Chapter 2: Linear Equations

Abstract:

Let A be an m× n matrix over a field 𝔽. Let b₁, … , b_m be elements of the field 𝔽. The system of equations

is called a system of linear equations. We also write where x and b are considered as column vectors. The system is said to be homogeneous if all the numbers b₁, … , b_m are equal to 0. The number n is called the number of unknowns, and m is called the number of equations. The system of homogeneous equations also admits the trivial solution x₁ = x₂ = … = x_n = 0…