homogeneous system of linear equations

A homogeneous system of linear equations is one with all LHSs linear in the variable, and all RHSs exactly 0:

a_1,1x₁ + ... + a_1,nx_n = 0
a_2,1x₁ + ... + a_2,nx_n = 0
...
a_m,1x₁ + ... + a_m,nx_n = 0

Of course, many systems of linear equations, such as

x+y=z
z-x=y

may be brought into the above form by manipulating them. The important thing is that no "constants" appear in the equations, or that any constants that do appear may be cancelled out within that equation (e.g. "x+y+4=z+4").

The importance of homogeneous systems of linear equations is that they provide a cornerstone for solving general systems of linear equations. A homogeneous system always has a solution: set all variables to 0. This is the trivial solution. It may, however, have more solutions. And if you squint at it from just the right angle, you see that its set of solutions is a vector space: adding solutions or multiplying a solution by a constant (a scalar) give solutions, too.

Suppose you know how to solve the general homogeneous system above, and find all the solutions. What about the general system

a_1,1x₁ + ... + a_1,nx_n = b₁
a_2,1x₁ + ... + a_2,nx_n = b₂
...
a_m,1x₁ + ... + a_m,nx_n = b_m

?

First off, it might not have any solutions:

x+y=1
2x+2y=2

It turns out that a solution might not exist iff the homogeneous system has infinitely many solutions (i.e. more than just the trivial all-0 solution).

But suppose it does, and we know just one particular solution (x_i=c_i)_i=1ⁿ for the general system. Then for any solution of the homogeneous system (x_i=y_i)_i=1ⁿ, it's easy to see that (x_i=c_i+y_i)_i=1ⁿ is also a solution to that general system. Nicer yet, you can get all the solutions in this way.

It is enough to know all solutions of the homogeneous system and just one solution of the general system.

Which brings us to the main point of the writeup: Many people get taught the method (called Gaussian elimination) for solving linear equations. That's not particularly interesting. And that's not the point of mathematics!

The point of mathematics is to make qualitative statements ("there is exactly one solution") about quantitative problems ("which values of x,y,z satisfy ..."?). The statement about the importance of homogeneous systems, while not particularly difficult, comes from the heart of mathematics: Given a more complicated situation (a general system), you can know something about what goes on there by solving a less complicated situation (a homogeneous system) and applying some mathematical knowledge.