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\author{luckystarr}
\pdfinfo{
  /Title (penn-state-math-220.pdf)
  /Creator (Cheatography)
  /Author (luckystarr)
  /Subject (Penn State: Math 220 Cheat Sheet)
}

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\begin{tabulary}{11cm}{L}
    \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{Penn State: Math 220 Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{luckystarr} via \textcolor{DarkBackground}{\uline{cheatography.com/59106/cs/15528/}}}
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  \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}}  \\
  \vspace{-2pt}luckystarr \\
  \uline{cheatography.com/luckystarr} \\
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  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Published 22nd April, 2018.\\
   Updated 22nd April, 2018.\\
   Page {\thepage} of \pageref{LastPage}.
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  Measure your website readability!\\
  www.readability-score.com
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\begin{multicols*}{3}

\begin{tabularx}{5.377cm}{x{1.89126 cm} x{3.08574 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{1.1}}  \tn
% Row 0
\SetRowColor{LightBackground}
A Matrix & row, columns \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
Coefficients Matrix & Just Left Hand Side \tn 
% Row Count 3 (+ 2)
% Row 2
\SetRowColor{LightBackground}
Augmented Matrix & Left and Right Hand Side \tn 
% Row Count 5 (+ 2)
% Row 3
\SetRowColor{white}
Solving Linear Systems & (1) Augmented Matrix\{\{nl\}\} (2) Row Operations\{\{nl\}\} (3) Solution to Linear System\{\{nl\}\}  The RHS is the solution \tn 
% Row Count 10 (+ 5)
% Row 4
\SetRowColor{LightBackground}
One Solution & Upper triangle with Augmented Matrix \tn 
% Row Count 12 (+ 2)
% Row 5
\SetRowColor{white}
No Solution & Last row is all zeros = RHS number \tn 
% Row Count 14 (+ 2)
% Row 6
\SetRowColor{LightBackground}
Infinitely Many Solutions & Last row (including RHS) is all zeros \tn 
% Row Count 16 (+ 2)
% Row 7
\SetRowColor{white}
Inconsistent & Has No Solution \tn 
% Row Count 17 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
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\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
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\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{1.1 Example(1)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/luckystarr_1524159342_Screen Shot 2018-04-19 at 1.31.20 PM.png}}} \tn 
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\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{1.74195 cm} x{3.23505 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{1.2}}  \tn
% Row 0
\SetRowColor{LightBackground}
Echelon Matrix & (1) Zero Rows at the bottom \{\{nl\}\} (2) Leading Entries are down and to the right \{\{nl\}\} (3) Zeros are below each leading entry \tn 
% Row Count 5 (+ 5)
% Row 1
\SetRowColor{white}
Reduced Echelon Matrix & (1) The leading entry of each nonzero row is 1 \{\{nl\}\} (2) Zeros are below AND above each 1 \tn 
% Row Count 9 (+ 4)
% Row 2
\SetRowColor{LightBackground}
Pivot Position & Location of Matrix that Corresponds to a leading 1 in REF \tn 
% Row Count 12 (+ 3)
% Row 3
\SetRowColor{white}
Pivot Column & Column in Matrix that contains a pivot \tn 
% Row Count 14 (+ 2)
% Row 4
\SetRowColor{LightBackground}
To get to EF & down and right \tn 
% Row Count 15 (+ 1)
% Row 5
\SetRowColor{white}
To get to REF & up and left \tn 
% Row Count 16 (+ 1)
% Row 6
\SetRowColor{LightBackground}
Free Variables & Variables that don't correspond to pivot columns \tn 
% Row Count 18 (+ 2)
% Row 7
\SetRowColor{white}
Consistent System & Pivot in every Column \tn 
% Row Count 20 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
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\par\addvspace{1.3em}

