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  "body": "\n## Statement\n\nLet $v_1, \\dots, v_n$ be vectors in [[the coordinate space|0XM8BZD76G]] $\\R^m$.\nLet $A$ be the $m \\times n$ matrix whose columns are $v_1, \\dots, v_n$.\n\nThen the [[linear span|0DTY8B9HMS]] of $v_1, \\dots, v_n$ is all of $\\R^m$ if and only if every row of $A$ contains a [[pivot|0BCMGDHZGR]] position.\n\n## Remarks\n\nIn particular, if $n \u003c m$, then the vectors cannot span $\\R^m$, since $A$ has at most $n$ pivot positions.\nA row without a pivot gives a vector $b$ for which the system $Ax = b$ has no solution.\n",
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