Initial commit of some initial documentation
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## Core latex/pdflatex auxiliary files:
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*.aux
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*.lof
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*.log
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*.lot
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*.fls
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*.out
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*.toc
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*.fmt
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*.fot
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*.cb
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*.cb2
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.*.lb
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## Intermediate documents:
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*.dvi
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*.xdv
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*-converted-to.*
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# these rules might exclude image files for figures etc.
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# *.ps
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# *.eps
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# *.pdf
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## Bibliography auxiliary files (bibtex/biblatex/biber):
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*.bbl
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*.bcf
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*.blg
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*-blx.aux
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*-blx.bib
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*.run.xml
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## Build tool auxiliary files:
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*.fdb_latexmk
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*.synctex
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*.synctex(busy)
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*.synctex.gz
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*.synctex.gz(busy)
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*.pdfsync
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## Build tool directories for auxiliary files
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# latexrun
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latex.out/
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## Auxiliary and intermediate files from other packages:
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# algorithms
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*.alg
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*.loa
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# achemso
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acs-*.bib
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# amsthm
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*.thm
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# beamer
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*.nav
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*.pre
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*.snm
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*.vrb
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# changes
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*.soc
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# comment
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*.cut
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# cprotect
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*.cpt
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# elsarticle (documentclass of Elsevier journals)
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*.spl
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# endnotes
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*.ent
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# fixme
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*.lox
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# feynmf/feynmp
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*.mf
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*.mp
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*.t[1-9]
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*.t[1-9][0-9]
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*.tfm
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#(r)(e)ledmac/(r)(e)ledpar
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*.end
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*.?end
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*.[1-9]
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*.[1-9][0-9]
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*.[1-9][0-9][0-9]
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*.[1-9]R
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*.[1-9][0-9]R
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*.[1-9][0-9][0-9]R
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*.eledsec[1-9]
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*.eledsec[1-9]R
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*.eledsec[1-9][0-9]
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*.eledsec[1-9][0-9]R
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*.eledsec[1-9][0-9][0-9]
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*.eledsec[1-9][0-9][0-9]R
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# glossaries
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*.acn
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*.acr
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*.glg
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*.glo
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*.gls
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*.glsdefs
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*.lzo
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*.lzs
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*.slg
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*.slo
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*.sls
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# uncomment this for glossaries-extra (will ignore makeindex's style files!)
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# *.ist
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# gnuplot
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*.gnuplot
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*.table
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# gnuplottex
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*-gnuplottex-*
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# gregoriotex
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*.gaux
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*.glog
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*.gtex
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# htlatex
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*.4ct
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*.4tc
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*.idv
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*.lg
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*.trc
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*.xref
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# hyperref
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*.brf
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# knitr
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*-concordance.tex
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# TODO Uncomment the next line if you use knitr and want to ignore its generated tikz files
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# *.tikz
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*-tikzDictionary
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# listings
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*.lol
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# luatexja-ruby
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*.ltjruby
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# makeidx
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*.idx
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*.ilg
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*.ind
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# minitoc
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*.maf
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*.mlf
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*.mlt
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*.mtc[0-9]*
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*.slf[0-9]*
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*.slt[0-9]*
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*.stc[0-9]*
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# minted
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_minted*
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*.pyg
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# morewrites
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*.mw
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# newpax
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*.newpax
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# nomencl
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*.nlg
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*.nlo
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*.nls
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# pax
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*.pax
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# pdfpcnotes
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*.pdfpc
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# sagetex
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*.sagetex.sage
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*.sagetex.py
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*.sagetex.scmd
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# scrwfile
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*.wrt
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# svg
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svg-inkscape/
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# sympy
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*.sout
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*.sympy
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sympy-plots-for-*.tex/
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# pdfcomment
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*.upa
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*.upb
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# pythontex
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*.pytxcode
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pythontex-files-*/
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# tcolorbox
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*.listing
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# thmtools
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*.loe
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# TikZ & PGF
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*.dpth
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*.md5
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*.auxlock
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# titletoc
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*.ptc
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# todonotes
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*.tdo
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# vhistory
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*.hst
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*.ver
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# easy-todo
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*.lod
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# xcolor
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*.xcp
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# xmpincl
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*.xmpi
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# xindy
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*.xdy
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# xypic precompiled matrices and outlines
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*.xyc
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*.xyd
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# endfloat
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*.ttt
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*.fff
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||||
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# Latexian
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TSWLatexianTemp*
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||||
## Editors:
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||||
# WinEdt
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*.bak
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*.sav
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||||
# Texpad
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||||
.texpadtmp
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||||
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||||
# LyX
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*.lyx~
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||||
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||||
# Kile
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||||
*.backup
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||||
# gummi
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||||
.*.swp
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# KBibTeX
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||||
*~[0-9]*
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# TeXnicCenter
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*.tps
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# auto folder when using emacs and auctex
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./auto/*
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*.el
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# expex forward references with \gathertags
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||||
*-tags.tex
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||||
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# standalone packages
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||||
*.sta
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||||
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# Makeindex log files
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||||
*.lpz
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||||
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||||
# xwatermark package
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||||
*.xwm
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||||
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||||
# REVTeX puts footnotes in the bibliography by default, unless the nofootinbib
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# option is specified. Footnotes are the stored in a file with suffix Notes.bib.