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\begin{tabularx}{5.377cm}{p{0.69678 cm} x{4.28022 cm} }
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\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{1.3}}  \tn
% Row 0
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RR\textasciicircum{}2\textasciicircum{} & Set of all vectors with 2 rows \tn 
% Row Count 1 (+ 1)
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\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{1.3 Example (2)}}  \tn
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\begin{tabularx}{5.377cm}{x{2.4885 cm} x{2.4885 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{1.4}}  \tn
% Row 0
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Vector Equation & x1{\bf{a1}}+x2{\bf{a2}}+x3{\bf{a3}} ={\bf{b}} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
Matrix Equation & A{\bf{x}}={\bf{b}} \tn 
% Row Count 3 (+ 1)
% Row 2
\SetRowColor{LightBackground}
If A is an m x n matrix the following are all true or all false & A{\bf{x}} = {\bf{b}}  has a solution for every {\bf{b}} in RR\textasciicircum{}m\textasciicircum{}\{\{nl\}\} Every {\bf{b}} in RR\textasciicircum{}m\textasciicircum{} is a lin. combo of columns in A \{\{nl\}\} Columns of A span RR\textasciicircum{}m\textasciicircum{} \{\{nl\}\} Matrix A has a pivot in every row (i.e. no row of zeros) \tn 
% Row Count 14 (+ 11)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{Anything in {\bf{Bold}} means it is a vector.}  \tn 
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\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/luckystarr_1524161852_Screen Shot 2018-04-19 at 2.11.53 PM.png}}} \tn 
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\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{1.4 Example (2)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/luckystarr_1524161878_Screen Shot 2018-04-19 at 2.12.23 PM.png}}} \tn 
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\begin{tabularx}{5.377cm}{x{2.4885 cm} x{2.4885 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{1.5}}  \tn
% Row 0
\SetRowColor{LightBackground}
Homogeneous & A{\bf{x}} = {\bf{0}} \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
Trivial Solution & A{\bf{x}} = {\bf{0}} if at lease one column is missing a pivot \tn 
% Row Count 4 (+ 3)
% Row 2
\SetRowColor{LightBackground}
Determine if homogenous Linear System has a non trivial solution & (1) Write as Augmented Matrix \{\{nl\}\} (2) Reduce to EF \{\{nl\}\} (3) Determine if there are any free variables(column w/o pivot) \{\{nl\}\} (4) If any free variables, than a non-trivial solution exists\{\{nl\}\} (5) Non-Trivial Solution can be found by further reducing to REF and solving for {\bf{x}} \tn 
% Row Count 19 (+ 15)
% Row 3
\SetRowColor{white}
If A{\bf{x}} = {\bf{0}} has one free variable & Than {\bf{x}} is a line that passes through the origin \tn 
% Row Count 22 (+ 3)
% Row 4
\SetRowColor{LightBackground}
If A{\bf{x}} = {\bf{0}} has two free variables & Than {\bf{x}} has a plane that passes through the origin \tn 
% Row Count 25 (+ 3)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{1.5 Example (1)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/luckystarr_1524169253_Screen Shot 2018-04-19 at 4.16.35 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{1.5 Example (2)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/luckystarr_1524169287_Screen Shot 2018-04-19 at 4.16.42 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
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\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{1.69218 cm} x{3.28482 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{1.7}}  \tn
% Row 0
\SetRowColor{LightBackground}
Linear Independence & No free Variables, none of the vectors are multiples of each other \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
To check ind/dep & reduce augmented matrix to EF and see if there are free variables(ie. every column must have a pivot to be linearly independent) \tn 
