1 files changed, 1101 insertions, 0 deletions

diff --git a/šola/la/kolokvij2.lyx b/šola/la/kolokvij2.lyx
new file mode 100644
index 0000000..c52bc97
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+++ b/šola/la/kolokvij2.lyx
@@ -0,0 +1,1101 @@
+#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 2cm
+\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 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 $\left(AB\right)^{T}=B^{T}+A^{T}$
+\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 $\det\left[\begin{array}{cc}
+a & b\\
+c & d
+\end{array}\right]=ad-bc$
+\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 $\det\left(P_{ij}A\right)=-detA,\quad\det\left(E_{i}\alpha A\right)=\alpha\det A$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\det\left(E_{ij}\alpha A\right)=\det A,\quad\det\left(AB\right)=\det A\det B$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\det\left[\begin{array}{cc}
+A & B\\
+0 & C
+\end{array}\right]=\det A\det C,\quad\det A^{T}=\det A$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+\det A^{n}=\left(\det A\right)^{n}\text{ velja tudi za inverz}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\det P_{ij}=-1,\quad\det E_{i}\left(\alpha\right)=\alpha,\quad\det E_{ij}\left(\alpha\right)=1$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\det\mathbb{R}^{3}$
+\end_inset
+
+: negativne diagonale prištejemo, pozitivne odštejemo
+\end_layout
+
+\begin_layout Paragraph
+Cramerjevo pravilo
+\end_layout
+
+\begin_layout Standard
+za rešitev sistema s kvadratno matriko koeficientov: 
+\begin_inset Formula $x_{i}=\frac{\det A_{i}\left(\vec{b}\right)}{\det A}$
+\end_inset
+
+, kjer je 
+\begin_inset Formula $A_{i}\left(\vec{b}\right)$
+\end_inset
+
+ matrika 
+\begin_inset Formula $A$
+\end_inset
+
+, ki ima namesto 
+\begin_inset Formula $i$
+\end_inset
+
+-tega stolpca 
+\begin_inset Formula $\vec{b}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Paragraph
+Inverz matrike
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $A_{ij}^{-1}=\frac{\det A_{ji}\left(-1\right)^{j+i}}{\det A}=\frac{1}{\det A}\tilde{A}^{T}$
+\end_inset
+
+, kjer je 
+\begin_inset Formula $\tilde{A}$
+\end_inset
+
+ kofaktorska matrika: 
+\begin_inset Formula $\tilde{A_{ij}}=\det A_{ji}\left(-1\right)^{i+j}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Paragraph
+Algebrske strukture
+\end_layout
+
+\begin_layout Standard
+grupoid: 
+\begin_inset Formula $\left(M\not=\emptyset,\circ:\text{M\ensuremath{\times M\to M}}\right)$
+\end_inset
+
+, 
+\series bold
+polgrupa
+\series default
+ je asociativen grupoid, 
+\series bold
+monoid
+\series default
+ je polgrupa z enoto, 
+\series bold
+grupa 
+\series default
+je monoid z inverzom za vsak element, 
+\series bold
+abelova grupa
+\series default
+ je komutativna.
+\end_layout
+
+\begin_layout Standard
+Desna enota: 
+\begin_inset Formula $a\circ e=a$
+\end_inset
+
+.
+ Če je leva in desna, je enota.
+ Grupoid ima kvečjemu eno enoto.
+ Če je več levih, desne ni.
+\end_layout
+
+\begin_layout Standard
+Desni inverz: 
+\begin_inset Formula $a\circ a^{-1}=e$
+\end_inset
+
+.
+ Če je levi in desni, je inverz.
+ Inverz je enoličen.
+ V monoidu je levi tudi desni.
+\end_layout
+
+\begin_layout Standard
+Ko je 
+\begin_inset Formula $\left(M,\circ\right)$
+\end_inset
+
+ grupoid in 
+\begin_inset Formula $N\subset M,N\not=\emptyset$
+\end_inset
+
+, je 
+\begin_inset Formula $N$
+\end_inset
+
+ 
+\series bold
+podgrupoid
+\series default
+, če 
+\begin_inset Formula $\forall a,b\in N:a\circ b\in N$
+\end_inset
+
+.
+ 
+\begin_inset Formula $N$
+\end_inset
+
+ podeduje 
+\begin_inset Formula $\circ$
+\end_inset
+
+ v 
+\begin_inset Formula $\circ_{N}:N\times N\to N$
+\end_inset
+
+.
+ 
+\begin_inset Formula $\circ_{N}$
+\end_inset
+
+ ohrani komutativnost in asociativnost.
+ Enota se ne ohrani vedno, inverzi se ne ohranijo vedno.
