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  "body": "\n## Statement\n\nLet $F$ be a field, let $A$ be an $m \\times n$ matrix over $F$, and let $b \\in F^m$.\nThe [[system|0GT4WWGWH2]] $Ax = b$ has a solution if and only if $\\rank A = \\rank [A \\mid b]$, where $[A \\mid b]$ is the [[augmented matrix|0T6MWGZSXA]] and $\\rank$ is the [[rank of a matrix|0ZC9YTXZRY]].\nIf a solution exists, then the solution set is an affine subspace of $F^n$ of dimension $n - \\rank A$.\nIn particular, the solution is unique if and only if $\\rank A = n$.\n\n## Remarks\n\nThe theorem is also called the Kronecker–Capelli theorem.\nIf $\\rank [A \\mid b] = \\rank A + 1$, then the system is [[inconsistent|0X6V9B1V10]].\nThe dimension $n - \\rank A$ is the number of free variables.\n\n## Explanations\n\n### L1 human\n\nThe theorem says that a system has a solution exactly when the right side adds no new independent information.\nIf adding the column $b$ raises the rank, then $b$ is not a combination of the columns of $A$, and no solution exists.\nWhen solutions exist, each unknown without a pivot is free, and it adds one dimension to the solution set.\n",
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