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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 |