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Homogeneous linear systems are consistent.

Created over 8 years ago, updated 24 days ago

Theorem: Every homogeneous linear system $Ax = 0$ is consistent.

Proof: The zero vector $x = \mathbf{0}$ always satisfies every equation because:
$$a_{i1} \cdot 0 + a_{i2} \cdot 0 + \cdots + a_{in} \cdot 0 = 0$$

for every row $i$.

This means a homogeneous system always has at least one solution (the trivial solution $x = 0$).

Important consequence: For homogeneous systems, the only question is whether the solution is unique (only $x = 0$) or whether there are infinitely many solutions (nontrivial solutions exist).