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% Packages
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%\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
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% Document Info
\author{anlumithe}
\pdfinfo{
  /Title (machine-learning-and-optimization.pdf)
  /Creator (Cheatography)
  /Author (anlumithe)
  /Subject (Machine Learning and Optimization Cheat Sheet)
}

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% Commands
\newcommand{\SetRowColor}[1]{\noalign{\gdef\RowColorName{#1}}\rowcolor{\RowColorName}} % Shortcut for row colour
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% Header and Footer
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\fancyhead{} % Set header to blank
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\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}}
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\end{tabulary}
\columnbreak
\begin{tabulary}{11cm}{L}
    \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{Machine Learning and Optimization Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{anlumithe} via \textcolor{DarkBackground}{\uline{cheatography.com/177005/cs/36970/}}}
\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}anlumithe \\
  \uline{cheatography.com/anlumithe} \\
  \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 8th February, 2023.\\
   Page {\thepage} of \pageref{LastPage}.
\end{tabulary}
\vfill
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\begin{tabulary}{5.8cm}{L}
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  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Sponsor}}  \\
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  \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*}{2}

\begin{tabularx}{8.4cm}{x{3.76 cm} x{4.24 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{Linear Algebra}}  \tn
% Row 0
\SetRowColor{LightBackground}
A+B & has result, only if A and B have same dim \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
A{\bf{x}}=x\_1*{\bf{a\_1}}+...+x\_n*{\bf{a\_n}} & result is vector \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
A*B & row of A times column of B, result is matrix \tn 
% Row Count 7 (+ 3)
% Row 3
\SetRowColor{white}
A\textasciicircum{}T\textasciicircum{} & row becomes column and vice versa \tn 
% Row Count 9 (+ 2)
% Row 4
\SetRowColor{LightBackground}
Properties of transpose & (1) if A is (m x n) A\textasciicircum{}T\textasciicircum{} is (n x m)\{\{nl\}\} (2) (A\textasciicircum{}T\textasciicircum{})\textasciicircum{}T\textasciicircum{}=A\{\{nl\}\} (3) (A+B)\textasciicircum{}T\textasciicircum{}=A\textasciicircum{}T\textasciicircum{}+B\textasciicircum{}T\textasciicircum{} \{\{nl\}\}(4) (AB)\textasciicircum{}T\textasciicircum{} = B\textasciicircum{}T\textasciicircum{}A\textasciicircum{}T\textasciicircum{} \{\{nl\}\} (5) (tA)\textasciicircum{}T\textasciicircum{}=tA\textasciicircum{}T\textasciicircum{} \tn 
% Row Count 16 (+ 7)
% Row 5
\SetRowColor{white}
Inverse of Matrix & AA\textasciicircum{}-1\textasciicircum{}=I=A\textasciicircum{}-1\textasciicircum{}A \tn 
% Row Count 17 (+ 1)
% Row 6
\SetRowColor{LightBackground}
Properties of invertible matrix & (A\textasciicircum{}-1\textasciicircum{})\textasciicircum{}-1\textasciicircum{}= A\{\{nl\}\}(AB)\textasciicircum{}-1\textasciicircum{}= B\textasciicircum{}-1\textasciicircum{}A\textasciicircum{}-1\textasciicircum{}\{\{nl\}\}(A\textasciicircum{}T\textasciicircum{})\textasciicircum{}-1\textasciicircum{}=(A\textasciicircum{}-1\textasciicircum{})\textasciicircum{}T\textasciicircum{} \tn 
% Row Count 21 (+ 4)
% Row 7
\SetRowColor{white}
Singular value decomposition & A = UΣV\textasciicircum{}T\textasciicircum{} \{\{nl\}\} what means that every \seqsplit{vector-matrix-multiplication} can be viewed as a 3 step process \{\{nl\}\} (1) rotation into space V \{\{nl\}\} (2) scaling by singular value \{\{nl\}\} (3) rotation into new space U \tn 
% Row Count 31 (+ 10)
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\par\addvspace{1.3em}

\vfill
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\begin{tabularx}{8.4cm}{x{3.76 cm} x{4.24 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{Linear Algebra (cont)}}  \tn
% Row 8
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Eigen value decomposition & A = QΛQ\textasciicircum{}-1\textasciicircum{}\{\{nl\}\} this is only possible, if A {\bf{{\emph{and}}}} Q are square matrices \tn 
% Row Count 4 (+ 4)
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\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{p{1.36 cm} x{6.64 cm} }
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\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{Linear regression}}  \tn
% Row 0
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Model & {\bf{y}} = X{\bf{θ\textasciicircum{}*\textasciicircum{}}}+{\bf{z}} \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
\mymulticolumn{2}{x{8.4cm}}{Risk} \tn 
% Row Count 2 (+ 1)
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\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{p{0.8 cm} p{0.8 cm} }
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\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{p{0.8 cm} p{0.8 cm} }
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\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{p{0.8 cm} p{0.8 cm} }
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\begin{tabularx}{8.4cm}{p{0.8 cm} p{0.8 cm} }
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\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{Neuronal Networks}}  \tn
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% Row Count 0 (+ 0)
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\par\addvspace{1.3em}


% That's all folks
\end{multicols*}

\end{document}