Solutions of Homogenous Systems

A system of the form:

Always consistent: $X=O$ is always a solution.

Solution Types

• Trivial solution
  $X = O$. Exists in every homogenous system.
• Non-trivial solution
  Any $X \neq O$ that satisfies the system.

Conditions

• $\text{Rank }A = n \iff$ unique solution (trivial only)
• $\text{Rank }A < n \implies$ infinitely many solutions (non-trivial solutions exist)

Null Space

The complete solution set of $AX = O$ is the null space of $A$.