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# Uncomment the next line to have this generated file ignored.
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||||
#*Notes.bib
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# Draw.io backup files
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*.bkp
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*.dtmp
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\documentclass[12pt,letterpaper]{report}
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\usepackage{init}
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\usepackage{import}
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\usepackage{dsfont}
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\newcommand{\cond}[1]{\text{cond}\left(#1\right)}
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\newcommand{\linear}[1]{\mathcal{L}\left(#1\right)}
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\newcommand{\Id}{\mathds{I}}
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\newcommand{\per}[1]{\text{per}\left(#1\right)}
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\setcounter{chapter}{-1}
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\author{Alexander J. Clarke}
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\title{Linear Algebra Theorems and Definitions}
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\begin{document}
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\maketitle
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\clearpage
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\begin{center}
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\thispagestyle{empty}
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\vspace*{\fill}
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All theorems, corollaries, lemmas, remarks, and asides are direct quotes from Linear Algebra, 4th Edition, by Stephen H. Friedberg, Arnold J. Insel, and Lawrence E. Spence
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\vspace*{\fill}
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\end{center}
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\tableofcontents
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\import{./}{chapter-0.tex}
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\import{chapter-1/}{chapter-1.tex}
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\import{chapter-2/}{chapter-2.tex}
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\import{chapter-3/}{chapter-3.tex}
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\import{chapter-4/}{chapter-4.tex}
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\import{chapter-5/}{chapter-5.tex}
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\import{chapter-6/}{chapter-6.tex}
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\import{chapter-7/}{chapter-7.tex}
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\end{document}
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\chapter{List of Symbols}
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\begin{align*}
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& A_{ij} & \text{the $ij$-th entry of the matrix $A$} \\
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& A^{-1} & \text{the inverse of the matrix $A$} \\
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& A^\dag & \text{the pseudoinverse of the matrix $A$} \\
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& A^* & \text{the adjoint of the matrix $A$} \\
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& \tilde{A}_{ij} & \text{the matrix $A$ with row $i$ and column $j$ deleted} \\
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& A^t & \text{the transpose of the matrix $A$} \\
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& (A|B) & \text{the matrix $A$ augmented by the matrix $B$} \\
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& B_1 \bigoplus \dots \bigoplus B_k & \text{the direct sum of matrices $B_1$ through $B_k$} \\
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& \mathcal{B}(V) & \text{the set of bilinear forms on $V$} \\
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& \beta^* & \text{the dual basis of $\beta$} \\
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& \beta_x & \text{the $T$-cyclic basis generated by $x$} \\
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& \C & \text{the field of complex numbers} \\
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& \C_i & \text{the $i$th Gerschgorin disk} \\
