7.1: The Syntax for Systems of Linear Equations

The following video explains the different syntax we use to describe linear systems.

from IPython.display import YouTubeVideo
YouTubeVideo("AQJeOg4ZoIk",width=640,height=360, cc_load_policy=True)

The following is a summary of the syntax shown in the video:

Linear Equation

\[ b = a_{1}x_{1} + a_{2}x_{2} + a_{3}x_{3} + \ldots + a_{n}x_{n} \nonumber \]

System of linear equations

\[b_1 = a_{11}x_1+a_{12}x_2+a_{13}x_3 + \ldots a_{1n} \nonumber \]

\[b_2 = a_{21}x_1+a_{22}x_2+a_{23}x_3 + \ldots a_{2n} \nonumber \]

\[b_3 = a_{31}x_1+a_{32}x_2+a_{33}x_3 + \ldots a_{3n} \nonumber \]

\[\vdots \nonumber \]

\[b_m = a_{m1}x_1+a_{m2}x_2+a_{m3}x_3 + \ldots a_{mn} \nonumber \]

System of linear equations (Matrix format)

\[\begin{split}
\left[
\begin{matrix}
b_1 \\
b_2 \\
b_3 \\
\vdots \\
b_m
\end{matrix}
\right]
=
\left[
\begin{matrix}
a_{11} & a_{12} & a_{13} & & a_{1n} \\
a_{21} & a_{22} & a_{23} & \ldots & a_{2n} \\
a_{31} & a_{32} & a_{33} & & a_{3n} \\
& \vdots & & \ddots & \vdots \\
a_{m1} & a_{m2} & a_{m3} & & a_{mn}
\end{matrix}
\right]
\left[
\begin{matrix}
x_1 \\
x_2 \\
x_3 \\
\vdots \\
x_m
\end{matrix}
\right] \end{split} \nonumber \]

System of linear equations (Augmented Form)

\[\begin{split}
\left[
\begin{matrix}
a_{11} & a_{12} & a_{13} & & a_{1n} \\
a_{21} & a_{22} & a_{23} & \ldots & a_{2n} \\
a_{31} & a_{32} & a_{33} & & a_{3n} \\
& \vdots & & \ddots & \vdots \\
a_{m1} & a_{m2} & a_{m3} & & a_{mn}
\end{matrix}
\, \middle\vert \,
\begin{matrix}
b_1 \\
b_2 \\
b_3 \\
\vdots \\
b_m
\end{matrix}
\right]
\end{split} \nonumber \]