Homogeneous Systems of Linear Equations

A linear equation is homogeneous when the constant term is 0. A linear system is homogeneous when all of its equations are homogeneous equations.

The solution to a homogeneous linear system is also the null space of the constant matrix

Gives

We also have:

Given a non-homogeneous linear system , the associated homogenous system is .

The solution to the associated homogeneous system tells us a lot about the original non-homogeneous system.

So

Note the solution to the associated homogeneous system

Where Is called the "particular solution"