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  "body": "\n## Statement\n\nTake a [[system of linear equations|0GT4WWGWH2]] over the real numbers that is [[consistent|0X6V9B1V10]]. Look at the [[pivot positions|0BCMGDHZGR]] of its coefficient matrix.\n\nIf each column of the coefficient matrix holds a pivot position, then the system has exactly one solution.\n\nIf at least one column of the coefficient matrix holds no pivot position, then the system has infinitely many solutions.\n\n## Remarks\n\nA column with a pivot position gives a basic variable. A column without one gives a free variable. So the two cases say: no free variable means one solution, and at least one free variable means infinitely many solutions.\n\nConsistency is needed. An inconsistent system has no solution, whatever its pivot columns are.\n",
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    "tag": "5MGKEK1K42",
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  "markdown": "https://math2.tamnd.com/node/5MGKEK1K42.md",
  "msc": [
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  "name": "Uniqueness of solutions of a consistent linear system by pivot columns",
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