% Row Count 8 (+ 5)
% Row 2
\SetRowColor{LightBackground}
To check if multiples & {\bf{u}} = c *  {\bf{v}} \{\{nl\}\} find value of c, then it is a multiple \{\{nl\}\} therefore linearly dependent \tn 
% Row Count 12 (+ 4)
% Row 3
\SetRowColor{white}
Linearly Dependent & If there are more columns than rows \tn 
% Row Count 14 (+ 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}{1.7 Example (1)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/luckystarr_1524170251_Screen Shot 2018-04-19 at 4.32.23 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.4885 cm} x{2.4885 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{1.8}}  \tn
% Row 0
\SetRowColor{LightBackground}
Every Matrix Transformation is a: & Linear Transformation \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
T({\bf{x}}) = & A({\bf{x}}) \tn 
% Row Count 3 (+ 1)
% Row 2
\SetRowColor{LightBackground}
If A is m x n Matrix, then the properties are & (1) T({\bf{u}} + {\bf{v}}) = T({\bf{u}}) + T({\bf{v}}) \{\{nl\}\} (2) T(c{\bf{u}}) = cT({\bf{u}}) \{\{nl\}\} (3) T({\bf{0}}) = {\bf{0}} \{\{nl\}\} (4) T(c{\bf{u}} + d{\bf{v}}) = cT({\bf{u}}) + dT({\bf{v}}) \tn 
% Row Count 11 (+ 8)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{1.8 Example (1)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/luckystarr_1524170910_Screen Shot 2018-04-19 at 4.47.23 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
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\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{1.8 Example (2)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/luckystarr_1524171086_Screen Shot 2018-04-19 at 4.47.23 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
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\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.4885 cm} x{2.4885 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{1.9}}  \tn
% Row 0
\SetRowColor{LightBackground}
RR\textasciicircum{}n\textasciicircum{} -{}-\textgreater{} RR\textasciicircum{}m\textasciicircum{} is said to be 'onto' & Equation T({\bf{x}}) =A{\bf{x}}={\bf{b}} has a unique solution or more than one solution \{\{nl\}\} each row has a pivot \tn 
% Row Count 6 (+ 6)
% Row 1
\SetRowColor{white}
RR\textasciicircum{}n\textasciicircum{} -{}-\textgreater{} RR\textasciicircum{}m\textasciicircum{} is said to be one-to-one & Equation T({\bf{x}}) =A{\bf{x}}={\bf{b}} has a unique solution or no solution \{\{nl\}\} each row has a pivot \tn 
% Row Count 11 (+ 5)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.38896 cm} x{2.58804 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{2.1}}  \tn
% Row 0
\SetRowColor{LightBackground}
Addition of Matrices & Can Add matrices if they have same \# of rows and columns \{\{nl\}\} (ie {\bf{A}}(3x4) and {\bf{B}}(3x4) so you can add them) \tn 
% Row Count 6 (+ 6)
% Row 1
\SetRowColor{white}
Multiply by Scalar & Multiply each entry by scalar \tn 
% Row Count 8 (+ 2)
% Row 2
\SetRowColor{LightBackground}
Matrix Multiplication ({\bf{A}} x {\bf{B}}) & Must each row of {\bf{A}} by each column of {\bf{B}} \tn 
% Row Count 11 (+ 3)
% Row 3
\SetRowColor{white}
Powers of a Matrix & Can compute powers by if the matrix has the same number of columns as rows \tn 
% Row Count 15 (+ 4)
% Row 4
\SetRowColor{LightBackground}
Transpose of Matrix & row 1 of A becomes column 1 of A \{\{nl\}\} row 2 of A becomes column 2 of A \tn 
% Row Count 19 (+ 4)
% Row 5
\SetRowColor{white}
Properties of Transpose & (1) if {\bf{A}} is m x n, then {\bf{A}}\textasciicircum{}T\textasciicircum{} is n x m \{\{nl\}\} (2)  ({\bf{A}}\textasciicircum{}T\textasciicircum{})\textasciicircum{}T\textasciicircum{} = {\bf{A}} \{\{nl\}\} (3) ({\bf{A}} + {\bf{B}})\textasciicircum{}T\textasciicircum{} = {\bf{A}} \textasciicircum{}T\textasciicircum{} + {\bf{B}} \textasciicircum{}T\textasciicircum{} \{\{nl\}\} (4) (t{\bf{A}})\textasciicircum{}T\textasciicircum{} = t{\bf{A}}\textasciicircum{}T\textasciicircum{} \{\{nl\}\} (5) ({\bf{A}} {\bf{B}})\textasciicircum{}T\textasciicircum{} = {\bf{B}}\textasciicircum{}T\textasciicircum{} {\bf{A}}\textasciicircum{}T\textasciicircum{} \tn 
