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  "body": "\n## Statement\n\nLet $F$ be a field and let $A$ be a [[matrix|0N1MY1T0W9]] over $F$.\nThe columns of $A$ are [[linearly independent|0CRMGPAH83]] if and only if each column of $A$ contains a [[pivot|0J0BBCMJ5W]] position.\n\n## Remarks\n\nEquivalently, the columns are dependent exactly when some column has no pivot.\nA column without a pivot gives a free variable of the system $Ax = 0$, and then this system has a nonzero solution.\n",
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