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  "body": "\n## Definition\n\nLet $F$ be a [[field|04C5ZBG1WT]].\nLet $A$ be an $m \\times n$ matrix over $F$ and let $b \\in F^m$.\nThe [[system of linear equations|0GT4WWGWH2]] $Ax = b$ in the unknown $x \\in F^n$ is *overdetermined* if $m \u003e n$, that is, if it has more equations than unknowns.\n\n## Remarks\n\nExample: the system $x = 1$, $y = 2$, $x + y = 3$ over $\\R$ has $m = 3$ equations and $n = 2$ unknowns, so it is overdetermined; it is consistent, with the solution $(x, y) = (1, 2)$.\nNon-example: the system $x + y = 1$, $x - y = 0$ has $m = n = 2$, so it is not overdetermined.\nIf $m \u003e n$, then the [[column space|03AJKXH1AV]] of $A$ has dimension at most $n \u003c m$, so it is a proper subspace of $F^m$.\nHence there are right-hand sides $b$ for which the system is [[inconsistent|0X6V9B1V10]].\nOver $\\R$, the consistent right-hand sides form a proper subspace of $\\R^m$, so a randomly chosen $b$ almost never gives a consistent system.\nOver $\\R$, the [[least squares|0BYTQ4X1HC]] solutions are the solutions of the normal equations $A^{T} A x = A^{T} b$, which are always consistent.\nA system with fewer equations than unknowns is called *underdetermined*.\n",
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