ELEMENTARY LINEAR ALGEBRA

ELEMENTARY LINEAR ALGEBRA

*K. R. Matthews

 

1. Introduction to linear equations

A linear equation in n unknowns x1. x2,…, xn is an equation of the form

a1x1 + a2x2 + … + anxn = b,

where a1, a2, …, an, b are given real numbers.

For example, with x and y instead of x1 and x2, the linear equation 2x + 3y = 6 describes the line passing through the points (3; 0) and (0; 2). 

Similarly, with x; y and z instead of x1; x2 and x3, the linear equation 2x + 3y + 4z = 12 describes the plane passing through the points (6; 0; 0); (0; 4; 0); (0; 0; 3).

A system of m linear equations in n unknowns x1; x2; …; xn is a family of linear equations

a11x1 + a12x2 + … + a1nxn = b1

a21x1 + a22x2 + … + a2nxn = b2

…

am1x1 + am2x2 + … + amnxn = bm.

We wish to determine if such a system has a solution, that is to find out if there exist numbers x1; x2; … ; xn which satisfy each of the equations simultaneously. We say that the system is consistent if it has a solution. Otherwise the system is called inconsistent.