Definition 57: Linear Equation

A linear equation in $𝐹$ over the variables ${𝑥}_1,{𝑥}_2,\dots ,{𝑥}_{𝑛}$ is an equation of the form:

$${𝑎}_1{𝑥}_1+{𝑎}_2{𝑥}_2+\cdots +{𝑎}_{𝑛}{𝑥}_{𝑛}=𝑏$$

for scalars ${𝑎}_1,{𝑎}_2,\dots ,{𝑎}_{𝑛},𝑏\in 𝐹$. A solution to a linear equation is an $𝑛$-tuple $\left({𝑠}_1,{𝑠}_2,\dots ,{𝑠}_{𝑛}\right)\in {𝐹}^{𝑛}$ such that ${𝑎}_1{𝑠}_1+{𝑎}_2{𝑠}_2+\cdots +{𝑎}_{𝑛}{𝑠}_{𝑛}=𝑏$. The solution set of a linear equation is the set of all solutions to that equation.

Definition 58: Linear System

A linear system (or system of linear equations) in $𝐹$ over the variables ${𝑥}_1,{𝑥}_2,\dots ,{𝑥}_{𝑛}$ is a finite set of linear equations over the same variables ${𝑥}_1,{𝑥}_2,\dots ,{𝑥}_{𝑛}$. The solution set of a linear system is the set of all $𝑛$-tuples $\left({𝑠}_1,{𝑠}_2,\dots ,{𝑠}_{𝑛}\right)\in {𝐹}^{𝑛}$ that are solutions to all equations in the system.