% Row Count 30 (+ 11)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.28942 cm} x{2.68758 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{2.2}}  \tn
% Row 0
\SetRowColor{LightBackground}
Singular matrix & A matrix that is NOT investable \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
Determinate of A (2 x 2) Matrix & det A = ad - bc \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
If A is invertable \& (nxn) & There will never be no solution or infinitely many solutions to A{\bf{x}} = {\bf{b}} \tn 
% Row Count 8 (+ 4)
% Row 3
\SetRowColor{white}
Properties of Invertable Matricies & ({\bf{A}}\textasciicircum{}-1\textasciicircum{})\textasciicircum{}-1\textasciicircum{} = {\bf{A}} \{\{nl\}\} (assuming {\bf{A}} \& {\bf{B}} are investable) (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 14 (+ 6)
% Row 4
\SetRowColor{LightBackground}
Finding Inverse Matrix & {[}A | I {]} -{}-\textgreater{} {[} I | A\textasciicircum{}-1\textasciicircum{}{]} {\emph{Use row operations}} \{\{nl\}\} STOP when you get a row of Zeros, it cannot be reduced \tn 
% Row Count 20 (+ 6)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{2.2 Example (1)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/luckystarr_1524173186_Screen Shot 2018-04-19 at 5.23.28 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{2.3 Invertable Matrix Theorem}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/luckystarr_1524173342_Screen Shot 2018-04-19 at 5.28.22 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.4885 cm} x{2.4885 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{2.8}}  \tn
% Row 0
\SetRowColor{LightBackground}
A subspace {\emph{S}} of RR\textasciicircum{}n\textasciicircum{} is a subspace is {\emph{S}} satisfies: & (1) {\emph{S}} contains zero vector \{\{nl\}\} (2) If {\bf{u}} \& {\bf{v}} are in {\emph{S}}, then {\bf{u}} + {\bf{v}} is also in {\emph{S}} \{\{nl\}\} (3) If {\emph{r}} is a real \# \& {\bf{u}} is in {\emph{S}}, then r{\bf{u}} is also in {\emph{S}} \tn 
% Row Count 9 (+ 9)
% Row 1
\SetRowColor{white}
Subspace RR\textasciicircum{}3\textasciicircum{} & Any Plane that Passes through the origin forms a subspace RR\textasciicircum{}3\textasciicircum{} \{\{nl\}\} Any set that contains nonlinear terms will NOT form a subspace RR\textasciicircum{}3\textasciicircum{} \tn 
% Row Count 16 (+ 7)
% Row 2
\SetRowColor{LightBackground}
Null Space (Nul A) & To determine in {\bf{u}} is in the Nul(A), check if:  A{\bf{u}} = {\bf{0}} \{\{nl\}\} If yes -{}-\textgreater{} then {\bf{u}} is in the Nullspace \tn 
% Row Count 22 (+ 6)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{2.8 Example (1)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/luckystarr_1524174156_Screen Shot 2018-04-19 at 5.41.50 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{2.8 Example (2)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/luckystarr_1524174181_Screen Shot 2018-04-19 at 5.41.56 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.18988 cm} x{2.78712 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{2.9}}  \tn
% Row 0
\SetRowColor{LightBackground}
Dimension of a non-zero Subspace & \# of vectors in any basis; it is the \# of linearly independent vectors \tn 
% Row Count 4 (+ 4)
% Row 1
\SetRowColor{white}
Dimension of a zero Subspace & is Zero \tn 
% Row Count 6 (+ 2)
% Row 2
\SetRowColor{LightBackground}
Dimension of a Column Space & \# of pivot columns \tn 
% Row Count 8 (+ 2)
% Row 3
\SetRowColor{white}
Dimension of a Null Space & \# of free variables in the solution A{\bf{x}}={\bf{0}} \tn 
% Row Count 11 (+ 3)
% Row 4
\SetRowColor{LightBackground}
Rank of a Matrix & \# of pivot columns \tn 
% Row Count 12 (+ 1)
% Row 5
\SetRowColor{white}