+\end_layout
+
+\begin_layout Standard
+Ko je 
+\begin_inset Formula $\left(M,\circ\right)$
+\end_inset
+
+ polgrupa, 
+\begin_inset Formula $N$
+\end_inset
+
+ podgrupoid, je 
+\series bold
+
+\begin_inset Formula $N$
+\end_inset
+
+ podpolgrupa
+\series default
+.
+\end_layout
+
+\begin_layout Standard
+Ko je 
+\begin_inset Formula $\left(M,\circ\right)$
+\end_inset
+
+ monoid in 
+\begin_inset Formula $N$
+\end_inset
+
+ podgrupoid, je 
+\begin_inset Formula $N$
+\end_inset
+
+ 
+\series bold
+podmonoid
+\series default
+, če vsebuje enoto 
+\begin_inset Formula $\left(M,\circ\right)$
+\end_inset
+
+ (da, prav tisto).
+\end_layout
+
+\begin_layout Standard
+Ko je 
+\begin_inset Formula $\left(M,\circ\right)$
+\end_inset
+
+ grupa in 
+\begin_inset Formula $N$
+\end_inset
+
+ podmonoid, je 
+\begin_inset Formula $N$
+\end_inset
+
+ 
+\series bold
+podgrupa
+\series default
+, če vsebuje inverze vseh svojih elementov.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $N\not=\emptyset$
+\end_inset
+
+ je 
+\series bold
+podgrupa 
+\series default
+
+\begin_inset Formula $\left(M,\circ\right)$
+\end_inset
+
+, ko 
+\begin_inset Formula $a,b\in N\Rightarrow a\circ b^{-1}\in N$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $GL_{n}$
+\end_inset
+
+ je grupa vseh obrnljivih matrik z množenjem matrik, 
+\begin_inset Formula $O_{n}$
+\end_inset
+
+ je grupa matrik, kjer 
+\begin_inset Formula $A^{T}=A^{-1}$
+\end_inset
+
+ (ortogonalne), 
+\begin_inset Formula $SL_{n}$
+\end_inset
+
+ je grupa matrik z 
+\begin_inset Formula $\det A=1$
+\end_inset
+
+, 
+\begin_inset Formula $SO_{n}$
+\end_inset
+
+ je grupa ortogonalnih matrik z 
+\begin_inset Formula $\det A=1$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Paragraph
+Homomorfizem
+\end_layout
+
+\begin_layout Standard
+grupoidov in polgrup 
+\begin_inset Formula $\left(M_{1},\circ_{1}\right),\left(M_{2},\circ_{2}\right)$
+\end_inset
+
+ je 
+\begin_inset Formula $f:M_{1}\to M_{2}\ni:\forall a,b\in M_{1}:\left(f\left(a\circ_{1}b\right)=f\left(a\right)\circ_{2}f\left(b\right)\right)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Homomorfizem monoidov mora imeti še lastnost 
+\begin_inset Formula $f\left(e_{1}\right)=e_{2}$
+\end_inset
+
+, homomorfizem grup pa lastnost 
+\begin_inset Formula $f\left(a^{-1}\right)=f\left(a\right)^{-1}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Kompozitum homomorfizmov je homomorfizem.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Izomorfizem
+\series default
+ je bijektiven homomorfizem.
+ Med izomorfnima grupama obstaja izomorfizem.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\left(M,+,\cdot\right)$
+\end_inset
+
+ je 
+\series bold
+bigrupoid
+\series default
+, ko sta 
+\begin_inset Formula $\left(M,+\right)$
+\end_inset
+
+ in 
+\begin_inset Formula $\left(M,\cdot\right)$
+\end_inset
+
+ grupoida.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Distributiven bigrupoid 
+\series default
+ima 
+\series bold
+po eno
+\series default
+ L in D distributivnost in je 
+\series bold
+polkolobar
+\series default
+, če je 
+\begin_inset Formula $\left(M,+\right)$
+\end_inset
+
+ komutativna polgrupa.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Kolobar
+\series default
+ je distri.
+ bigrupoid, kjer je 
+\series bold
+
+\begin_inset Formula $\left(M,+\right)$
+\end_inset
+
+ 
+\series default
+abelova grupa.
+\end_layout
+
+\begin_layout Standard
+Pri 
+\series bold
+asociativnem kolobarju 
+\series default
+je 
+\begin_inset Formula $\left(M,\cdot\right)$
+\end_inset
+
+ polgrupa.
+ Lemut pravi, da je to pogoj že za kolobarje, Cimprič pa ne.
+\end_layout
+
+\begin_layout Standard
+Pri 
+\series bold
+asociativnem kolobarju z enoto 
+\series default
+je 
+\begin_inset Formula $\left(M,\cdot\right)$
+\end_inset
+
+ monoid.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Obseg
+\series default
+ je kolobar z enoto za množenje 
+\series bold
+
+\begin_inset Formula $1$
+\end_inset
+
+ 
+\series default
+in inverzom za množenje za vsak neničeln element (
+\begin_inset Formula $0$
+\end_inset
+
+ je enota za 
+\begin_inset Formula $+$
+\end_inset
+
+).
