Glossary

Linear equation and linear system

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

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