The Rank Theorem & Matrix A has {\emph{n}} columns: rank A (\# pivots) + dim Nul A (\# free var.) = {\emph{n}} \tn 
% Row Count 16 (+ 4)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{dim = dimension; var. = variable}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{2.9 Refrence}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/luckystarr_1524174695_Screen Shot 2018-04-19 at 5.51.03 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.4885 cm} x{2.4885 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{3.1}}  \tn
% Row 0
\SetRowColor{LightBackground}
Calculating Determinant of Matrix A is another way to tell if a linear system of equations has a solution & (1) Det(A) not =0, then A{\bf{x}}={\bf{b}} has a unique solution \{\{nl\}\} (2) Det(A) =0, then A{\bf{x}}={\bf{b}} has no solutions or inf many \tn 
% Row Count 7 (+ 7)
% Row 1
\SetRowColor{white}
If A{\bf{x}} not= {\bf{0}} & A\textasciicircum{}-1\textasciicircum{} exist \tn 
% Row Count 8 (+ 1)
% Row 2
\SetRowColor{LightBackground}
If A{\bf{x}} = {\bf{0}} & A\textasciicircum{}-1\textasciicircum{} Does NOT exist \tn 
% Row Count 9 (+ 1)
% Row 3
\SetRowColor{white}
Cofactor Expansion & Use row/column w/ most zeros \tn 
% Row Count 11 (+ 2)
% Row 4
\SetRowColor{LightBackground}
If Matrix A has an upper or lower triangle of zeros & The det(A) is the multiplication down the diagonals \tn 
% Row Count 14 (+ 3)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{3.1 Reference (1)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/luckystarr_1524354905_Screen Shot 2018-04-21 at 7.49.41 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{3.1 Example (1)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/luckystarr_1524354939_Screen Shot 2018-04-21 at 7.49.52 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{3.1 Reference (2)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/luckystarr_1524354971_Screen Shot 2018-04-21 at 7.51.05 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{3.1 Example (2)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/luckystarr_1524355003_Screen Shot 2018-04-21 at 7.53.44 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.43873 cm} x{2.53827 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{3.2}}  \tn
% Row 0
\SetRowColor{LightBackground}
Determinate Property 1 & If a multiple of 1 row of A is added to another row to produce Matrix B, then det(B)=det(A) \tn 
% Row Count 5 (+ 5)
% Row 1
\SetRowColor{white}
Determinate Property 2 & If 2 rows of A are interchanged to produce B, then det(B)=-det(A) \tn 
% Row Count 9 (+ 4)
% Row 2
\SetRowColor{LightBackground}
Determinate Property 3 & If one row of A is multiplied to produce B, then det(B)=k*det(A) \tn 
% Row Count 13 (+ 4)
% Row 3
\SetRowColor{white}
Assuming both A \& B are n x n Matrices & (1) det(A\textasciicircum{}T\textasciicircum{}) = det(A) \{\{nl\}\} (2) det(AB) = det(A)*det(B) \{\{nl\}\} (3) det(A\textasciicircum{}-1\textasciicircum{}) = 1/det(A) \{\{nl\}\} (4) det(cA) = c\textasciicircum{}n\textasciicircum{} det(A) \{\{nl\}\} (5) det(A\textasciicircum{}r\textasciicircum{}) = (detA)\textasciicircum{}r\textasciicircum{} \tn 
% Row Count 21 (+ 8)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{1.69218 cm} x{3.28482 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{3.3 AKA Cramer's Rule}}  \tn
% Row 0
\SetRowColor{LightBackground}
Cramer's Rule & Can be used to find the solution to a linear system of equations A{\bf{x}}={\bf{b}} when A is an investable square matrix \tn 
% Row Count 5 (+ 5)
% Row 1
\SetRowColor{white}
Def. of Cramer's Rule & Let A be an n x n invertible matrix. For any {\bf{b}} in RR\textasciicircum{}n\textasciicircum{}, the unique solution {\bf{x}} of A{\bf{x}}={\bf{b}} has entries given by \{\{nl\}\} xi = detAi({\bf{b}})/det(A)   {\emph{i = 1,2,...n}} \tn 
% Row Count 12 (+ 7)
% Row 2
\SetRowColor{LightBackground}
Ai({\bf{b}}) & is the matrix A w/ column i replaced w/ vector {\bf{b}} \tn 