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Komutativen kolobar
+\series default
+ ima komutativno množenje.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Polje 
+\series default
+je komutativen obseg.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Podbigrupoid 
+\series default
+je 
+\begin_inset Formula $N\subset M$
+\end_inset
+
+, zaprta za 
+\begin_inset Formula $+$
+\end_inset
+
+ in 
+\begin_inset Formula $\cdot$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Podkolobar
+\series default
+ je 
+\begin_inset Formula $N\subset M$
+\end_inset
+
+, da je 
+\begin_inset Formula $N$
+\end_inset
+
+ podgrupa 
+\begin_inset Formula $\left(M,+\right)$
+\end_inset
+
+ in podgrupoid 
+\begin_inset Formula $\left(M,\cdot\right)$
+\end_inset
+
+ – 
+\begin_inset Formula $N$
+\end_inset
+
+ zaprta za odštevanje in množenje.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Podobseg
+\series default
+ je podkolobar, kjer je 
+\begin_inset Formula $N\backslash\left\{ 0\right\} $
+\end_inset
+
+ podgrupa 
+\begin_inset Formula $\left(M\backslash\left\{ 0\right\} ,\cdot\right)$
+\end_inset
+
+.
+ 
+\begin_inset Formula $0$
+\end_inset
+
+ namreč ni obrnljiva – 
+\begin_inset Formula $N$
+\end_inset
+
+
+ zaprta za 
+\begin_inset Formula $-$
+\end_inset
+
+ in deljenje.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Homomorfizem kolobarjev
+\series default
+ je 
+\begin_inset Formula $f:M_{1}\to M_{2}\ni:f\left(a+_{1}b\right)=f\left(a\right)+_{2}f\left(b\right)\wedge f\left(a\cdot_{1}b\right)=f\left(a\right)\cdot_{2}f\left(b\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Homomorfizem kolobarjev z enoto 
+\series default
+dodatno 
+\begin_inset Formula $f\left(1_{1}\right)=1_{2}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Vektorski prostor
+\end_layout
+
+\begin_layout Standard
+je Abelova grupa z množenjem s skalarjem.
+ 
+\begin_inset Formula $F$
+\end_inset
+
+ je polje, za prostor 
+\begin_inset Formula $\left(V,+,\cdot\right)$
+\end_inset
+
+ nad 
+\begin_inset Formula $F$
+\end_inset
+
+ velja:
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $\left(V,+\right)$
+\end_inset
+
+ je Abelova grupa
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $\alpha\cdot\left(a+b\right)=\alpha\cdot a+\alpha\cdot b,\quad\left(\alpha+\beta\right)\cdot a=\alpha\cdot a+\beta\cdot a$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $\left(\alpha\cdot\beta\right)\cdot a=\alpha\cdot\left(\beta\cdot a\right),\quad1\cdot a=a$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Direktna vsota vektorskih prostorov
+\series default
+ je vektorski prostor.
+ 
+\begin_inset Formula $V_{1}\oplus V_{2}$
+\end_inset
+
+ so pari 
+\begin_inset Formula $\left(v_{1},v_{2}\right)$
+\end_inset
+
+.
+ 
+\begin_inset Formula $\left(v_{1},v_{2}\right)+\left(v_{1}',v_{2}'\right)=\left(v_{1}+v_{1}',v_{2}+v_{2}'\right)$
+\end_inset
+
+, 
+\begin_inset Formula $\alpha\cdot\left(v_{1},v_{2}\right)=\left(\alpha\cdot v_{1},\alpha\cdot v_{2}\right)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Paragraph
+Vektorski podprostor
+\end_layout
+
+\begin_layout Standard
+je 
+\begin_inset Formula $W\subseteq V,W\not=\emptyset$
+\end_inset
+
+, zaprta za seštevanje in množenje s skalarjem.
+ Oziroma taka, da vsebuje vse svoje linearne kombinacije — 
+\begin_inset Formula $\forall a,b\in W\forall\alpha,\beta\in F:\alpha a+\beta b\in W$
+\end_inset
+
+.
+ Vsak podprostor vsebuje 0.
+ 
+\series bold
+Presek podprostorov
+\series default
+ je tudi sam podprostor.
+ 
+\series bold
+Vsota podprostorov
+\series default
+ (
+\begin_inset Formula $W_{1}+W_{2}=\left\{ w_{1}+w_{2};w_{1}\in W_{1},w_{2}\in W_{2}\right\} $
+\end_inset
+
+) je tudi sama podprostor.
+\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