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  "body": "\n## Statement\n\nLet $A$ be a matrix with columns $v_1, v_2, \\dots, v_n$.\nThen the following five conditions are equivalent.\n\n1. The columns of $A$ are [[linearly dependent|0CRMGPAH83]].\n2. At least one of the vectors $v_1, \\dots, v_n$ is a [[linear combination|0V8F3E48E6]] of the other ones.\n3. Some column of $A$ has no [[pivot|0BCMGDHZGR]] position.\n4. The [[homogeneous equation|0KP0QNXPRT]] $Ax = 0$ has infinitely many solutions, so it has a nonzero solution.\n5. There are scalars $c_1, \\dots, c_n$, not all zero, with $c_1 v_1 + \\cdots + c_n v_n = 0$.\n\n## Remarks\n\nCondition 5 is the definition of dependence written out for the columns of $A$.\nIt says the same as condition 4, because $Ax$ is the combination of the columns with weights taken from $x$.\nCondition 3 lets one test dependence by row reduction.\n",
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