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& \cond{A} & \text{the condition number of the matrix $A$} \\
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& C^n(\R) & \text{set of functions $f$ on $\R$ with $f^{(n)}$ continuous} \\
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& C^\infty & \text{set of functions with derivatives of every order} \\
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& C(\R) & \text{the vector space of continuous functions on $\R$} \\
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& C([0,1]) & \text{the vector space of continuous functions on $[0,1]$} \\
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& C_x & \text{the $T$-cyclic subspaces generated by $x$} \\
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& D & \text{the derivative operator on $C^\infty$} \\
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& \ldet{A} & \text{the determinant of the matrix $A$} \\
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& \delta_{ij} & \text{the Kronecker delta} \\
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& \ldim{V} & \text{the dimension of $V$} \\
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& e^A & \lim_{m \to \infty} \left(I + A + \frac{A^2}{2!} + \dots + \frac{A^m}{m!}\right) \\
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& e_i & \text{the $i$th standard vector of $\F^n$} \\
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\end{align*}
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\begin{align*}
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& E_\lambda & \text{the eigenspace of $T$ corresponding to $\lambda$} \\
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& \F & \text{a field} \\
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& f(A) & \text{the polynomial $f(x)$ evaluated at the matrix $A$} \\
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& F^n & \text{the set of $n$-tuples with entries in a field $\F$} \\
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& f(T) & \text{the polynomial $f(x)$ evaluated at the operator $T$} \\
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& \mathcal{F}(S,\F) & \text{the set of functions from $S$ to a field $\F$} \\
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& H & \text{space of continuous complex functions on $[0, 2\pi]$} \\
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& I_n \text{ or } I & \text{the $n \times n$ identity matrix} \\
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& \Id_V \text{ or } \Id & \text{the identity operator on $V$} \\
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& K_\lambda & \text{generalized eigenspace of $T$ corresponding to $\lambda$} \\
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& K_\phi & \{x\ |\ (\phi(T))^p(x) = 0 \text{, for some positive integer $p$}\} \\
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& L_A & \text{left-multiplication transformation by matrix $A$} \\
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& \lim_{m \to \infty}A_m & \text{the limit of a sequence of matrices} \\
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& \linear{V} & \text{the space of linear transformations from $V$ to $V$} \\
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& \linear{V, W} & \text{the space of linear transformations from $V$ to $W$} \\
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& M_{m \times n}(\F) & \text{the set of $m \times n$ matrices with entries in $\F$} \\
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& v(A) & \text{the column sum of the matrix $A$} \\
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& v_j(A) & \text{the $j$th column sum of the matrix $A$} \\
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& N(T) & \text{the null space of $T$} \\
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& \nullity{T} & \text{the dimension of the null space of $T$} \\
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& O & \text{the zero matrix} \\
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& \per{M} & \text{the permanent of the $2 \times 2$ matrix $M$} \\
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& P(\F) & \text{the space of polynomials with coefficients in $\F$} \\
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||||
& P_n(\F) & \text{the polynomials in $P(\F)$ of degree at most $n$} \\
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||||
& \phi_\beta & \text{the standard representation with respect to basis $\beta$} \\
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& \R & \text{the field of real numbers} \\
