Linear equation and linear system

By Rodrigo Ribeiro6 de August de 20207 de August de 2020

Linear equation

Definition: A linear equation on the variables $x_1,x_2,\dots, x_n$ is an equation of the type

$a_1x_1 + a_2x_2 + \cdots + a_nx_n = b$ ,

where $a_1,a_2,\dots, a_n$ and $b$ are complex numbers. We call the numbers $a_1,a_2,\dots, a_n$ the coefficients of the equation.
Example:

$2x_1 + \sqrt{7}x_2 -\frac{x_n}{3} = 1$

Linear system

Definition: A linear system or system of linear equations is a collection of linear equation.

A linear system on $n$ variables and $m$ equations is a list of $m$ equations and each equation has $n$ variables. In this case we say the linear system has order $m\times n$ .

In symbols, such linear system looks like:

$a_{1,1}x_1 + a_{1,2}x_2 + \cdots + a_{1,n}x_n = b_1$
$a_{2,1}x_1 + a_{2,2}x_2 + \cdots + a_{n,n}x_n = b_2$
$\vdots$
$a_{m,1}x_1 + a_{m,2}x_2 + \cdots + a_{m,n}x_n = b_m$