% Row Count 14 (+ 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}{3.3 Example (1)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/luckystarr_1524355947_Screen Shot 2018-04-21 at 8.11.55 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.4885 cm} x{2.4885 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{5.1}}  \tn
% Row 0
\SetRowColor{LightBackground}
A{\bf{u}}=λ{\bf{u}} & A is an nxn matrix. A nonzero vector {\bf{u}} is an eigenvector of A if there exists such a scalar λ \tn 
% Row Count 5 (+ 5)
% Row 1
\SetRowColor{white}
To determine if λ is an eigenvalue & reduce {[}(A-λI)|0{]} to echelon form and see if it has any free variables. \{\{nl\}\} yes -\textgreater{} λ is Eigenvalue \{\{nl\}\} no -\textgreater{} λ is not eigenvalue \tn 
% Row Count 12 (+ 7)
% Row 2
\SetRowColor{LightBackground}
To determine if given vector is an eigenvector & Ax=λ{\bf{x}} \tn 
% Row Count 15 (+ 3)
% Row 3
\SetRowColor{white}
Eigenspace of A = & Nullspace of (A-λI) \tn 
% Row Count 16 (+ 1)
% Row 4
\SetRowColor{LightBackground}
Eigenvalues of triangular Matrix & entries along diagonal *you CANNOT row reduce a matrix to find its eigenvalues \tn 
% Row Count 20 (+ 4)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{5.1 Example (1)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/luckystarr_1524356577_Screen Shot 2018-04-21 at 8.17.50 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{5.1 Example (2)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/luckystarr_1524356601_Screen Shot 2018-04-21 at 8.19.45 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{5.1 Example (3)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/luckystarr_1524356626_Screen Shot 2018-04-21 at 8.20.51 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.4885 cm} x{2.4885 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{5.2}}  \tn
% Row 0
\SetRowColor{LightBackground}
If λ is an eigenvalue of a Matrix A & then (A-λI){\bf{x}}={\bf{0}} will have a nontrivial solution \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
A nontrivial solution will exist & if det(A-λI)=0 (Characteristic Equation) \tn 
% Row Count 6 (+ 3)
% Row 2
\SetRowColor{LightBackground}
A is nxn Matrix. A is invertible if and only if & (1) The \# 0 is NOT an λ of A \{\{nl\}\} (2) The det(A) is not zero \tn 
% Row Count 10 (+ 4)
% Row 3
\SetRowColor{white}
Similar Matrices & If nxn Matrices A and B are similar, then they have the same characteristic polynomial (same λ) with same multiplicities \tn 
% Row Count 17 (+ 7)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{5.2 Example (1)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/luckystarr_1524357115_Screen Shot 2018-04-21 at 8.27.48 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.33919 cm} x{2.63781 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{5.3}}  \tn
% Row 0
\SetRowColor{LightBackground}
A matrix A written in diagonal form & A=PDP\textasciicircum{}-1\textasciicircum{} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
Power of Matrix & A\textasciicircum{}k\textasciicircum{} = Diagonal matrix and \#'s on diagonal get raised to the k \tn 
% Row Count 5 (+ 3)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{Determining if Matrix is Diagonalizable} \tn 
% Row Count 6 (+ 1)
% Row 3
\SetRowColor{white}
λ of a nxn matrix & n distinct (or real) λ then matrix is diagonalizable \{\{nl\}\} less than n λ, it may or may not be diagonalizable; it will be if \# of linearly dependent eigenvectors = n \tn 
% Row Count 14 (+ 8)
% Row 4
\SetRowColor{LightBackground}
eigenvectors of nxn matrix & n linearly independent eigenvectors, then diagonalizable \{\{nl\}\} less than n linearly independent eigenvectors, then matrix is NOT diagonlizable \tn 
% Row Count 21 (+ 7)
% Row 5
\SetRowColor{white}
{\emph{D}} & matrix w/ λ down diagonal \tn 
% Row Count 23 (+ 2)
% Row 6
\SetRowColor{LightBackground}
{\emph{P}} & columns of P have linearly n linearly independent eigenvectors \tn 
% Row Count 26 (+ 3)
% Row 7
