2 files changed, 806 insertions, 0 deletions
diff --git a/šola/la/kolokvij1.lyx b/šola/la/kolokvij1.lyx
new file mode 100644
index 0000000..429701d
--- /dev/null
+++ b/šola/la/kolokvij1.lyx
@@ -0,0 +1,805 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass article
+\begin_preamble
+\usepackage{siunitx}
+\usepackage{pgfplots}
+\usepackage{listings}
+\usepackage{multicol}
+\sisetup{output-decimal-marker = {,}, quotient-mode=fraction, output-exponent-marker=\ensuremath{\mathrm{3}}}
+\end_preamble
+\use_default_options true
+\begin_modules
+enumitem
+\end_modules
+\maintain_unincluded_children false
+\language slovene
+\language_package default
+\inputencoding auto
+\fontencoding global
+\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_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_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\leftmargin 1cm
+\topmargin 0cm
+\rightmargin 1cm
+\bottommargin 2cm
+\headheight 1cm
+\headsep 1cm
+\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
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Title
+List s formulami za 1.
+ kolokvij Linearne algebre
+\end_layout
+
+\begin_layout Author
+
+\noun on
+Anton Luka Šijanec
+\end_layout
+
+\begin_layout Date
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+today
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+newcommand
+\backslash
+euler{e}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{multicols}{2}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\vec{u}\cdot\vec{u}=\vert\vert\vec{u}\vert\vert^{2}$
+\end_inset
+
+, 
+\begin_inset Formula $\left(\alpha\vec{u}+\beta\vec{v}\right)\cdot\vec{w}=\alpha\left(\vec{u}\cdot\vec{w}\right)+\beta\left(\vec{v},\vec{w}\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Paralelogramska ident.: 
+\begin_inset Formula $\vert\vert\vec{u}+\vec{v}\vert\vert^{2}+\vert\vert\vec{u}-\vec{v}\vert\vert^{2}=2\vert\vert\vec{u}\vert\vert^{2}+2\vert\vert\vec{v}\vert\vert^{2}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Ploščina paralelograma: 
+\begin_inset Formula $\vert\vert\vec{u}\times\vec{v}\vert\vert=\vert\vert\vec{u}\vert\vert\vert\vert\vec{v}\vert\vert\sin\phi$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $<\vec{u},\vec{v}>=\vert\vert\vec{u}\vert\vert\vert\vert\vec{v}\vert\vert\cos\phi$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $<\vec{u}\times\vec{v},\vec{u}>=0$
+\end_inset
+
+, 
+\begin_inset Formula $\vert\vert\vec{u}\times\vec{v}\vert\vert^{2}+<\vec{u},\vec{v}>^{2}=\vert\vert\vec{u}\vert\vert^{2}\vert\vert\vec{v}\vert\vert^{2}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Vol.
+ ppd.: 
+\begin_inset Formula $[u,v,w]=<u\times v,w>=\vert\vert u\times v\vert\vert\cdot\vert\vert w\vert\vert\cos\phi$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\vec{u}\times\vec{u}=0$
+\end_inset
+
+, 
+\begin_inset Formula $\vec{u}\times\vec{v}=-\left(\vec{v}\times\vec{u}\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Linearnost 
+\begin_inset Formula $\times$
+\end_inset
+
+: 
+\begin_inset Formula $\left(\alpha\vec{u}+\beta\vec{v}\right)\times\vec{w}=\alpha\left(\vec{u}\times\vec{w}\right)+\beta\left(\vec{v}\times\vec{w}\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $[u,v,w]=-[u,w,v]$
+\end_inset
+
+, 
+\begin_inset Formula $[u,v,w]=[w,u,v]=[v,w,u]$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\vec{r}=\vec{r_{0}}+t\vec{p},t\in\mathbb{R}\Longleftrightarrow x=x_{0}+tp_{1},y=y_{0}+tp_{2},z=z_{0}+tp_{3}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\Longleftrightarrow t=\frac{x-x_{0}}{p_{1}}=\frac{y-y_{0}}{p_{2}}=\frac{z-z_{0}}{p_{3}}$
+\end_inset
+
+ :Normalna enačba 
+\begin_inset Formula $\mathbb{R}^{3}$
+\end_inset
+
+ premice
+\end_layout
+
+\begin_layout Standard
+Projekcija 
+\begin_inset Formula $\vec{r_{1}}$
+\end_inset
+
+ na 
+\begin_inset Formula $\vec{r_{0}}+t\vec{p}$
+\end_inset
+
+ 
+\begin_inset Formula $\coloneqq$
+\end_inset
+
+ 
+\begin_inset Formula $\vec{r_{1}'}=\vec{r_{0}}+t'\vec{p}$
+\end_inset
+
+ in 
+\begin_inset Formula $<\vec{r_{1}'}-\vec{r_{1}},\vec{p}>=0$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Dvojni 
+\begin_inset Formula $\times$
+\end_inset
+
+: 
+\begin_inset Formula $\vec{a}\times\left(\vec{b}\times\vec{c}\right)=\vec{b}\left(\vec{a}\cdot\vec{c}\right)-\vec{c}\left(\vec{a}\cdot\vec{b}\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Dvojni 
+\begin_inset Formula $\times$
+\end_inset
+
+: 
+\begin_inset Formula $\left(\vec{a}\times\vec{b}\right)\times\vec{c}=\vec{b}\left(\vec{a}\cdot\vec{c}\right)-\vec{a}\left(\vec{b}\cdot\vec{c}\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Norm.
