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# Homogeneous linear systems are consistent.

Put content here**Theorem:** Every homogeneous linear system $Ax = 0$ is **consistent**.
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**Proof:** The zero vector $x = \mathbf{0}$ always satisfies every equation because:
$$a_{i1} \cdot 0 + a_{i2} \cdot 0 + \cdots + a_{in} \cdot 0 = 0$$
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for every row $i$.
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This means a homogeneous system always has at least one solution (the trivial solution $x = 0$).
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**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).

# Parents

* Linear systems of equations