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& \rank{A} & \text{the rank of the matrix $A$} \\
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& \rank{T} & \text{the rank of the linear transformation $T$} \\
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& \rho(A) & \text{the row sum of the matrix $A$} \\
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||||
& \rho_i(A) & \text{the $i$th row sum of the matrix $A$} \\
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||||
& R(T) & \text{the range of the linear transformation $T$} \\
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||||
& S_1 + S_2 & \text{the sum of sets $S_1$ and $S_2$} \\
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||||
& \lspan{S} & \text{the span of the set $S$} \\
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||||
& S^\perp & \text{the orthogonal complement of the set $S$} \\
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||||
& [T]_\beta & \text{the matrix representation of $T$ in basis $\beta$} \\
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||||
& [T]_\beta^\gamma & \text{the matrix representation of $T$ in bases $\beta$ and $\gamma$} \\
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||||
& T^{-1} & \text{the inverse of the linear transformation $T$} \\
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||||
\end{align*}
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\begin{align*}
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& T^\dag & \text{the pseudoinverse of the linear transformation $T$} \\
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||||
& T^* & \text{the adjoint of the linear operator $T$} \\
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||||
& T_0 & \text{the zero transformation} \\
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||||
& T^t & \text{the transpose of the linear transformation $T$} \\
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||||
& T_\theta & \text{the rotation transformation by $\theta$} \\
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||||
& T_W & \text{the restriction of $T$ to a subspace $W$} \\
|
||||
& \ltr{A} & \text{the trace of the matrix $A$} \\
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||||
& V^* & \text{the dual space of the vector space $V$} \\
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||||
& V/W & \text{the quotient space of $V$ modulo $W$} \\
|
||||
& W_1 + \dots + W_k & \text{the sum of subspaces $W_1$ through $W_k$} \\
|
||||
& \sum_{i=1}^k W_i & \text{the sum of subspaces $W_i$ through $W_k$} \\
|
||||
& W_1 \bigoplus W_2 & \text{the direct sum of subspaces $W_1$ and $W_2$} \\
|
||||
& W_1 \bigoplus \dots \bigoplus W_k & \text{the direct sum of subspaces $W_1$ through $W_k$} \\
|
||||
& \norm{x} & \text{the norm of the vector $\vec{x}$} \\
|
||||
& [x]_\beta & \text{the coordinate vector of $x$ relative to $\beta$} \\
|
||||
& \langle x, y \rangle & \text{the inner product of $\vec{x}$ and $\vec{y}$} \\
|
||||
& \Z_2 & \text{the field consisting of $0$ and $1$} \\
|
||||
& \overline{\vec{z}} & \text{the complex conjugate of $\vec{z}$} \\
|
||||
& \vec{0} & \text{the zero vector} \\
|
||||
\end{align*}
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||||
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||||
\section{Bases and Dimension}
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||||
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||||
\chapter{Vector Spaces}
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||||
\subimport{./}{introduction.tex}
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||||
\subimport{./}{vector-spaces.tex}
|
||||
\subimport{./}{subspaces.tex}
|
||||
\subimport{./}{linear-combinations-and-systems-of-linear-equations.tex}
|
||||
\subimport{./}{linear-dependence-and-linear-independence.tex}
|
||||
\subimport{./}{bases-and-dimension.tex}
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||||
\subimport{./}{maximal-linearly-independent-subsets.tex}
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||||
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||||
\section{Introduction}
|
||||
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||||
\section{Linear Combinations and Systems of Linear Equations}
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||||
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||||
\section{Linear Dependence and Linear Independence}
|
||||
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||||
\section{Maximal Linearly Independent Subsets}
