\documentclass[10pt,a4paper]{article} % Packages \usepackage{fancyhdr} % For header and footer \usepackage{multicol} % Allows multicols in tables \usepackage{tabularx} % Intelligent column widths \usepackage{tabulary} % Used in header and footer \usepackage{hhline} % Border under tables \usepackage{graphicx} % For images \usepackage{xcolor} % For hex colours %\usepackage[utf8x]{inputenc} % For unicode character support \usepackage[T1]{fontenc} % Without this we get weird character replacements \usepackage{colortbl} % For coloured tables \usepackage{setspace} % For line height \usepackage{lastpage} % Needed for total page number \usepackage{seqsplit} % Splits long words. %\usepackage{opensans} % Can't make this work so far. Shame. Would be lovely. \usepackage[normalem]{ulem} % For underlining links % Most of the following are not required for the majority % of cheat sheets but are needed for some symbol support. \usepackage{amsmath} % Symbols \usepackage{MnSymbol} % Symbols \usepackage{wasysym} % Symbols %\usepackage[english,german,french,spanish,italian]{babel} % Languages % Document Info \author{mk\_mk\_mk} \pdfinfo{ /Title (algebra-and-geometry-class-27.pdf) /Creator (Cheatography) /Author (mk\_mk\_mk) /Subject (Algebra and Geometry - Class 27 Cheat Sheet) } % Lengths and widths \addtolength{\textwidth}{6cm} \addtolength{\textheight}{-1cm} \addtolength{\hoffset}{-3cm} \addtolength{\voffset}{-2cm} \setlength{\tabcolsep}{0.2cm} % Space between columns \setlength{\headsep}{-12pt} % Reduce space between header and content \setlength{\headheight}{85pt} % If less, LaTeX automatically increases it \renewcommand{\footrulewidth}{0pt} % Remove footer line \renewcommand{\headrulewidth}{0pt} % Remove header line \renewcommand{\seqinsert}{\ifmmode\allowbreak\else\-\fi} % Hyphens in seqsplit % This two commands together give roughly % the right line height in the tables \renewcommand{\arraystretch}{1.3} \onehalfspacing % Commands \newcommand{\SetRowColor}[1]{\noalign{\gdef\RowColorName{#1}}\rowcolor{\RowColorName}} % Shortcut for row colour \newcommand{\mymulticolumn}[3]{\multicolumn{#1}{>{\columncolor{\RowColorName}}#2}{#3}} % For coloured multi-cols \newcolumntype{x}[1]{>{\raggedright}p{#1}} % New column types for ragged-right paragraph columns \newcommand{\tn}{\tabularnewline} % Required as custom column type in use % Font and Colours \definecolor{HeadBackground}{HTML}{333333} \definecolor{FootBackground}{HTML}{666666} \definecolor{TextColor}{HTML}{333333} \definecolor{DarkBackground}{HTML}{8FC9BA} \definecolor{LightBackground}{HTML}{F1F8F6} \renewcommand{\familydefault}{\sfdefault} \color{TextColor} % Header and Footer \pagestyle{fancy} \fancyhead{} % Set header to blank \fancyfoot{} % Set footer to blank \fancyhead[L]{ \noindent \begin{multicols}{3} \begin{tabulary}{5.8cm}{C} \SetRowColor{DarkBackground} \vspace{-7pt} {\parbox{\dimexpr\textwidth-2\fboxsep\relax}{\noindent \hspace*{-6pt}\includegraphics[width=5.8cm]{/web/www.cheatography.com/public/images/cheatography_logo.pdf}} } \end{tabulary} \columnbreak \begin{tabulary}{11cm}{L} \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{Algebra and Geometry - Class 27 Cheat Sheet}}}} \\ \normalsize{by \textcolor{DarkBackground}{mk\_mk\_mk} via \textcolor{DarkBackground}{\uline{cheatography.com/209409/cs/45002/}}} \end{tabulary} \end{multicols}} \fancyfoot[L]{ \footnotesize \noindent \begin{multicols}{3} \begin{tabulary}{5.8cm}{LL} \SetRowColor{FootBackground} \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}} \\ \vspace{-2pt}mk\_mk\_mk \\ \uline{cheatography.com/mk-mk-mk} \\ \end{tabulary} \vfill \columnbreak \begin{tabulary}{5.8cm}{L} \SetRowColor{FootBackground} \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}} \\ \vspace{-2pt}Not Yet Published.\\ Updated 15th November, 2024.\\ Page {\thepage} of \pageref{LastPage}. \end{tabulary} \vfill \columnbreak \begin{tabulary}{5.8cm}{L} \SetRowColor{FootBackground} \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Sponsor}} \\ \SetRowColor{white} \vspace{-5pt} %\includegraphics[width=48px,height=48px]{dave.jpeg} Measure your website readability!\\ www.readability-score.com \end{tabulary} \end{multicols}} \begin{document} \raggedright \raggedcolumns % Set font size to small. Switch to any value % from this page to resize cheat sheet text: % www.emerson.emory.edu/services/latex/latex_169.html \footnotesize % Small font. \begin{multicols*}{3} \begin{tabularx}{5.377cm}{X} \SetRowColor{DarkBackground} \mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{SECTION 1}} \tn % Row 0 \SetRowColor{LightBackground} \mymulticolumn{1}{x{5.377cm}}{{\bf{{\emph{C}}}}, Fields, Lists, {\bf{{\emph{F\textasciicircum{}n\textasciicircum{}}}}}, Vector Spaces} \tn % Row Count 1 (+ 1) \hhline{>{\arrayrulecolor{DarkBackground}}-} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{5.377cm}{X} \SetRowColor{DarkBackground} \mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Definition of {\bf{{\emph{complex numbers}}}} (n∈{\bf{{\emph{C}}}})}} \tn % Row 0 \SetRowColor{LightBackground} \mymulticolumn{1}{x{5.377cm}}{{\bf{{\emph{C}}}} = \{ a+bi : a,b ∈ {\bf{{\emph{R}}}} \}} \tn % Row Count 1 (+ 1) % Row 1 \SetRowColor{white} \mymulticolumn{1}{x{5.377cm}}{{\bf{{\emph{addition in C}}}} (a+bi) + (c+di) = (a+c) + (b+d)i} \tn % Row Count 3 (+ 2) % Row 2 \SetRowColor{LightBackground} \mymulticolumn{1}{x{5.377cm}}{{\bf{{\emph{multiplication in C}}}} (a+bi)(c+di) = (ac-bd) + (ad+bc)i} \tn % Row Count 5 (+ 2) \hhline{>{\arrayrulecolor{DarkBackground}}-} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{5.377cm}{X} \SetRowColor{DarkBackground} \mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Properties of Fields}} \tn % Row 0 \SetRowColor{LightBackground} \mymulticolumn{1}{x{5.377cm}}{{\bf{commutativity}} a+b = b+a ∧ ab = ba ∀a,b ∈ {\bf{F}}} \tn % Row Count 2 (+ 2) % Row 1 \SetRowColor{white} \mymulticolumn{1}{x{5.377cm}}{{\bf{associativity}} (a+b)+c = a+(b+c) ∧ (ab)c = a(bc) ∀a,b,c ∈ {\bf{F}}} \tn % Row Count 4 (+ 2) % Row 2 \SetRowColor{LightBackground} \mymulticolumn{1}{x{5.377cm}}{{\bf{identities}} c+0 = c ∧ 1c = c ∀c ∈ {\bf{F}}} \tn % Row Count 6 (+ 2) % Row 3 \SetRowColor{white} \mymulticolumn{1}{x{5.377cm}}{{\bf{additive inverse}} ∀a ∈ {\bf{F}} ∃b ∈ {\bf{F}} : a+b = 0} \tn % Row Count 8 (+ 2) % Row 4 \SetRowColor{LightBackground} \mymulticolumn{1}{x{5.377cm}}{{\bf{multiplicative inverse}} a ∈ {\bf{F}}, a ≠ 0, ∃b ∈ {\bf{F}} : a+b = 0} \tn % Row Count 10 (+ 2) % Row 5 \SetRowColor{white} \mymulticolumn{1}{x{5.377cm}}{{\bf{distributive property}} c(a+b) = ca + cb ∀a,b,c ∈ {\bf{F}}} \tn % Row Count 12 (+ 2) \hhline{>{\arrayrulecolor{DarkBackground}}-} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{5.377cm}{X} \SetRowColor{DarkBackground} \mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Definition of {\bf{{\emph{list L with length n}}}}}} \tn % Row 0 \SetRowColor{LightBackground} \mymulticolumn{1}{x{5.377cm}}{n∈{\bf{{\emph{N}}}}, {\bf{L}} : \{ 1,2,…,n \} → Elements} \tn % Row Count 1 (+ 1) % Row 1 \SetRowColor{white} \mymulticolumn{1}{x{5.377cm}}{{\bf{L}} = (x\textasciitilde{}1\textasciitilde{},...,x\textasciitilde{}n\textasciitilde{})} \tn % Row Count 2 (+ 1) \hhline{>{\arrayrulecolor{DarkBackground}}-} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{5.377cm}{X} \SetRowColor{DarkBackground} \mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Definition of {\bf{{\emph{F\textasciicircum{}n\textasciicircum{}}}}}}} \tn % Row 0 \SetRowColor{LightBackground} \mymulticolumn{1}{x{5.377cm}}{{\bf{{\emph{F\textasciicircum{}n\textasciicircum{}}}}} = \{ (x\textasciitilde{}1\textasciitilde{},...,x\textasciitilde{}{\emph{n}}\textasciitilde{}) : x\textasciitilde{}j\textasciitilde{} ∈ {\bf{{\emph{F}}}} for j = 1,...,{\emph{n}}\}} \tn % Row Count 2 (+ 2) % Row 1 \SetRowColor{white} \mymulticolumn{1}{x{5.377cm}}{{\bf{{\emph{addition in F\textasciicircum{}n\textasciicircum{}}}}} (x\textasciitilde{}1\textasciitilde{},...,x\textasciitilde{}{\emph{n}}\textasciitilde{}) + (y\textasciitilde{}1\textasciitilde{},...,y\textasciitilde{}{\emph{n}}\textasciitilde{}) = (x\textasciitilde{}1\textasciitilde{}+y\textasciitilde{}1\textasciitilde{},...,x\textasciitilde{}{\emph{n}}\textasciitilde{}+y\textasciitilde{}{\emph{n}}\textasciitilde{})} \tn % Row Count 4 (+ 2) % Row 2 \SetRowColor{LightBackground} \mymulticolumn{1}{x{5.377cm}}{{\bf{{\emph{zero vector}}}} {\emph{0}} = (0,...,0)} \tn % Row Count 5 (+ 1) % Row 3 \SetRowColor{white} \mymulticolumn{1}{x{5.377cm}}{{\bf{{\emph{additive inverse}}}} For {\emph{x}} ∈ {\bf{{\emph{F\textasciicircum{}n\textasciicircum{}}}}}, {\emph{-x}} ∈ {\bf{{\emph{F\textasciicircum{}n\textasciicircum{}}}}} = (-x\textasciitilde{}1\textasciitilde{},...,-x\textasciitilde{}{\emph{n}}\textasciitilde{})} \tn % Row Count 7 (+ 2) % Row 4 \SetRowColor{LightBackground} \mymulticolumn{1}{x{5.377cm}}{{\bf{{\emph{scalar multiplication in F\textasciicircum{}n\textasciicircum{}}}}} c ∈ {\bf{{\emph{F}}}}, c(x\textasciitilde{}1\textasciitilde{},...,x\textasciitilde{}{\emph{n}}\textasciitilde{}) = (cx\textasciitilde{}1\textasciitilde{},...,cx\textasciitilde{}{\emph{n}}\textasciitilde{})} \tn % Row Count 9 (+ 2) \hhline{>{\arrayrulecolor{DarkBackground}}-} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{5.377cm}{X} \SetRowColor{DarkBackground} \mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Definition of addition and scalar multiplication}} \tn % Row 0 \SetRowColor{LightBackground} \mymulticolumn{1}{x{5.377cm}}{{\bf{{\emph{addition on a set V}}}} {\emph{+}} : ({\emph{u}},{\emph{v}} ∈ {\bf{{\emph{V}}}}) → {\emph{u}} +{\emph{v}}} \tn % Row Count 2 (+ 2) % Row 1 \SetRowColor{white} \mymulticolumn{1}{x{5.377cm}}{{\bf{{\emph{scalar multiplication on a set V}}}} {\emph{*}} : (c ∈ {\bf{{\emph{F}}}}, {\emph{v}} ∈ {\bf{{\emph{V}}}}) → c{\emph{v}}} \tn % Row Count 4 (+ 2) \hhline{>{\arrayrulecolor{DarkBackground}}-} \end{tabularx} \par\addvspace{1.3em} \begin{tabularx}{5.377cm}{X} \SetRowColor{DarkBackground} \mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Defintion of Vector Space}} \tn % Row 0 \SetRowColor{LightBackground} \mymulticolumn{1}{x{5.377cm}}{Vector Space = {\emph{V}} + ({\emph{+}}) + ({\emph{*}})} \tn % Row Count 1 (+ 1) % Row 1 \SetRowColor{white} \mymulticolumn{1}{x{5.377cm}}{{\bf{commutativity}} {\emph{u}}+{\emph{v}} = {\emph{v}}+{\emph{u}}} \tn % Row Count 2 (+ 1) % Row 2 \SetRowColor{LightBackground} \mymulticolumn{1}{x{5.377cm}}{{\bf{associativity}} ({\emph{u}}+{\emph{v}})+{\emph{w}} = {\emph{u}}+({\emph{v}}+{\emph{w}}) ∧ (ab){\emph{v}} = a(b{\emph{v}})} \tn % Row Count 4 (+ 2) % Row 3 \SetRowColor{white} \mymulticolumn{1}{x{5.377cm}}{{\bf{additive identity}} ∃{\emph{0}} ∈ {\emph{V}} : {\emph{v}}+{\emph{0}} = {\emph{v}}} \tn % Row Count 6 (+ 2) % Row 4 \SetRowColor{LightBackground} \mymulticolumn{1}{x{5.377cm}}{{\bf{additive inverse}} ∀{\emph{v}} ∈ {\emph{V}}, ∃{\emph{w}} ∈ {\emph{V}} : {\emph{v}}+{\emph{w}} = {\emph{0}}} \tn % Row Count 8 (+ 2) % Row 5 \SetRowColor{white} \mymulticolumn{1}{x{5.377cm}}{{\bf{multiplicative identity}} 1{\emph{v}} = {\emph{v}}} \tn % Row Count 9 (+ 1) % Row 6 \SetRowColor{LightBackground} \mymulticolumn{1}{x{5.377cm}}{{\bf{distributive property}} a({\emph{v}}+{\emph{w}}) = a{\emph{v}} + a{\emph{w}} ∀a ∈ {\bf{F}}} \tn % Row Count 11 (+ 2) % Row 7 \SetRowColor{white} \mymulticolumn{1}{x{5.377cm}}{{\emph{v}} ∈ {\emph{V}} = vector ∨ point} \tn % Row Count 12 (+ 1) \hhline{>{\arrayrulecolor{DarkBackground}}-} \end{tabularx} \par\addvspace{1.3em} % That's all folks \end{multicols*} \end{document}