\in\mathbb{R}$
+\end_inset
+
+ limita 
+\begin_inset Formula $f$
+\end_inset
+
+ v 
+\begin_inset Formula $a$
+\end_inset
+
+,
+ če 
+\begin_inset Formula $\forall\varepsilon>0\exists\delta=\delta\left(a,\varepsilon\right)>0\forall x\in D,x\not=a:\left|\left|x-a\right|\right|<\delta\Rightarrow\left|fx-L\right|<\varepsilon$
+\end_inset
+
+ in pišemo 
+\begin_inset Formula $\lim_{x\to a}fx=L$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Remark*
+Medtem ko imamo pri funkcijah ene spremenljivke levo in desno limito,
+ je tu obnašanje bolj zapleteno,
+ saj obstaja veliko različnih načinov približevanja k 
+\begin_inset Formula $a$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Definition*
+Naj bo 
+\begin_inset Formula $D\subseteq\mathbb{R}^{k}$
+\end_inset
+
+ in 
+\begin_inset Formula $f:D\to\mathbb{R}$
+\end_inset
+
+ funkcija in 
+\begin_inset Formula $a\in D$
+\end_inset
+
+.
+ 
+\begin_inset Formula $f$
+\end_inset
+
+ je zvezna v 
+\begin_inset Formula $a$
+\end_inset
+
+,
+ če 
+\begin_inset Formula $\forall\varepsilon>0\exists\delta=\delta\left(a,\varepsilon\right)>0\forall x\in D:\left|\left|x-a\right|\right|<\delta\Rightarrow\left|fx-fa\right|<\varepsilon$
+\end_inset
+
+.
+ 
+\begin_inset Formula $f$
+\end_inset
+
+ je zvezna,
+ če je zvezna na vsaki točki svojega definicijskega območja.
+\end_layout
+
+\begin_layout Remark*
+Če je 
+\begin_inset Formula $a$
+\end_inset
+
+ stekališče 
+\begin_inset Formula $D$
+\end_inset
+
+,
+ je 
+\begin_inset Formula $f$
+\end_inset
+
+ zvezna v 
+\begin_inset Formula $a\Leftrightarrow\lim_{x\to a}fx=fa$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Corollary*
+Če je 
+\begin_inset Formula $a$
+\end_inset
+
+ izolirana točka 
+\begin_inset Formula $D$
+\end_inset
+
+,
+ je 
+\begin_inset Formula $f$
+\end_inset
+
+ zvezna v 
+\begin_inset Formula $a$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Definition*
+Naj bo 
+\begin_inset Formula $f:D\subseteq\mathbb{R}^{k}\to\mathbb{R}$
+\end_inset
+
+ funkcija,
+ 
+\begin_inset Formula $Z=fD$
+\end_inset
+
+ njena zaloga vrednosti in 
+\begin_inset Formula $g:Z\to\mathbb{R}$
+\end_inset
+
+ funkcija.
+ Kompozitum ali sestavljena funkcija 
+\begin_inset Formula $f$
+\end_inset
+
+ in 
+\begin_inset Formula $g$
+\end_inset
+
+ je funkcija 
+\begin_inset Formula $k$
+\end_inset
+
+ spremenljivk 
+\begin_inset Formula $g\circ f:D\to\mathbb{R}$
+\end_inset
+
+,
+ definirana s predpisom 
+\begin_inset Formula $\left(g\circ f\right)x=gfx$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Theorem*
+Naj bo 
+\begin_inset Formula $f$
+\end_inset
+
+ funkcija 
+\begin_inset Formula $k$
+\end_inset
+
+ spremenljivk,
+ zvezna v 
+\begin_inset Formula $a\in\mathbb{R}^{k}$
+\end_inset
+
+ in 
+\begin_inset Formula $g$
+\end_inset
+
+ funkcija ene spremenljivke,
+ zvezna v 
+\begin_inset Formula $fa\in\mathbb{R}$
+\end_inset
+
+.
+ Tedaj je 
+\begin_inset Formula $g\circ f$
+\end_inset
+
+ zvezna v 
+\begin_inset Formula $a$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Proof
+Izberimo poljuben 
+\begin_inset Formula $\varepsilon$
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document
diff --git a/šola/la/teor.lyx b/šola/la/teor.lyx
new file mode 100644
index 0000000..440fc84
--- /dev/null
+++ b/šola/la/teor.lyx
@@ -0,0 +1,2233 @@
+#LyX 2.4 created this file. For more info see https://www.lyx.org/
+\lyxformat 620
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass article
+\begin_preamble
+\usepackage{hyperref}
+\usepackage{siunitx}
+\usepackage{pgfplots}
+\usepackage{listings}
+\usepackage{multicol}
+\sisetup{output-decimal-marker = {,}, quotient-mode=fraction, output-exponent-marker=\ensuremath{\mathrm{3}}}
+\usepackage{amsmath}
+\usepackage{tikz}
+\newcommand{\udensdash}[1]{%
+    \tikz[baseline=(todotted.base)]{
+        \node[inner sep=1pt,outer sep=0pt] (todotted) {#1};
+        \draw[densely dashed] (todotted.south west) -- (todotted.south east);
+    }%
+}%
+\DeclareMathOperator{\Lin}{\mathcal Lin}
+\DeclareMathOperator{\rang}{rang}
+\DeclareMathOperator{\sled}{sled}
+\DeclareMathOperator{\Aut}{Aut}
+\DeclareMathOperator{\red}{red}
+\DeclareMathOperator{\karakteristika}{char}
+\DeclareMathOperator{\Ker}{Ker}
+\usepackage{algorithm,algpseudocode}
+\providecommand{\corollaryname}{Posledica}
+\end_preamble
+\use_default_options true
+\begin_modules
+enumitem
+theorems-ams
+\end_modules
+\maintain_unincluded_children no
+\language slovene
+\language_package default
+\inputencoding auto-legacy
+\fontencoding auto
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_roman_osf false
+\font_sans_osf false
+\font_typewriter_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\float_placement class
+\float_alignment class
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry true
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification false
+\use_refstyle 1
+\use_formatted_ref 0
+\use_minted 0
+\use_lineno 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\leftmargin 2cm
+\topmargin 2cm
+\rightmargin 2cm
+\bottommargin 2cm
+\headheight 2cm
+\headsep 2cm
+\footskip 1cm
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style german
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tablestyle default
+\tracking_changes false
+\output_changes false
+\change_bars false
+\postpone_fragile_content false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\docbook_table_output 0
+\docbook_mathml_prefix 1
+\end_header
+
+\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