Homogeneous Linear Systems

Definition

• A homogeneous linear system is defined by $A \mathbf{x} = \mathbf{0}$, where $A$ is the coefficient matrix and $\mathbf{x}$ is the unknown vector.

Structure

• Fewer equations than unknowns: m < n (where $m$ = number of equations, $n$ = number of unknowns).
• This often implies the presence of free variables.

Solutions

• The trivial solution $\mathbf{x} = \mathbf{0}$ always satisfies $A \mathbf{x} = \mathbf{0}$.
• If there are more unknowns than equations (underdetermined) and rank($A$) < $n$, there can be nontrivial solutions.

Quick takeaway

• In a homogeneous system with m < n, expect potential nontrivial solutions; analyze rank to determine the solution space.