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||||
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||||
\section{Subspaces}
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||||
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||||
\section{Vector Spaces}
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||||
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||||
\chapter{Linear Transformations and Matrices}
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||||
\subimport{./}{linear-transformations-null-spaces-and-ranges.tex}
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||||
\subimport{./}{the-matrix-representation-of-a-linear-transformation.tex}
|
||||
\subimport{./}{composition-of-linear-transformations-and-matrix-multiplication.tex}
|
||||
\subimport{./}{invertibility-and-isomorphisms.tex}
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||||
\subimport{./}{the-change-of-coordinate-matrix.tex}
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||||
\subimport{./}{dual-spaces.tex}
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||||
\subimport{./}{homogeneous-linear-differential-equations-with-constant-coefficients.tex}
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||||
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||||
\section{Compositions of Linear Transformations and Matrix Multiplication}
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||||
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||||
\section{Dual Spaces}
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||||
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||||
\section{Homogeneous Linear Differential Equations with Constant Coefficients}
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||||
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||||
\section{Invertibility and Isomorphisms}
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||||
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||||
\section{Linear Transformations, Null Spaces, and Ranges}
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||||
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||||
\section{The Change of Coordinate Matrix}
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||||
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||||
\section{The Matrix Representation of a Linear Transformation}
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||||
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||||
\chapter{Elementary Matrix Operations and Systems of Linear Equations}
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||||
\subimport{./}{elementary-matrix-operations-and-elementary-matrices.tex}
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||||
\subimport{./}{the-rank-of-a-matrix-and-matrix-inverses.tex}
|
||||
\subimport{./}{systems-of-linear-equations-theoretical-aspects.tex}
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||||
\subimport{./}{systems-of-linear-equations-computational-aspects.tex}
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|
||||
\section{Elementary Matrix Operations and Elementary Matrices}
|
||||
@@ -0,0 +1 @@
|
||||
\section{Systems of Linear Equations -- Computational Aspects}
|
||||
@@ -0,0 +1 @@
|
||||
\section{Systems of Linear Equations -- Theoretical Aspects}
|
||||
@@ -0,0 +1 @@
|
||||
\section{The Rank of a Matrix and Matrix Inverses}
|
||||
@@ -0,0 +1 @@
|
||||
\section{A Characterization of the Determinant}
|
||||
@@ -0,0 +1,6 @@
|
||||
\chapter{Determinants}
|
||||
\subimport{./}{determinants-of-order-2.tex}
|
||||
\subimport{./}{determinants-of-order-n.tex}
|
||||
\subimport{./}{properties-of-determinants.tex}
|
||||
\subimport{./}{summary-important-facts-about-determinants.tex}
|
||||
\subimport{./}{a-characterization-of-the-determinant.tex}
|
||||
@@ -0,0 +1 @@
|
||||
\section{Determinants of Order 2}
|
||||
@@ -0,0 +1 @@
|
||||
\section{Determinants of Order $n$}
|
||||
@@ -0,0 +1 @@
|
||||
\section{Properties of Determinants}
|
||||
@@ -0,0 +1 @@
|
||||
\section{Summary -- Important Facts about Determinants}
|
||||
@@ -0,0 +1,5 @@
|
||||
\chapter{Diagonalization}
|
||||
\subimport{./}{eigenvalues-and-eigenvectors.tex}
|
||||
\subimport{./}{diagonalizability.tex}
|
||||
\subimport{./}{matrix-limits-and-markov-chains.tex}
|
||||
\subimport{./}{invariant-subspaces-and-the-cayley-hamilton-theorem.tex}
|
||||
@@ -0,0 +1 @@
|
||||
\section{Diagonalizability}
|
||||
@@ -0,0 +1 @@
|
||||
\section{Eigenvalues and Eigenvectors}
|
||||
@@ -0,0 +1 @@
|
||||
\section{Invariant Subspaces and the Cayley-Hamilton Theorem}
