56. Homogeneous System
A linear system is homogeneous if its right-hand side is the zero vector:
Every homogeneous system is consistent — it always has at least the trivial solution .
56.1. When are there non-trivial solutions?
The non-trivial solutions form the null space of (equivalently, the kernel of the linear map ):
By the Rank–Nullity Theorem, for an matrix :
So:
| Solution set | |
|---|---|
| only the trivial solution | |
| a subspace of dimension — infinitely many solutions |
Notably: an homogeneous system has a non-trivial solution iff is singular (i.e., ).
Example
Solve for .
Row-reduce: . So , nullity .
From row 1: , so . With free:
Two-parameter family of solutions — a plane through the origin in .
56.2. Connection to general systems
If has solution (a particular solution), the full solution set is:
— the particular solution plus any solution to the associated homogeneous system. This is why homogeneous systems are the “structural skeleton” of solution sets.