3.158 min read

Homogeneous and Inhomogeneous Systems

The homogeneous system is $A\mathbf{x} = \mathbf{0}$ — right-hand side is the zero vector. It is always consistent (trivial solution $\mathbf{x} = \mathbf{0}$ ), and its solution set is the null space $\text{Null}(A)$ — a subspace.

The inhomogeneous system is $A\mathbf{x} = \mathbf{b}$ with $\mathbf{b} \neq \mathbf{0}$ . When consistent, its solution set is $\mathbf{x}_p + \text{Null}(A)$ — an affine set.

The connection: the inhomogeneous solution set is a "shift" of the homogeneous solution set. Understanding $\text{Null}(A)$ is half the battle in solving any system.

Formal View

Definition 3.15

Homogeneous system:

A\mathbf{x} = \mathbf{0}

; solution set is

\text{Null}(A)

(a subspace). Inhomogeneous system:

A\mathbf{x} = \mathbf{b}

with

\mathbf{b} \neq \mathbf{0}

; solution set (if nonempty) is

\mathbf{x}_p + \text{Null}(A)

(an affine set).

Why This Matters

The homogeneous/inhomogeneous split is universal — it appears in differential equations, optimization, and control.

In differential equations, the homogeneous solution is the "transient" and the particular solution is the "steady state"
Residuals in least squares are vectors in the null space of the constraint matrix

Learning Resources

Homogeneous and particular solutions

MIT OpenCourseWare

General solution = particular + homogeneous.

51 min

Quiz

Question 1

The homogeneous system $A\mathbf{x} = \mathbf{0}$ always has at least one solution.

Common Mistakes

Thinking the homogeneous solution $\mathbf{0}$ is the only solution — there may be a whole subspace of solutions (the null space).