Homogeneous System

56. Homogeneous System

A linear system is homogeneous if its right-hand side is the zero vector:

𝐴 \vec{𝑥} = 𝟎

Every homogeneous system is consistent — it always has at least the trivial solution $\vec{𝑥} = 𝟎$ .

56.1. When are there non-trivial solutions?

The non-trivial solutions form the null space of $𝐴$ (equivalently, the kernel of the linear map $𝑇 (𝑥) = 𝐴 𝑥$ ):

Sol (𝐴 𝑥 = 𝟎) = null (𝐴) = ker (𝑇)

By the Rank–Nullity Theorem, for an $𝑚 \times 𝑛$ matrix $𝐴$ :

dim (null (𝐴)) = 𝑛 - rank (𝐴)

So:

$rank (𝐴)$	Solution set
$= 𝑛$	only the trivial solution $\vec{𝑥} = 𝟎$
$< 𝑛$	a subspace of dimension $𝑛 - rank (𝐴) > 0$ — infinitely many solutions

Notably: an $𝑛 \times 𝑛$ homogeneous system has a non-trivial solution iff $𝐴$ is singular (i.e., $det (𝐴) = 0$ ).

Example

Solve $𝐴 \vec{𝑥} = 𝟎$ for $𝐴 = [\begin{matrix} 1 & 2 & - 1 \\ 2 & 4 & - 2 \end{matrix}]$ .

Row-reduce: $𝑅_{2} \to 𝑅_{2} - 2 𝑅_{1} = (0, 0, 0)$ . So $rank = 1$ , nullity $= 3 - 1 = 2$ .

From row 1: $𝑥_{1} + 2 𝑥_{2} - 𝑥_{3} = 0$ , so $𝑥_{1} = - 2 𝑥_{2} + 𝑥_{3}$ . With $𝑥_{2}, 𝑥_{3}$ free:

\vec{𝑥} = 𝑥_{2} [\begin{matrix} - 2 \\ 1 \\ 0 \end{matrix}] + 𝑥_{3} [\begin{matrix} 1 \\ 0 \\ 1 \end{matrix}]

Two-parameter family of solutions — a plane through the origin in $ℝ^{3}$ .

56.2. Connection to general systems

If $𝐴 \vec{𝑥} = \vec{𝑏}$ has solution ${\vec{𝑥}}_{𝑝}$ (a particular solution), the full solution set is:

{\vec{𝑥}}_{𝑝} + null (𝐴) = {{\vec{𝑥}}_{𝑝} + {\vec{𝑥}}_{ℎ} : {\vec{𝑥}}_{ℎ} \in null (𝐴)}

— the particular solution plus any solution to the associated homogeneous system. This is why homogeneous systems are the “structural skeleton” of solution sets.

56. Homogeneous System

56.1. When are there non-trivial solutions?

56.2. Connection to general systems

56.3. See also