+ e.
+ ravnine: 
+\begin_inset Formula $n_{1}x+n_{2}y+n_{3}z=d=n_{1}x_{0}+n_{2}y_{0}+n_{3}z_{0}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\Longleftrightarrow<\vec{r}-\vec{r_{0}},\vec{n}>=0$
+\end_inset
+
+, saj je 
+\begin_inset Formula $\forall\vec{r}\in$
+\end_inset
+
+ravnine
+\begin_inset Formula $:\left(\vec{r}-\vec{r_{0}}\right)\bot\vec{n}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Parametrična e.
+ ravnine: 
+\begin_inset Formula $\vec{r}=\vec{r_{0}}+s\vec{p}+t\vec{q},s\in\mathbb{R},q\in\mathbb{R}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Proj.
+ 
+\begin_inset Formula $\vec{r_{1}}=\vec{r_{0}}+s\vec{p}+t\vec{q}$
+\end_inset
+
+ velja 
+\begin_inset Formula $<\vec{r_{1}'}-\vec{r_{1},\vec{p}}>=0=<\vec{r_{1}'}-\vec{r_{1},\vec{q}}>$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+sistem: 
+\begin_inset Formula $s\vec{p}\cdot\vec{p}+t\vec{q}\cdot\vec{p}=\left(\vec{r_{1}}-\vec{r_{0}}\right)\cdot\vec{p}$
+\end_inset
+
+; 
+\begin_inset Formula $s\vec{p}\cdot\vec{q}+t\vec{q}\cdot\vec{q}=\left(\vec{r_{1}}-\vec{r_{0}}\right)\cdot\vec{q}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Hiperravnino v 
+\begin_inset Formula $\mathbb{R}^{n}$
+\end_inset
+
+ določa 
+\begin_inset Formula $n$
+\end_inset
+
+ linearno neodvisnih vektorjev.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+Posplošena rešitev: 
+\begin_inset Formula $\min\sum_{k=1}^{m}\left(a_{k,1}x_{1}+\ldots+a_{1,n}x_{n}-b_{k}\right)^{2}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Formula $\Longleftrightarrow\min\vert\vert x_{1}\vec{a_{1}}+\ldots+x_{n}\vec{a_{n}}-\vec{b}\vert\vert^{2}$
+\end_inset
+
+ (proj 
+\begin_inset Formula $\vec{b}$
+\end_inset
+
+ na hiperravnino)
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\left(AB\right)^{T}=B^{T}+A^{T}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $E_{ij}\left(\alpha\right)\coloneqq\texttt{i+=\ensuremath{\alpha}j}$
+\end_inset
+
+, 
+\begin_inset Formula $P_{ij}\coloneqq\texttt{i,j=j,i}$
+\end_inset
+
+, 
+\begin_inset Formula $E_{i}\left(\alpha\right)\coloneqq\texttt{i*=\ensuremath{\alpha}}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $E_{ij}\left(\alpha\right)^{-1}=E_{ij}\left(\alpha\right)$
+\end_inset
+
+,
+\begin_inset Formula $P_{ij}^{-1}=P_{ji}$
+\end_inset
+
+,
+\begin_inset Formula $E_{i}\left(\beta\right)^{-1}=E_{i}\left(\beta^{-1}\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\nexists A_{m,n}^{-1}\Leftrightarrow A=0\Leftrightarrow m\not=n\Leftrightarrow\det A=0\Leftrightarrow A$
+\end_inset
+
+ ima 
+\begin_inset Formula $\vec{0}$
+\end_inset
+
+ vrstico/stolpec
+\end_layout
+
+\begin_layout Paragraph
+Karakterizacija obrnljivih matrik
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{multicols}{2}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument 1
+status open
+
+\begin_layout Plain Layout
+label=
+\begin_inset Formula $\Leftrightarrow$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula $\exists A^{-1}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $\exists B\ni:BA=I$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $\exists B\ni:AB=I$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $\left(AX=0\Longrightarrow X=0\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+stolpci so ogrodje
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $\text{RKSO}\left(A\right)=I$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $\forall\vec{b}\exists\vec{x}\ni:A\vec{x}=\vec{b}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $A=$
+\end_inset
+
+ produkt E.