\SetRowColor{white}
Finding {\emph{P}} & solve A-λI and plug in the λ values. Reduce to EF, solve for {\bf{x}}, \& find eigenvector \tn 
% Row Count 31 (+ 5)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{5.3 Example (1)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/luckystarr_1524357862_Screen Shot 2018-04-21 at 8.41.19 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{5.3 Example (2)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/luckystarr_1524357887_Screen Shot 2018-04-21 at 8.43.09 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{5.3 Example (3)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/luckystarr_1524357911_Screen Shot 2018-04-21 at 8.43.40 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.4885 cm} x{2.4885 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{6.1}}  \tn
% Row 0
\SetRowColor{LightBackground}
Length of vector {\bf{x}} & ||{\bf{x}}|| = sqrt(x1\textasciicircum{}2\textasciicircum{}+x2\textasciicircum{}2\textasciicircum{}) \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
Length fo vector {\bf{x}} in RR\textasciicircum{}2\textasciicircum{} & ||{\bf{x}}|| = sqrt({\bf{x}} • {\bf{x}}) \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
The Unit Vector & {\bf{u}} = v/||v|| \tn 
% Row Count 5 (+ 1)
% Row 3
\SetRowColor{white}
Two vectors {\bf{u}} \& {\bf{v}} in RR\textasciicircum{}n\textasciicircum{}, the distance between {\bf{u}} \& {\bf{v}} & ||{\bf{u}} - {\bf{v}}|| \tn 
% Row Count 9 (+ 4)
% Row 4
\SetRowColor{LightBackground}
Two vectors {\bf{u}} \& {\bf{v}} are orthogonal if and only if & ||u+v||\textasciicircum{}2\textasciicircum{}= ||u||\textasciicircum{}2\textasciicircum{} +||v||\textasciicircum{}2\textasciicircum{} \{\{nl\}\} {\bf{u}} • {\bf{v}} = 0 \tn 
% Row Count 12 (+ 3)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.73735 cm} x{2.23965 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{6.2}}  \tn
% Row 0
\SetRowColor{LightBackground}
The distance from {\bf{y}} to the line through {\bf{u}} \& the origin & ||{\bf{z}}|| = ||{\bf{y}} - {\bf{y-hat}}|| \tn 
% Row Count 3 (+ 3)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{6.2 Example (1)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/luckystarr_1524359014_Screen Shot 2018-04-21 at 8.58.39 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{6.2 Example (2)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/luckystarr_1524359033_Screen Shot 2018-04-21 at 8.58.58 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{6.2 Example (3)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/luckystarr_1524359059_Screen Shot 2018-04-21 at 9.00.14 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{6.2 Example (4)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/luckystarr_1524359081_Screen Shot 2018-04-21 at 9.00.21 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{6.2 Example (5)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/luckystarr_1524359103_Screen Shot 2018-04-21 at 9.01.49 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{6.2 Example (6)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/luckystarr_1524359127_Screen Shot 2018-04-21 at 9.02.35 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{6.2 Example (7)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/luckystarr_1524359150_Screen Shot 2018-04-21 at 9.02.44 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{6.2 Reference (1)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/luckystarr_1524359232_6.2 Reference.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{6.2 Reference (2)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/luckystarr_1524359262_6.2 Reference 2.