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 $\boldsymbol{A}\boldsymbol{x}=\boldsymbol{b}$ $b$ . 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.

On a grid, the line through the origin of solutions of x1 + 2x2 = 0, the kernel, and a parallel line of solutions of x1 + 2x2 = 3, obtained by shifting the kernel by a particular solution.

$\boldsymbol{A}\boldsymbol{x}=\boldsymbol{0}$ (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 $\boldsymbol{x}_p+\ker \boldsymbol{A}$ $ker A$ , for any one particular solution $\boldsymbol{x}_p$ , or empty.

With $\boldsymbol{b}=\boldsymbol{0}$ both lines pass through the origin, so $\boldsymbol{x}=\boldsymbol{0}$ always solves the system. Make the rows parallel for infinitely many solutions, or switch $\boldsymbol{b}$ on.

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¶

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: 20 October 2026 03:02