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  "body": "\n## Definition\n\nLet $F$ be a field.\nA [[system of linear equations|0GT4WWGWH2]] over $F$ is *homogeneous* when each of its constant terms is $0$.\nIn matrix form, a homogeneous system is an equation $Ax = 0$, where $A$ is an $m \\times n$ matrix over $F$ and $x = (x_1, \\dots, x_n)^T$.\n\n## Remarks\n\nThe tuple $(0, \\dots, 0)$ is always a solution, the *trivial solution*.\nThe solution set of $Ax = 0$ is a [[linear subspace|00YC1D652K]] of $F^n$, the [[null space|0PHDYNR24D]] of $A$.\nIf a system $Ax = b$ has a solution $p$, then its solution set is $\\{p + v : Av = 0\\}$.\nA homogeneous system with more unknowns than equations has a nontrivial solution.\n\n## Explanations\n\n### L1 human\n\nA homogeneous system has zero on the right side of every equation.\nSetting every unknown to zero always works, so the question is whether other solutions exist.\nThe sum of two solutions is again a solution, and so is any multiple of a solution.\n",
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