Homogeneous and Non-Homogeneous Systems Explained | Generated by AI
In the context of a homogeneous system like Ax = 0, “homogeneous” specifically means that the right-hand side is zero.
To clarify the distinction:
Homogeneous system: Ax = 0 (the constant term is zero)
Non-homogeneous system: Ax = b (where b ≠ 0, the constant term is non-zero)
The word “homogeneous” essentially means “uniform” or “of the same kind.” In this case, it refers to having zero on the right-hand side of all equations.
Key implications of a homogeneous system:
A homogeneous system always has at least one solution: the trivial solution x = 0 (the zero vector). This is guaranteed because substituting zero into any equation gives 0 = 0, which is always true.
The interesting question for a homogeneous system is whether it has non-trivial solutions (solutions other than x = 0). This happens when the matrix A is singular (not invertible), which means its columns are linearly dependent and the system has infinitely many solutions.
In contrast, a non-homogeneous system Ax = b may have no solution, exactly one solution, or infinitely many solutions—it doesn’t automatically have a solution at all.