+ M.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{multicols}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\exists A^{-1}\Longleftrightarrow\exists B\ni:BA=I\Longleftrightarrow\exists B\ni:AB=I\Longleftrightarrow$
+\end_inset
+
+ stolpci so LN 
+\begin_inset Formula $\Longleftrightarrow\left(AX=0\Longrightarrow X=0\right)\Longleftrightarrow$
+\end_inset
+
+stolpci so ogrodje
+\begin_inset Formula $\Longleftrightarrow\text{RKSO}\left(A\right)=$
+\end_inset
+
+
+\begin_inset Formula $I\Longleftrightarrow\forall\vec{b}\exists\vec{x}\ni:A\vec{x}=\vec{b}\Longleftrightarrow A=$
+\end_inset
+
+produkt E.M.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Matrični zapis sistema: 
+\begin_inset Formula $A\vec{x}=\vec{b}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Najkrajša rešitev sistema 
+\begin_inset Formula $\vec{x_{0}}\Leftarrow\vert\vert A\vec{x_{0}}-\vec{b}\vert\vert=\min\vert\vert A\vec{x}-\vec{b}\vert\vert$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+...
+ je običajna rešitev 
+\begin_inset Formula $A^{T}A\vec{x}=A^{T}\vec{b}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Desno množenje z E.
+ M.
+ je manipulacija stoplcev.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $M/A\coloneqq D-CA^{-1}B$
+\end_inset
+
+, 
+\begin_inset Formula $M/D\coloneqq A-BD^{-1}C$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+M^{-1}=\left[\begin{array}{cc}
+A & B\\
+C & D
+\end{array}\right]^{-1}=
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+=\left[\begin{array}{cc}
+A^{-1}+A^{-1}B\left(M/A\right)^{-1}CA^{-1} & -A^{-1}B\left(M/A\right)^{-1}\\
+-\left(M/A\right)^{-1}CA^{-1} & \left(M/A\right)^{-1}
+\end{array}\right]=
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+=\left[\begin{array}{cc}
+\left(M/D\right)^{-1} & -\left(M/D\right)^{-1}BD^{-1}\\
+-D^{-1}C\left(M/D\right)^{-1} & D^{-1}+D^{-1}C\left(M/D\right)^{-1}BD^{-1}
+\end{array}\right]
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\det\left[\begin{array}{cc}
+a & b\\
+c & d
+\end{array}\right]=ad-bc$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\vec{a}\times\vec{b}=\left|\begin{array}{ccc}
+\vec{i} & \vec{j} & \vec{k}\\
+a_{1} & a_{2} & a_{3}\\
+b_{1} & b_{2} & b_{3}
+\end{array}\right|$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $A_{i,j}\coloneqq A$
+\end_inset
+
+ brez 
+\begin_inset Formula $i$
+\end_inset
+
+te vrstice in 
+\begin_inset Formula $j$
+\end_inset
+
+tega stolpca
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\det[a]=a$
+\end_inset
+
+, 
+\begin_inset Formula $\det A=\sum_{k=1}^{n}\left(-1\right)^{k+1}a_{1,k}\det A_{1,j}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Razvoj po 
+\begin_inset Formula $i$
+\end_inset
+
+ti vrstici: 
+\begin_inset Formula $\det A=\sum_{j=1}^{n}\left(-1\right)^{i+j}a_{ij}\det A_{ij}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Razvoj po 
+\begin_inset Formula $j$
+\end_inset
+
+tem stolpcu: 
+\begin_inset Formula $\det A=\sum_{i=1}^{n}\left(-1\right)^{i+j}a_{ij}\det A_{ij}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\det$
+\end_inset
+
+ trikotne matrike: 
+\begin_inset Formula $\prod_{i=1}^{n}a_{ii}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Trikotna matrika ima pod ali nad diagonalo same ničle.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\det\left(P_{ij}A\right)=-detA$
+\end_inset
+
+, 
+\begin_inset Formula $\det\left(E_{i}\alpha A\right)=\alpha\det A$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\det\left(E_{ij}\alpha A\right)=\det A$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\det\left(AB\right)=\det A\det B$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Za možne napake ne odgovarjam.
+ Srečno!
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{multicols}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document
diff --git a/šola/p1/dn/.gitignore b/šola/p1/dn/.gitignore
new file mode 100644
index 0000000..e93ee15
--- /dev/null
+++ b/šola/p1/dn/.gitignore
@@ -0,0 +1 @@
+*Testi/