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  "body": "\n## Statement\n\nLet $\\mathbf v_1, \\dots, \\mathbf v_n$ and $\\mathbf b$ be vectors in $\\mathbb R^m$.\nForm the [[augmented matrix|0T6MWGZSXA]] whose columns are $\\mathbf v_1, \\dots, \\mathbf v_n$ followed by $\\mathbf b$.\nThen $\\mathbf b$ is a [[linear combination|0V8F3E48E6]] of $\\mathbf v_1, \\dots, \\mathbf v_n$ if and only if the [[linear system|0GT4WWGWH2]] with this augmented matrix is [[consistent|0X6V9B1V10]].\n\nMoreover, each solution $(c_1, \\dots, c_n)$ of the system gives weights with $c_1\\mathbf v_1 + \\dots + c_n\\mathbf v_n = \\mathbf b$.\n\n## Remarks\n\nThe unknowns of the system are the weights $c_1, \\dots, c_n$.\nEach row of the system is one coordinate of the vector equation $c_1\\mathbf v_1 + \\dots + c_n\\mathbf v_n = \\mathbf b$.\n",
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