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{6.3 Example (1.1)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/luckystarr_1524415053_Screen Shot 2018-04-22 at 12.35.49 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{6.3 Example (1.2)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/luckystarr_1524415077_Screen Shot 2018-04-22 at 12.35.57 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{6.3 Example (2)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/luckystarr_1524415100_Screen Shot 2018-04-22 at 12.36.25 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{6.3 Example (3)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/luckystarr_1524415126_Screen Shot 2018-04-22 at 12.36.50 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{6.3 Example (4)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/luckystarr_1524415150_Screen Shot 2018-04-22 at 12.37.04 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.4885 cm} x{2.4885 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{6.4}}  \tn
% Row 0
\SetRowColor{LightBackground}
Gram- Schmidt Process Overview & take a given set of vectors \& transform them into a set of orthogonal or orthonormal vectors \tn 
% Row Count 5 (+ 5)
% Row 1
\SetRowColor{white}
Given {\bf{x1}} \& {\bf{x2}}, produce {\bf{v1}} \& {\bf{v2}} where the v's are perp. to each other & (1) Let {\bf{v1}}={\bf{x1}} \{\{nl\}\} (2) Find {\bf{v2}}; {\bf{v2}}={\bf{x2}} - {\bf{x2hat}} \tn 
% Row Count 10 (+ 5)
% Row 2
\SetRowColor{LightBackground}
{\bf{x2 hat}} & (x2•v1)/(v1•v1) * v1 \tn 
% Row Count 12 (+ 2)
% Row 3
\SetRowColor{white}
Orthogonal Basis & \{{\bf{v1}},{\bf{v2}},...,{\bf{vn}}\} \tn 
% Row Count 14 (+ 2)
% Row 4
\SetRowColor{LightBackground}
Orthonormal Basis & \{{\bf{v1}}/||{\bf{v1}}||, {\bf{v2}}/ ||{\bf{v2}}||,..., {\bf{vn}}/||{\bf{vn}}||\} \tn 
% Row Count 18 (+ 4)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{6.4 Reference (1)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/luckystarr_1524415853_G-S process for RR3.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{6.4 Example (1)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/luckystarr_1524415873_Screen Shot 2018-04-22 at 12.46.00 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.4885 cm} x{2.4885 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{7.1}}  \tn
% Row 0
\SetRowColor{LightBackground}
Symmetric Matrix & A square matrix where A\textasciicircum{}T\textasciicircum{}=A \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
If A is a symmetric Matrix & then eigenvectors associated w/ distinct eigenvalues are orthogonal \{\{nl\}\} If a matrix is symmetrical, it has an orthogonal \& orthonormal basis of vectors \tn 
% Row Count 10 (+ 8)
% Row 2
\SetRowColor{LightBackground}
Orthogonal matrix is a square matrix w/ orthonormal columns & (1) Matrix is square \{\{nl\}\} (2) Columns are orthogonal \{\{nl\}\} (3) Columns are unit vectors \tn 
% Row Count 15 (+ 5)
% Row 3
\SetRowColor{white}
If Matrix P has orthonormal columns & P\textasciicircum{}T\textasciicircum{}P=I \tn 
% Row Count 17 (+ 2)
% Row 4
\SetRowColor{LightBackground}
If P is a nxn orthogonal matrix & P\textasciicircum{}T\textasciicircum{}=P\textasciicircum{}-1\textasciicircum{} \tn 
% Row Count 19 (+ 2)
% Row 5
\SetRowColor{white}
A=PDP\textasciicircum{}T\textasciicircum{} & A must be symmetric, P must be normalized \tn 
% Row Count 22 (+ 3)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{7.1 Reference (1)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/luckystarr_1524416693_7.1 Reference.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{7.1 Example (1)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/luckystarr_1524416725_Screen Shot 2018-04-22 at 1.00.37 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{7.1 Example (2.1)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/luckystarr_1524416750_Screen Shot 2018-04-22 at 1.03.39 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{7.1 Example (2.2)}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/luckystarr_1524416771_Screen Shot 2018-04-22 at 1.03.55 PM.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}


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

\end{document}