|
||||
@@ -0,0 +1 @@
|
||||
\section{Matrix Limits and Markov Chains}
|
||||
@@ -0,0 +1 @@
|
||||
\section{Bilinear and Quadratic Forms}
|
||||
@@ -0,0 +1,12 @@
|
||||
\chapter{Inner Product Spaces}
|
||||
\subimport{./}{inner-products-and-norms.tex}
|
||||
\subimport{./}{the-gram-schmidt-orthogonalization-process-and-orthogonal-complements.tex}
|
||||
\subimport{./}{the-adjoint-of-a-linear-operator.tex}
|
||||
\subimport{./}{normal-and-self-adjoint-operators.tex}
|
||||
\subimport{./}{unitary-and-orthogonal-operators-and-their-matrices.tex}
|
||||
\subimport{./}{orthogonal-projections-and-the-spectral-theorem.tex}
|
||||
\subimport{./}{the-singular-value-decomposition-and-the-pseudoinverse.tex}
|
||||
\subimport{./}{bilinear-and-quadratic-forms.tex}
|
||||
\subimport{./}{einsteins-special-theory-of-relativity.tex}
|
||||
\subimport{./}{conditioning-and-the-rayleigh-quotient.tex}
|
||||
\subimport{./}{the-geometry-of-orthogonal-operators.tex}
|
||||
@@ -0,0 +1 @@
|
||||
\section{Conditioning and the Rayleigh Quotient}
|
||||
@@ -0,0 +1 @@
|
||||
\section{Einstein's Special Theory of Relativity}
|
||||
@@ -0,0 +1 @@
|
||||
\section{Inner Products and Norms}
|
||||
@@ -0,0 +1 @@
|
||||
\section{Normal and Self-Adjoing Operators}
|
||||
@@ -0,0 +1 @@
|
||||
\section{Orthogonal Projections and the Spectral Theorem}
|
||||
@@ -0,0 +1 @@
|
||||
\section{The Adjoint of a Linear Operator}
|
||||
@@ -0,0 +1 @@
|
||||
\section{The Geometry of Orthogonal Operators}
|
||||
@@ -0,0 +1 @@
|
||||
\section{The Gram-Schmidt Orthogonalization Process and Orthogonal Complements}
|
||||
@@ -0,0 +1 @@
|
||||
\section{The Singular Value Decomposition and the Pseudoinverse}
|
||||
@@ -0,0 +1 @@
|
||||
\section{Unitary and Orthogonal Operators and Their Matrices}
|
||||
@@ -0,0 +1,5 @@
|
||||
\chapter{Canonical Forms}
|
||||
\subimport{./}{the-jordan-canonical-form-i.tex}
|
||||
\subimport{./}{the-jordan-canonical-form-ii.tex}
|
||||
\subimport{./}{the-minimal-polynomial.tex}
|
||||
\subimport{./}{the-rational-canonical-form.tex}
|
||||
@@ -0,0 +1 @@
|
||||
\section{The Jordan Canonical Form I}
|
||||
@@ -0,0 +1 @@
|
||||
\section{The Jordan Canonical Form II}
|
||||
@@ -0,0 +1 @@
|
||||
\section{The Minimal Polynomial}
|
||||
@@ -0,0 +1 @@
|
||||
\section{The rational Canonical Form}
|
||||
@@ -0,0 +1,59 @@
|
||||
\ProvidesPackage{init}
|
||||
|
||||
\usepackage{import}
|
||||
\usepackage[utf8]{inputenc}
|
||||
\usepackage{pgfplots}
|
||||
\usepackage[english]{babel}
|
||||
\usepackage{amsthm}
|
||||
\usepackage{thmtools}
|
||||
\usepackage{hyperref}
|
||||
\usepackage{cancel}
|
||||
\usepackage{mathtools}
|
||||
\usepackage{amsmath}
|
||||
\usepackage{amsfonts}
|
||||
\usepackage{amssymb}
|
||||
\usepackage{graphicx}
|
||||
\usepackage{relsize}
|
||||
\usepackage{listings}
|
||||
\graphicspath{ {./images/} }
|
||||
\usepackage{array}
|
||||
\usepackage{tikz}
|
||||
\usetikzlibrary{arrows}
|
||||
\usepackage[left=2cm, right=2.5cm, top=2.5cm, bottom=2.5cm]{geometry}
|
||||
\usepackage{enumitem}
|
||||
\usepackage{mathrsfs}
|
||||
|
||||
% Math Functions
|
||||
\newcommand{\limx}[2]{\displaystyle\lim\limits_{#1 \to #2}}
|
||||
\newcommand{\st}{\ \text{s.t.}\ }
|
||||
\newcommand{\abs}[1]{\left\lvert #1 \right\rvert}
|
||||
\newcommand{\dotp}{\dot{\mathcal{P}}}
|
||||
\newcommand{\dotq}{\dot{\mathcal{Q}}}
|
||||
\newcommand{\Int}[1]{\text{int}\left(#1\right)}
|
||||
\newcommand{\cl}[1]{\text{cl}\left(#1\right)}
|
||||
\newcommand{\bd}[1]{\text{bd}\left(#1\right)}
|
||||
\newcommand{\lr}[1]{\left(#1\right)}
|
||||
\newcommand{\lspan}[1]{\text{span}\left(#1\right)}
|
||||
\newcommand{\ldim}[1]{\text{dim}\left(#1\right)}
|
||||
\newcommand{\nullity}[1]{\text{nullity}\left(#1\right)}
|
||||
\newcommand{\rank}[1]{\text{rank}\left(#1\right)}
|
||||
\newcommand{\ldet}[1]{\text{det}\left(#1\right)}
|
||||
\newcommand{\ltr}[1]{\text{tr}\left(#1\right)}
|
||||
\newcommand{\norm}[1]{\left\lVert#1\right\rVert}
|
||||
\DeclareMathOperator{\sign}{sgn}
|
||||
\renewcommand{\qedsymbol}{$\blacksquare$}
|
||||
|
||||
% Special Sets
|
||||
\newcommand{\R}{\mathbb{R}}
|
||||
\newcommand{\N}{\mathbb{N}}
|
||||
\newcommand{\Q}{\mathbb{Q}}
|
||||
\newcommand{\C}{\mathbb{C}}
|
||||
\newcommand{\Z}{\mathbb{Z}}
|
||||
\newcommand{\F}{\mathbb{F}}
|
||||
|
||||
% Theorem Styles
|
||||
\declaretheorem[numberwithin=subsection, style=definition]{theorem, definition, notation, lemma, corollary, remark, example}
|
||||
|
||||
% Formatting
|
||||
\setlist[enumerate]{font=\bfseries}
|
||||
|
||||
Reference in New Issue
Block a user