What echelon form reveals

Homogeneous systems and parameterised solutions

Among all systems, those whose right-hand side is zero play a special role. They are called homogeneous, and they give the structure of every solution set.

The homogeneous system

A system is homogeneous when all its equations are equal to zero on the right:

{ 2x + 3y -  z = 0
{  x -  y + 2z = 0

Such a system is never inconsistent: x = y = z = 0 is always a solution. It is called the trivial (or zero) solution. So the real question is not "is there a solution?" but "are there others besides the zero one?".

The answer lies in the rank:

r = n   ->  the zero solution is the ONLY one
r < n   ->  there are infinitely many non-zero solutions

What the solutions look like

The solution set of a homogeneous system is not some arbitrary cloud: it passes through the origin and is closed under addition and under multiplication by a number. If u and v are solutions, so are u + v and 3u.

Geometrically it is always one of these objects:

    {0}                line through O          plane through O
     •                      /                  ________
   (the single            /  •                /_______/
    point O)            /                    (a whole plane)

  n - r = 0            n - r = 1              n - r = 2

This is the first example of a vector subspace — the notion studied in the course on vector spaces. Its dimension is exactly n - r.

The link with the full system

Here is the result that structures the whole chapter. Take a system A X = B and its associated homogeneous system A X = 0. If X0 is one particular solution of the full system, then:

  solutions of A X = B   =   X0   +   all solutions of A X = 0
                             ^^                  ^^
                    one particular solution   the homogeneous part

In other words: general solution = particular solution + homogeneous solution. Geometrically, the solution set is the homogeneous line (or plane) translated by X0:

        solutions of AX = 0        solutions of AX = B
             /                          /
            /  (line through O)        /   (same direction,
           /                          /     shifted by X0)
          O                          X0

Two systems with the same left-hand side therefore have parallel solution sets: same shape, same dimension, different position.

Writing infinitely many solutions properly

Take x + 2y - z = 0 with three unknowns and a single pivot: n - r = 2 parameters. Set y = s and z = t, so x = -2s + t:

(x ; y ; z) = (-2s + t ; s ; t)
            = s(-2 ; 1 ; 0) + t(1 ; 0 ; 1)      for any real s, t

This is the right way to write it: it exhibits two vectors whose combinations give exactly the solutions. These two vectors form a basis of the solution plane. Answering "there are infinitely many solutions" without parameterising them means stopping halfway.

Summary