Homogeneous and Inhomogeneous Systems¶

Definition¶

Over $\mathbb{R}$ or $\mathbb{C}$, a linear system $\boldsymbol{A}\boldsymbol{x}=\boldsymbol{b}$ is homogeneous if $\boldsymbol{b}=\boldsymbol{0}$$0$. It is inhomogeneous if $\boldsymbol{b}\neq \boldsymbol{0}$$0$.

Interpretation¶

The standard matrix form gathers all equations into $\boldsymbol{A}\boldsymbol{x}=\boldsymbol{b}$$\mathbf{b}$. Homogeneity is the special choice in which the right-hand side is the zero vector, so the zero vector is automatically a solution.

A homogeneous system asks which vectors are mapped to zero. An inhomogeneous system asks which vectors are mapped to a specified nonzero right-hand side.

$\boldsymbol{A}\boldsymbol{x}=\boldsymbol{0}$ (here $x_1+2x_2=0$$0$): the solutions form the subspace $\ker \boldsymbol{A}$, so $\boldsymbol{0}$ is always a solution and solutions can be added and scaled; for $\boldsymbol{A}\boldsymbol{x}=\boldsymbol{b}$ (here $x_1+2x_2=3$$3$) the solution set is $\boldsymbol{x}_p+\ker \boldsymbol{A}$$\ker \mathbf{A}$, for any one particular solution $\boldsymbol{x}_p$, or empty.

Key Results¶

• A homogeneous system always has at least the trivial solution $\boldsymbol{x}=\boldsymbol{0}$$0$.
• If a homogeneous system has more than one solution, then it has infinitely many solutions.
• An inhomogeneous system may have no solution, one solution, or infinitely many solutions.

Examples¶

• $\boldsymbol{A}\boldsymbol{x}=\boldsymbol{0}$ is homogeneous.
• $\boldsymbol{A}\boldsymbol{x}=\boldsymbol{b}$ with $\boldsymbol{b}\neq \boldsymbol{0}$ is inhomogeneous.

Prerequisites¶

• Standard Form of a Linear System
• zero vector

Related Concepts¶

• System of Linear Equations
• Kernel
• Solution Set
• Linear Combination
• Rank

Sources¶

• Math II: Linear Algebra · Week 4

Keywords: homogeneous linear system, inhomogeneous linear system, nonhomogeneous system, inhomogeneous system Symbols: complex numbers, real numbers

Last updated: 27 October 2026 11:52