. .
$$
\begin{array}{c|ccc}
\text{from \\ to} & A & B & C \\ \hline
A & 1/3 & 1/6 & 1/2 \\
B & 1/4 & 1/2 & 1/4 \\
C & 3/5 & 1/5 & 1/5
\end{array}
$$
test
$$
\left(\begin{array}{}
\rowcolor{blue} a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{array}\right)
$$
test
$$
\color{blue} blue
$$
another
$$
\color{#efa333} blue
$$
another
$$
\color{#0000ff} blue
$$
Oh here we go
$$
\left\{ \begin{array}{}
3x & -&2y & +& 5z & = 4 \\
& & 5y & -& 6z & =7 \\
& & & & 8z & =2
\end{array}\right.
$$
$$
\displaystyle \left\{ \begin{array}{}
\cos(t)x & + & \sin(t)y& = 1 \\
-\sin(t)x & + & \cos(t)y & = 0
\end{array}\right.
$$
$$
(LS) \xrightarrow[]{(R_{1})\to 4(R_{1})} (LS') \xrightarrow[]{(R_{2})\to (R_{2})+3(R_{1})} (LS'')
$$
$$
(LS) \xrightarrow[]{(R_{n})\to (R_{n}) + a (R_m) } {(LS^\ast)} \xrightarrow[]{(R_{p})\to b (R_{p})} (LS'')
$$
Here is a tikz circle
```tikz
\begin{document}
\begin{tikzpicture}
\draw (0,0) circle (1in);
\end{tikzpicture}
\end{document}
```
Let us see more
```tikz
\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd} [sep = .5 cm]
& & & A \arrow [dl, dotted] \arrow [rr, "/" {marking}] \arrow [dd] & & B \arrow [dl, dashed, red, "w_2"] \arrow [dd] \arrow [rr] & & 0 \\
& & C \arrow [rr, "w_1" {description, right}, crossing over] & & D \arrow [rr, crossing over] & & 0 \\
& 0 \arrow [rr] & & A^{\prime} \arrow [dl] \arrow [rr] & & B^{\prime} \arrow [dl] \\
0 \arrow [rr] & & C^{\prime} \arrow [rr] \arrow [from = uu, crossing over] & & D^{\prime} \arrow [from = uu, crossing over]\\
\end{tikzcd}
\end{document}
```
Let's see:
```tikz
\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd} [sep = .5 cm]
&\parbox{2cm}{\raggedright an augmented matrix} \arrow[d]& \\
& \parbox{2cm}{\raggedright EF of an augmented matrix}\arrow[dl] \arrow[dr]& \\
\parbox{2.2cm}{\raggedright pivot in the augmented column}\arrow[d] & & \parbox{3cm}{\raggedright no pivot in the augmented column} \arrow[dl] \arrow[d] \\
\parbox{2cm}{\raggedright no solution} & \parbox{2cm}{\raggedright yes, there are free variables} \arrow[d]& \parbox{2cm}{\raggedright no, there is no free variables} \arrow[d] \\
& \parbox{2.5cm}{\raggedright multiple solutions (and infinitely many if over R or C)} & \parbox{2cm}{\raggedright unique solution}
\end{tikzcd}
\end{document}
```
```tikz
\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
A\ar{rr}{x} \ar{dr}{10} & & B\ar{dd}{2} \\
& C \ar{ur}{z} \ar{dl}{w} & \\
D\ar{uu}{y} \ar{rr}{u} & & E \ar{ul}{4}
\end{tikzcd}
\end{document}
```
$$
\colorbox{lightblue}{1}
$$
```tikz
\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
A \ar{r}{x_1}\ar{dr}{1}& B\ar{r}{x_2}\ar{dr}{2} & C\ar{r}{x_3}\ar{dr}{3} & D\ar{r}{x_4}\ar{dr}{4} & E \ar{d}{x_5} \\
F\ar{u}{x_{10}} & G\ar{l}{x_9}\ar{u}{10} & H\ar{l}{x_8}\ar{u}{11} & I\ar{l}{x_7}\ar{u}{12} & J\ar{l}{x_6}
\end{tikzcd}
\end{document}
```
```tikz
\begin{document}
\begin{tikzpicture}
\draw[blue,fill,opacity = 0.1] (0,0)--(3,0)--(4,0.8)--(1,0.8)--cycle;
\node[blue, anchor= west, text width = 3cm] at (3.4,0.4) {plane spanned by the columns of \(A\)};
\draw[thick, opacity = 0.5, blue, ->] (0,0) -- (2.5,0);
\draw[thick, opacity = 0.5, blue, ->] (0,0) -- (0.8,0.64);
\draw[thick, black, ->] (0,0)--(2.4,0.5) node[anchor=west]{\(\vec u\) ?};
\draw[thick, black, ->] (0,0)--(1,2) node[anchor=west]{\(\vec u\) ?};;
\end{tikzpicture}
\end{document}
```
```tikz
\begin{document}
\begin{tikzpicture}
\tikzset{>=latex}
\draw[gray] (-3,0)--(3,0);
\draw[gray] (0,-2)--(0,4);
\node[blue, rotate = -26.57, anchor=south] at (1,2) {\(x + 2y = 5\)};
\draw[blue,thick] (-3,4)--(3,1);
\node[red, rotate = -26.57, anchor=south] at (1,-0.5) {\(x + 2y = 0\)};
\draw[red,thick] (-3,1.5)--(3,-1.5);
\foreach \i in {1,2,3,4}
{
\draw[red,thick, opacity = 0.3] (-3,1.5 + 0.5* \i)--(3,-1.5 + 0.5* \i);
}
\foreach \i in {-2,...,2}
{
\draw[black, very thick, ->, opacity = 0.1, dotted] (0 + \i,0 + -0.5*\i)--(0+\i,2.5-0.5*\i);
}
\draw[black, very thick, ->] (0,0)--(0,2.5);
\node[anchor = west] at (0,1.25) {\(\vec x_p = {0 \choose 5/2}\)};
\end{tikzpicture}
\end{document}
```